R Gourdon

M Jean

Marc Lancelin

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

M Obadia

Hassan Khalil

Ali Jamil Khalil

Keywords: thermal techniques, Scanning Thermal Microscopy, micro and nanoscale heat transfer, thermal conductance, thermal conductivity, phonon transmission coefficient, microresistors, water meniscus

Two techniques to characterize the thermal properties of materials and to analyze the heat transfer at the micro/nanoscales have been studied and are presented in this manuscript. The first technique is the Scanning Thermal Microscopy (SThM), an Atomic Force Microscopy (AFM)-based technique. Operating in its active mode, the AFM probe integrates a resistive element that is electrically heated. Used in AFM contact mode, it allows the localized thermal excitation of the material to be studied. The determination of the sample thermal properties requires the analysis of the probe thermal response through the modeling of the probe-sample system including its surrounding. With the help of a review of SThM studies, the current scientific questions and the analytical models used to analyze the probesample system are explored. Special attention is given to the probe-sample thermal interaction that conditions the tip-sample interface temperature. Reducing the probe size requires a more in depth study of the physics of the heat transfer mechanisms of this interaction. In this work, a new methodology for studying and specifying the heat rate exchanged between a probe and a sample through thermal conduction through water meniscus has been established. The methodology is based on the analysis of the dependence of force-distance curves on probe temperature obtained in ambient air. It was applied with samples of different thermal properties, surface roughness and wettability to three resistive probes different in size and heater configurations: Wollaston, KNT and doped silicon (DS) probes. Whatever the probe and the sample are, the contribution of water meniscus in the probe-sample interaction has been shown to be lower than the one of air. Moreover, for the Wollaston and KNT probes, the parameters of a model based on a description of the probe-sample system using a network of thermal resistances are identified from measurements performed under ambient and vacuum conditions. The thermal conductances at the solid-solid contact were determined for various samples. This allowed identifying the phonon transmission coefficient in the case of KNT probe and a nonmetallic sample. Experimental results and numerical simulations for measurements performed under ambient conditions have demonstrated that the thermal interaction parameters of the heat conduction through air strongly depend on the sample thermal conductivity. Improvements of the current calibration methodology for thermal conductivity measurement have been proposed. The variation of sample roughness during the calibration and measurements may be an issue. Moreover, the sensitivity to sample thermal conductivity for the Wollaston and KNT probes is shown to be strongly reduced for thermal conductivities larger than 10 and a few W.m -1 .K -1 respectively. The second technique developed in this thesis is a less local thermal analysis method. It operates by contact, requiring the implementation of the sample surface with a network of resistive wire probes. One wire of the network is heated by an alternating current at frequency f and has the role of heating source, continuous and at 2f frequency, for the sample. The other wires are sensors of temperature (method "2ω"). A 2D analytical model, based on the principle of thermal-waves, was developed to identify the effective thermal properties of anisotropic samples from experimental measurements with the technique. This model can be reduced to a simplified 1D model in the case of diffusive, homogeneous and isotropic materials. Finite element simulations and this model were used to design the experimental setup and validate the method with pure silicon. The results obtained at sample temperatures ranging from ambient to 500 K are consistent with literature.

This thesis is the result of many experiences I encountered during the period of my work at the CETHIL laboratory. I wish to acknowledge many remarkable individuals who helped me achieve the works that are presented in this manuscript.

First, I would like to thank all the members of the jury for their participation in the defense and for their comments. I thank the reviewers Oleg Kolosov and Jonathan Weaver for their careful reading of the manuscript and their reports on the whole study. I thank Sebastian Volz, the president of the jury, for useful discussions. I also thank Nathalie Trannoy and Olivier Bourgeois for their efforts during the examination of the manuscript. I enjoyed meeting all of you either during past conferences or during the defense and I hope our paths will cross in the near future.

I would like to express my gratitude to my PhD supervisors, Séverine Gomès and Stéphane Lefèvre.

Séverine, you already proved that you're an excellent researcher and a great advisor. I will be always grateful for the opportunities and the freedom you gave me during the work.

During my PhD journey, I was a member of the MINT group (Micro and Nano Scale Heat Transfer group). The meetings and the discussions I had with the different group members gave me the opportunity to enhance my knowledge in the different current studies in our domain. Many thanks go to Rodolphe (Vaillon), the coordinator of this group, for his great sense of organization and management. I'm obliged here to thank Olivier C (Chapuis) for long and fruitful scientific discussions and for making me more curious and motivated. I also thank Olivier M (Merchiers) whom I consider a friend in our lab. I have to thank the other members of our group for the great moments we had during the meetings (Trang, Wassim, Farida ("je prends ce que je veux!!"), Ahmad, Mouhannad, Sylvie, Etienne, David, Olivier D, Carolina, Elyes, Yanqing (Ivy)….). Special thanks go to Trang for being funny all time (ah bon!) and to Wassim for his beautiful loud laugh. The atmosphere in the group was always warm and lovely.

I would like to thank some people, with whom I collaborate, for their warm welcome: Antoine and Pascal from ILM Lyon, Annie and Pierre from CLYM Lyon, and Nicolas Horny from GRESPI Reims. I appreciate the great moments I had with different persons in the lab. Thanks to Jocelyn Bonjour, the director of the lab, for his understanding and listening ear. Thanks to Florence, the right hand, for her support and administrative help all the time. I also thank the other members in the secretary office who always helped me with different tasks. I especially mention Corinne D, Sophie, and Christine D. I also thank the technical staff in our lab (even though they don't care about those 'nanoscale stuff'): Nico (et ses 7 laux et son équipe de Grenoble), Joël (le grand), Christophe Ducat (malgré que tu es Stéphanois), Bernard, Loïc, Bertrand… I won't forget the special moments I had with the actual and former PhD students and postdoctoral researchers. For these moments, I thank Léon (and his English classy gestures), Aurélia (et son dragon ball!!), Auline, Lucie, Sébastian, Mathieu Labat, Pablo, Kim (oppa gangnam style !!), Zakaria Slimani, Damien (et son esprit de syndicats), Yun-Xin, Anna, Wei, Kévin (ses doodles qui se convergent jamais!!), Christophe Daverat, Pierrick (et son esprit 50 % corse -50 % breton), Céline, Thibault B, Marcella, Christophe Kinkelin (toujours le dernier pour finir un repas!!), Daha ("muscles toi Daha")… I would like to thank my former officemates, Roula ("oufffffff"-and her great meals) and Matthieu Zinet ("heuresement il y a la voiture de Matthieu!!")… Infinite thanks to both of them for being such good friends all time. I would like to wish the best for the new ones Sandra (Allez Auxerre!! -Kapo et Cissé allez allez) and Chi-Kien (et son robot!!).

I was lucky outside the lab to meet and to be surrounded by great people who became close friends. Many thanks go to those Lebanese friends who made the ambiance funny and joyful.

Thanks to Srour (and his superb meals -open a restaurant man!!), Mortada (long football matches), Sabbah (and his African style meals -Come On You Gunners COYG!!), Mozannar (and his technology equipments), Mouazzen (and his endless guarantees), Zeinab (stop being always late girl!! Fortunately, you're funny to compensate that), Banjak (stop lifting weights man!! You're already a beast!!), Ankouni (and his delicious burgers), Tarhini (and his "lost village"!!), Inas and Nour, Hodroge, Hammoud, Khashman, Fardoun (and his hoummous), Damergi… Big thanks to Romain and Tere, who were amazing during all of this period… Romain, you made me fall in love with Olympique Lyonnais (Allez OL!) and with rugby. Superb you're man!! Tere, muchas gracias señorita for being supportive all the time. Many thanks go to for Tristan (with his French charm -'ton nom est déjà Amoureux, qu'est-ce qu'on peut ajouter ?!') and Ainhoa (ijo polaco!!). I also thank the couple Sylvain (and his political analysis) and Pepa (you eat a lot girl really!!) for the beautiful moments we had in Lyon. Special, huge, and unique thanks go to the special Esti (Estitxu) for her endless support and her beautiful spirit. Hat tip. Eskerrik asko. Que bonito Candanchu Con O Sin Nieveeeeee!! I would like to thank my parents (Loubna and Ahmad) for all their infinite support and encouragement. Due to their suffering and the sacrifices they made in their lives, I was able later to follow my ambitions. Hopefully, one day I will be able to give them back just a little from what they gave me all time. I also thank my cute siblings, my sister Fatima (Fattoumaand her fiancé too 'oh!') and my brother Hussein (Nayno) for being the greatest people I can have. Many thanks also go to my big family including Grandparents, Cousins, Aunts and Uncles (lots of love to you…).

Résumé

Deux techniques de caractérisation thermique des matériaux et d'analyse du transfert de chaleur aux micro-et nano-échelles ont été étudiées et sont présentées dans ce mémoire. La première technique est la microscopie thermique à sonde locale (SThM). La pointe d'un microscope à force atomique intègre un élément résistif. Utilisée en mode contact, cette pointe, chauffée par effet joule, permet l'excitation thermique localisée de l'échantillon. La détermination des propriétés thermiques de l'échantillon nécessite l'analyse de la réponse de cette pointe avec un modèle du système sonde-échantillon et de son environnement. Un état de l'art général des études réalisées en SThM permet de poser les questions scientifiques actuellement traitées dans le domaine ainsi que les modèles utilisés pour analyser le système pointe-échantillon. Une attention particulière est accordée à l'interaction thermique sondeéchantillon. Sa compréhension nécessite d'être approfondie. L'étude ici présentée tient compte des propriétés thermiques, de la rugosité et de la mouillabilité de la surface de différents échantillons. Une nouvelle méthodologie est établie pour la spécification du transfert de chaleur échangée par conduction thermique au travers du ménisque d'eau formé au contact sonde-échantillon. Cette méthodologie est basée sur l'analyse de la dépendance à la température de la sonde des courbes de force-distance obtenues à l'air ambiant. Elle est appliquée à trois sondes de taille, forme et constitution différentes: la sonde Wollaston, la sonde KNT et une sonde en silicium dopé. Quels que soient la sonde et l'échantillon, la contribution du ménisque d'eau à l'interaction est montrée être inférieure à celle de l'air. Par ailleurs, pour les sondes Wollaston et KNT, les paramètres d'une modélisation, basée sur une description du système sonde-échantillon avec un réseau de conductances thermiques, sont identifiés à partir de mesures effectuées à l'ambiante et sous vide. La conductance thermique au contact solide-solide est déterminée pour différents échantillons. Cela a permis d'identifier le coefficient de transmission de phonons dans le cas de la sonde KNT et d'échantillons nonmétalliques. Nos résultats expérimentaux et de simulations numériques pour les mesures effectuées sous air démontrent que les paramètres utilisés pour décrire la conduction thermique via l'air dépendent fortement de la conductivité thermique de l'échantillon. Aussi des améliorations de la méthodologie actuelle d'étalonnage pour la mesure de la conductivité thermique sont proposées. Il est en outre montré que la variation de la rugosité de l'échantillon lors de l'étalonnage et de mesures doit être prise en compte. La sensibilité à la conductivité thermique pour les sondes Wollaston et KNT est part ailleurs montrée fortement réduite pour les matériaux de conductivité thermique supérieure à 10 et quelques W.m -1 .K -1 respectivement. La seconde technique développée est une méthode d'analyse thermique moins locale nécessitant l'instrumentation de la surface de l'échantillon avec un réseau de sondes résistives filiformes. L'un des fils du réseau, chauffé par un courant alternatif à la fréquence f, a le rôle de source excitatrice continue et à la fréquence 2f de l'échantillon. Les autres fils sont des capteurs de la température (méthode «2ω»). Un modèle analytique 2D, basé sur le principe des ondes thermiques et développé pour identifier les propriétés thermiques d'échantillons anisotropes est présenté. Ce modèle peut être réduit à un modèle simplifié 1D dans le cas de matériaux diffusifs, homogènes et isotropes. Des simulations par éléments finis et avec ce modèle ont été utilisées pour dimensionner le montage expérimental et valider la méthode sur un échantillon de silicium pur. Les résultats obtenus à des températures de l'échantillon variant de l'ambiante à 500 K corroborent ceux de la littérature.

Mots-clés: méthodes thermiques, microscopie thermique à sonde locale, transfert de chaleur aux micro-et nano-échelles, conductance thermique, conductivité thermique, coefficient de transmission des phonons, méthode à fils résistifs déposés, ménisque d'eau G atom-atom Thermal conductance of atom-atom contact W.K -1

G c

Contact thermal conductance W.K -1

G ep

Volumetric phonon-electron coupling constant W.m -3 .K -1

G eq

Equivalent thermal conductance G eq =(G c G s )/(G c +G s ) W.K -1

G p

Probe thermal conductance W.K -1

G p-w

Probe-water thermal conductance W.K -1

G w-s

Water-sample thermal conductance W.K -1

G Q Quantum thermal conductance W.K -1

G rad

Thermal conductance of radiation W.K -1

G s

Thermal conductance of the sample W.K -1

G ss

Thermal conductance of the solid-solid contact W.K -1

G total, meniscus

The total conductance of the water meniscus W.K -1

G w

Thermal conductance through the water meniscus W.K - Modeling the tip-sample system as given by Gotsmann et al. [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. .................... Figure 1.11. Modeling the scattering at the interface through the phonon reflection coefficient r 12 [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF] 1.14. Variation the heater temperature of a silicon probe during the approach and retract curves [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF] (Frequency of 1 Hz -P = 10

List of Figures

.27. Variation of the probe electrical resistance when the tip contacts the sample as a function of the sample thermal conductivities. The simulations accounts for three conductivities of the probe materials [START_REF] Juszczyk | DC Experiments in Quantitative Scanning Thermal Microscopy[END_REF]

List of tables

General introduction

The understanding of heat transfer and the improvement of some industrial production processes require the development of sophisticated thermal metrologies. For micro and nanotechnologies in particular, these metrologies should be accurate in terms of measurement of the thermal parameters and spatial resolutions. A perfect example is thermal management in electronics [START_REF] Shakouri | Nanoscale thermal transport and microrefrigerators on a chip[END_REF]. It is indeed a crucial aspect for the design of integrated circuits due to the integration density achieved nowadays. Everything from the scale of the printed circuit board (a few cm) to that of elementary transistor (a few nm) must be designed to limit power consumption and to facilitate heat dissipation [START_REF] Chandrakasan | Design of high-performance microprocessor circuits[END_REF][START_REF] Borkar | Low power design challenges for the decade[END_REF]. All micro and nanosystems currently fabricated with microelectronics technology tools such as thin film based devices or micro and nanostructured components also raise similar issues. Moreover, these systems require a precise knowledge of the thermal properties of the constituting materials or of those considered for optimization. Another field is thermoelectric energy conversion, which is one of the ways that might help facing the energy challenge [START_REF] Venkatasubramanian | Thin-film thermoelectric devices with high roomtemperature figures of merit[END_REF]. Searching for a larger thermoelectric figure of merit ZT is of paramount importance. One common feature of recently-discovered thermoelectric materials with ZT>1 is that most of them have very low lattice thermal conductivities [START_REF] Snyder | Complex thermoelectric materials[END_REF]. The thermal properties of these new materials need to be characterized. More generally, the development of nanoscale metrologies is required to study of the effects of the nanostructuration process. This type of processes is used to confer revolutionary physical properties to materials. The understanding of the thermal impacts is the key that will allow the predictive modeling of the heat transfer in next-generation systems. It is essential for the design stage and for the improvement in terms of quality and security of systems.

From a fundamental point of view, there has been an urgent need to understand the physics of the heat transfer at the micro and nanoscales. In the case of heat transfer within nanostructures and nanostructured materials or of heat exchange between two systems, the macroscopic thermal models are not valid if the dimensions of the zones of thermal contact and systems become comparable to the mean free path of the heat carriers of the two systems. Several parameters can affect the heat transfer between two systems and they are not fully understood yet. They include the thermal properties of systems, the roughness of material surfaces, the wettability of surfaces, the temperature of materials, etc…

In terms of spatial resolution, the far field optical methods used to characterize the thermal properties of materials are limited by the diffraction limit [START_REF] Volz | Microscale and nanoscale heat transfer[END_REF]. In these techniques, the signal detection is performed at distances larger than the wavelength. In order to achieve a better spatial resolution in the determination of thermal properties and studying the heat transfer at submicron scales, Scanning Thermal Microscopy SThM has been developed for more than 25 years [START_REF] Gomès | Scanning thermal microscopy : A review[END_REF][START_REF] Gomès | Thermometry at the nanoscale -Fundamentals and selected Applications[END_REF]. The major part of this manuscript is dedicated to the characterization of the measurement with this technique. Commercialized resistive probes of different spatial resolutions were used to study the thermal system composed by the probe interacting with the sample in SThM (called "probe-sample thermal system" in the manuscript). Such approach would provide a better understanding of the technique and seek its improvement. Another thermal technique, less local because requiring the deposit of resistive wire-probes on the sample surfaces was also developed. It is presented in the last part of the manuscript.

The first chapter introduces the SThM technique. It presents the three resistive probes used in this work. For each of these probes, a state of the art of researches is given in terms of the modeling of the probe-sample thermal system. The chapter presents the experimental and modeling investigations that studied the heat losses to the environment while working under ambient conditions. The heat transfer mechanisms of the heat transfer between the probe and the sample are presented: thermal radiation, thermal conduction at the solid-solid contact, thermal conduction through water meniscus formed at the probe-sample contact and through air. Based on literature researches, each heat transfer mechanism is detailed and analyzed in order to bring out the parameters that affect the mechanism. The tasks to deal with in this manuscript and the experimental conditions are defined.

The second chapter is dedicated to the study of the heat transfer through water meniscus with the Wollaston wire probe (a micrometric probe). Two potential applications related to the results of the study are first presented. The chapter gives then a new methodology to determine the characteristic dimensions of the meniscus, an effective dimensional parameter of the probe apex and the meniscus thermal conductance. The last section of this chapter discusses some preliminary results of experiments performed in an environmental scanning electron microscope (ESEM) in order to observe the water meniscus at the contact between a probe and a sample. This was motivated by the current development of a SThM-ESEM combined system at CETHIL.

Based on the comparison of experimental results obtained with results of simulation for various experimental conditions, a quantitative analysis of the conduction through the solidsolid contact and air is given in the third chapter for the Wollaston probe. The Scanning Thermal Microscopy (SThM) with its better spatial resolution than the photothermal methods is used nowadays to characterize the thermophysical properties of materials. However the characterization of materials using SThM probes is still challenging and requires a better understanding and a proper modeling of the probe-sample system. Thus, any interpretation of the measurements must be accompanied with an appropriate modeling of the probe-sample system to identify precisely the properties of the considered materials.

The investigation of the probe-sample system involves studying the heat transfers through three elements: 1) the probe 2) the sample and 3) their surrounding environment.

Therefore, this chapter summarizes, to present, a state of the art of the research studies that involves the SThM to characterize the sample thermal conductivity as well the heat transfer mechanisms between the probe and sample. It gives, in order:

-An introduction of the technique and the description of the resistive probes that are involved in this work.

-The thermal modeling of the probe-sample system currently proposed and used for the interpretation of SThM measurements obtained with these probes.

-The research works performed for studying, describing and identifying the heat losses from the probe to the environment medium.

-The heat mechanisms that exist between the probe and the sample depending on the conditions of experiments.

Based on this commented review, the scientific tasks to deal with in the upcoming chapters are introduced. [START_REF] Williams | Photothermal imaging with sub-100-nm spatial resolution[END_REF] and later in Atomic Force Microscopy (AFM) [START_REF] Nonnenmacher | Scanning probe microscopy of thermal conductivity and subsurface properties[END_REF] and Scanning Near-field Optical Microscope (SNOM) configurations [START_REF] Goodson | Near-field optical thermometry[END_REF]. However, SThM is mostly used nowadays in an AFM configuration that allows obtaining simultaneous topographical and thermal images. Moreover, the forces that exist between the probe and the sample at the contact can be measured. A schematic of SThM that is based on AFM is presented in Figure 1.1. The probes used in this work operate on an AFM-based SThM technique.

Operational modes of resistive SThM probes

With the resistive probe, the information about the probe temperature is determined through the measurements of the probe electrical resistivity.

SThM resistive probes can be used in two modes:

a)

Passive mode: This mode is mainly used to map the temperature at the surface of active/heating samples to study local heat sources as an example [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. An electrical current sufficiently small, to avoid a significant additional heating of the probe, flows through the probe while the probe voltage is measured during the sample scanning.

b) Active mode: The probe is simultaneously the heater and the sensor. Either DC or AC (or both) current is injected in the probe.

In both modes, the electrical resistance of the thermosensitive element and thus its temperature is controlled through an electronic unit (Thermal Unit control in Figure 1.1 that can be a Wheastone bridge as an example). However, the probe temperature measured through this unit is the mean temperature of the whole resistive element. As will be seen later, the estimation of the heat transferred to the sample requires the determination of the tip apex temperature. This temperature cannot be deduced directly from the measurements and its determination requires an appropriate description of the probe-sample thermal system.

Resistive probes description 1.3.1 The Wollaston wire probe

The Wollaston probe has been used widely to characterize solid materials and to study the interaction between the probe and the sample [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF]. The resistive element of this probe is a wire of 5 µm in diameter and 200 µm in length. It is made of Pt 90%/ Rh 10% and bent in a V-shape (Figure 1.2). It is obtained by electrochemical etching of the silver shell of the Wollaston wire of 75 µm in diameter [START_REF] Dinwiddie | Thermal conductivity contrast imaging with a scanning thermal microscope[END_REF]. Table 1.1 summarizes the properties of the Wollaston wire probe.

The KNT probe (the palladium or GLA probe)

The KNT probe is basically made for creating hot spots on surfaces and for characterization of polymer layers and thin films [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. It consists of a thin film Pd as resistor and pads made of gold deposited on a cantilever as shown in Figure 1.3. The cantilever was first made of Silicon Dioxide (SiO 2 ) and then changed to Silicon Nitride (Si 3 N 4 ) [START_REF] Dobson | New Methods for Calibrated Scanning Thermal Microscopy (SThM)[END_REF]. The cantilever length, width, thickness are 150 µm, 60 µm and 0.4 µm respectively. The tip height and thickness are respectively 10 µm and 40nm and its width varies between 1.5 and 6 µm. The tip radius is smaller than 100 nm [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF][START_REF] Dobson | New Methods for Calibrated Scanning Thermal Microscopy (SThM)[END_REF]. Figure 1.4 shows SEM images of the considered probe.

Silicon nanoprobes

The silicon nanoprobes were first developed by IBM for high data-rate storage systems and high-speed nanoscale lithography applications [START_REF] Despont | VLSI-NEMS chip for parallel AFM data storage[END_REF][START_REF] Binnig | Ultrahigh-density atomic force microscopy data storage with erase capability[END_REF]. This type of probe can be also used for other application such as the characterization of polymers [START_REF] Gotsmann | Thermally activated nanowear modes of a polymer surface induced by a heated tip[END_REF]. Later, Nelson and King [START_REF] Nelson | Measuring material softening with nanoscale spatial resolution using heated silicon probes[END_REF] developed similar silicon probes but with pyramidal tips . As seen in Figure 1.5, the cantilever consists of two micrometric legs with high doping level and a low doped resistive element platform. The tip of a nanometric radius curvature has a pyramidal shape (conical in case of IBM probe) is mounted on top of the resistive element. Table 1.2 gives the dimensions of the silicon probes of IBM and the AN-300 commercialized by Anasys Instruments. Due to the change of the probes configuration and size, each type of probe requires a proper approach of the measurements for a better characterization of the thermal properties of the samples. The models mainly used to describe the probe-sample system are given in the next section.

Approaches of the measurement

For resistive probes, the analysis and interpretation of measurements requires a specific modeling linking the electrical measurement to the sample thermal properties. The establishment of this modeling requires the estimation of the heat flux transferred from the probe to the sample that is related to the tip apex temperature. The tip apex temperature is often determined by solving the heat equation within the probe.

The Wollaston probe

In his thesis, Lefèvre [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF] divided through symmetry the probe-sample system into two parts and each part was modeled as a thermal fin as shown in Figure 1.6.

After writing the heat balance on an element dx of the probe as illustrated in Figure 1.7, the heat equation is: Here, θ= T-T a is the relative temperature at abscissa x of the fin. p, S, ρ, λ and α are respectively the perimeter, section, electrical resistivity, thermal conductivity and thermal resistivity of the Pt90/Rh10 wire. h denotes heat transfer coefficient of the losses to the ambient environment.

2 2 2 2 1 (1 ) hp I x S S a t θ ρ θ θ aθ λ λ ∂ ∂ - + + = ∂ ∂ (1.1)
Since the section of the Pt90/Rh10 wire is much smaller compared to that of the Wollaston wire, the extremities are assumed to be heat sinks (T=T a at x=0). Besides, the mean temperature of the probe is usually determined through electronic means (Wheatstone Bridge). Then the term 1 aθ + in Equation (1.1) is replaced by 1 aθ + in first approximation.

The probe can be heated through DC or AC current.

a) DC regime

When the probe is heated through DC current, the time taken by the probe to reach its equilibrium does not exceed 3 ms [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF]. Once this time is exceeded, Equation (1.1) is: When the probe is far from the contact, the boundary condition at the tip apex is:

( ) 0 2 L x x θ ∂ = = ∂ (1.3)
When the probe is in contact with the sample,

( ) ( ) 2 2 eq L L S x G x x θ λ θ ∂ = = = ∂ (1.4)
where G eq (W.K -1 ) represents the equivalent thermal conductance between the sample thermal conductance G s and the probe-sample contact conductance G c : Let m²=hp/λS and Ω=G eq /G p where G p =λS/L is the probe thermal conductance. Solving Equation (1.2) with the two conditions (Equations (1.3) and (1.4)), the probe Joule power when in contact is was used in the calibration of SThM probes for the measurements of the thermal conductivity of bulk samples [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Gomès | Characterization of the thermal conductivity of insulating thin films by scanning thermal microscopy[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF]. In these works, the heat losses through convection were not taken into account (h=0). Under this condition, the resolution of Equation (1.2) leads to:

s c eq s c G G G G G = + (1.5)
3 4 4( ) G G P c s P G G G G G G c p s p c c s ∆ = + + (1.8)
Later in this manuscript, we will show to what extent the coefficient h could affect the determination of the thermal conductivity of bulk samples and the thermal interaction parameters between the probe and the sample.

b) Alternating regime

When an AC current (alternating current) I=I 0 cos(ωt) heats the probe, the Joule power dissipated in the Pt/Rd wire is:

2 0 (1 cos(2 t)) 2 p RI P ω = + (1.9)
Then, the temperature in the probe is: The probe voltage can be written as follows:

0 0 2 0 0 2 0 0 (1 ) cos( t) cos( ) cos(3 t ) 2 2 p DC R I R I V RI R I t ω ω β θ β θ aθ ω ω φ ω φ = = + + + + + (1.11)
Under the assumption that the other voltage harmonics are negligible.

Practically, the terms at the first harmonic are deliberately suppressed using a Wheatstone bridge since they are much larger than the terms at the third harmonic

0 0 2 3 cos(3 t ) 2 R I V RI ω ω β θ ω φ = = + .
Through this expression, the alternating temperature θ 2ω can be experimentally determined. Through a proper modeling, this temperature can be related to the thermal properties of the sample. This method was first developed by Cahill (well-known 3ω method) [START_REF] Cahill | Thermal conductivity measurement from 30 to 750 K: the 3ω method[END_REF] and was adapted by Lefèvre [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF] to the Wollaston wire probe in the AC regime. For sake of simplification, the complex writing is used (

0 i t I I e ω =
) and the temperature in the probe is written as: Parameters identified with these approaches:

Measuring 2ω
θ through the measurements of V 3ω as a function of the angular frequency ω permits a determination of the equivalent conductance G eq for a specific sample [START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF]. Thus, with appropriate description of the sample conductance G s , one can determine the contact conductance G c for a sample of well-known thermal conductivity. This cannot be done in the DC regime where the determined G c is equivalent among different samples of well-known thermal conductivity. However, an issue concerning the thermal expansion of the probe while working under the AC regime was reported [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF]. In fact, the contact between the probe and the sample is not stable. That could be the reason behind the dispersion of the measurements obtained by Lefèvre [START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF].Based on these observations, Chapuis [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF] suggested that it is better using the DC regime for distances of separation d s smaller than 50 nm.

The palladium probe

Puyoo [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF] used the old palladium probe (with SiO 2 cantilever) in the AC regime to determine the thermal conductivity of silicon nanowires embedded in silicon dioxide sample. In his approach of measurement, he adapted the model of Lefèvre [START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF] to the palladium probe by describing the probe as two thermal fins composed of two adjacent materials (Pd and SiO 2 ) as shown in Figure 1.9. The term of diffusion was just taken for the part of SiO 2 since this part has a larger section compared to that of Pd (S SiO2 /S Pd ∼ 25). The same boundary conditions as in Equations (1.3) and (1.4) were taken and the measurements were performed.

Since the dimensions at the tip apex are down to distances compared to the mean free path of material, the thermal conductivity of the probe apex can differ from the one estimated in the diffusive regime [START_REF] Prasher | Predicting the thermal resistance of nanosized constrictions[END_REF]. In fact, in the case of Silicon Nitride Si 3 N 4 cantilever, the phonon mean free path at room temperature is of the order of 20 nm [START_REF] Watari | Effect of grain size on the thermal conductivity of Si3N4[END_REF] and the tip radius is about 50 nm.

Tovee et al. [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF] suggested that heat transfer in the probe based on the diffusive regime is a reasonable approximation. They related their approach to the fact that the Knudsen number is smaller in the case of SiO 2 (old cantilever) -Kn≃0.2. However, the thermal resistance of the probe R p determined experimentally in vacuum environment (R p = 8 × 10 4 K.W -1 ) is larger than expected with numerical simulations (R p = 5.35 × 10 4 K.W -1 [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]). This could be related to the section decreasing at the probe apex and thus to an additional thermal resistance induced by the boundary scattering within the probe apex to be taken into account [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF][START_REF] Shivaprasad | Temperature coefficient of resistance of thin palladium films[END_REF].

These effect were not taken into account in the numerical simulations as discussed by the authors [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF].

An additional issue that faces the description of the probe-sample system is the spreading resistance of the sample R s . In fact, the radius of the tip-sample mechanical contact might be smaller than the mean free path of heat carriers in the sample and thus an appropriate description of R sp is needed as will be discussed in section 1.4.

Silicon nanoprobes

The classical description of the heat conduction within the tip of these probes based on the diffusive regime cannot be applied. This is due to the small dimensions and shape of the probe and to the fact that the mean free path of the tip material (Si) is comparable to the tip dimensions. As an example, the mean free path of phonons in Silicon is around 200 nm at room temperature and the radius of the Si tip apex is around 30 nm. Therefore, due to the diffuse scattering of the phonons at the tip boundaries, the thermal resistance of the nanostructures (nano-tip in our case) is increased [START_REF] Chen | Phonon heat conduction in nanostructures[END_REF]. While working in an air medium with these probes, most of the heat is transferred from the probe cantilever to the air and thus to the substrate [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF].

In their works based on numerical simulations, Nelson and King [START_REF] Nelson | Modeling and simulation of the interface temperature between a heated silicon tip and a substrate[END_REF] modeled the overall thermal resistance in the tip is as two parts: a conical resistance R cone (for diffusive part) and a hemispherical resistance R sphere (nanoscaled apex part). The radius of the sphere was assumed the same of the tip apex and the heat transfer in the hemisphere was assumed one-dimensional in Cartesian coordinates [START_REF] Nelson | Modeling and simulation of the interface temperature between a heated silicon tip and a substrate[END_REF] and the conduction to the ambient air was neglected. This condition would include less error while working under ultra-vacuum environment where heat conduction through the air is eliminated. The thermal resistance due the phonon scattering at the tip-sample contact is given as a function of an interval of the thermal boundary resistance [START_REF] Swartz | Thermal boundary resistance[END_REF] and depends on the sample material [START_REF] Nelson | Modeling and simulation of the interface temperature between a heated silicon tip and a substrate[END_REF]. R b is defined as "the ratio of the temperature discontinuity at the interface to the power per unit area flowing across that interface (K.m 2 .W -1 ) [START_REF] Swartz | Thermal boundary resistance[END_REF]".

(Kapitza resistance) R b (m 2 .K.W -1 )
Furthermore, Gotsmann et al. [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF] proposed an analytical model in which the tip-sample system is modeled as a series of thermal resistances as described in Figure 1.10. The temperature at the sample surface can be given through the coefficient Δ: 17) Here, R sp , R interface , R tip are respectively the thermal resistances of the sample, the interface and the tip. R tip denotes the diffusive part of the tip resistance and the part linked to boundary scattering should be included in R interface [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. Under UHV conditions (~ 10 -8 -10 -9 mbar), R interface depends on the probe and sample materials and the scattering physics at the interface.

0 0 s s heater tip interface s T T R T T R R R - ∆ = = - + + (1.
The coefficient Δ in Equation (1.17) varies between 0.3 and 0.7 for polymers (R s >> R interface ), about 10 -3 -10 -4 for metals (R s << R interface ) [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. The described model does not take the losses at the tip apex to the surrounding medium into account and is more accurate for the interpretation of measurements performed under UHV conditions. Moreover, Gotsmann et al. [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF] modeled R interface through a phonon reflection coefficient r 12 as shown in Figure 1.11. In this approach, R interface is dependent on the sample material and on the scattering at the interface: As seen in this section, each probe has its own geometrical dimensions and its configuration of the heater. This requires an appropriate model of the probe-sample system for each of these probes described above that takes into account the configuration of the heater and the effect of the decreasing sizes.

The modeling of the probe-sample interaction is related to different parameters in order to quantify the mechanisms of heat transport between the probe and the sample.

Heat losses through the air

While working under ambient conditions, a fraction of the heat is lost from the probe to the medium environment. When the dimensions go down to the micro-scales, the heat transfer coefficient due to convection becomes smaller as the buoyancy forces decrease [START_REF] Guo | Size effect on microscale single-phase flow and heat transfer[END_REF]. Thus, the heat losses to the air are due to the mechanism of conduction [START_REF] Hu | Investigation of the natural convection boundary condition in microfabricated structures[END_REF]. The equivalent heat transfer coefficient h to these losses, which depends on the probe size, is larger than that at macro-scales due to larger surface to volume ratio.

In order to estimate h for the Wollaston wire probe, Chapuis [START_REF] Chapuis | Heat transfer between a hot AFM tip and a cold sample: impact of the air pressure[END_REF] measured in AC regime the amplitude of the second harmonic θ 2ω of the probe temperature at different levels of pressure P for a probe away from any contact. The theoretical θ 2ω =f(h) (see Equation (1.14)) and the measurement θ 2ω =f(P) permitted to have the graph h=f(P). The graph was fitted with the theory of Lee [START_REF] Lees | Kinetic-Theory Description of Conductive Heat Transfer from a Fine Wire[END_REF] of the resolution of the Boltzmann equation for the heat transfer through gas molecules between two co-axial cylinders. The obtained value was approximately 3500 W.m -2 .K -1 . In fact, this value is close to the value given by the correlation given by Taine and

Petit [START_REF] Taine | Transferts thermiques[END_REF] for the case of a long vertical cylinder of diameter D and length L:

0.05 0.93( ( ) ) D Nu Ra D L = (For Ra(D)D/L < 0.05) (1.19)
Based on this correlation and for the dimensions of the Wollaston wire probe, h is around 2800 W.m -2 .K -1 .

These values are similar too to the values determined by Thierry et al. [START_REF] Thiery | Two omega method for active thermocouple microscopy[END_REF] for welded Wollaston wires of different diameters (5 µm, 1.3 and 0.5) by using the two omega method.

The identified values for h are 2300, 6500 and 13000 W.m -2 .K -1 for wire diameter of 5, 1.3 and 0.5 µm respectively.

For the palladium wire, Puyoo [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF] identified h by comparing the probe signal while working under ambient environment and under vacuum conditions (10 -5 mbar ) where no transfer to the air is assumed. The identified value was about 6600 W.m -2 .K -1 .

Chapter 1 -SThM: State of the art and exposition of the issues Kim and King [41] used numerical simulations and FE model to determine the heat losses to the air from the silicon cantilever of the silicon probes described in part 1.2.3. The estimated coefficient h around the leg was found about 2000 W.m -2 .K -1 and about 7000 W.m -2 .K -1 around the heater for a steady state heating.

All the values of h in the studies mentioned above show that h is enhanced largely at the small scales comparing to the values at the macro scales. We should mention that these studies identified the values of h when the probe is out of contact of the sample. Such values are susceptible to some changes when the probe comes to contact with the sample due to a restricted area of exchange between the probe and the environment medium. This is often neglected in SThM as discussed by Raphael [START_REF] Raphaël | Contribution à la microscopie thermique à sonde locale en mode alternatif: Caractérisation de la réponse et de l'interaction sonde échantillon[END_REF]. Moreover, as can be seen, a high dispersion during the evaluation of h can be noticed. The dispersion of the values of h is related to the variety of the models of estimating h and the boundary conditions at the probe surface.

Moreover, the models do not overcome the change in the probe parameters. Lefèvre [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF] estimated the error to be less than 3 % while evaluating the heat dissipated within the probe through Joule effect between two cases: h=1000 W.m -2 .K -1 and h=0.

Thermal spreading resistance

As seen in the section of the interaction modeling, the exchange between the probe in contact with the sample can be modeled through an equivalent conductance G eq (Equation (1.5)). One of the components of G eq is the thermal spreading resistance R s in the sample. R s depends on the description of the thermal boundary condition taken into account at the thermal contact and the sample geometry and dimensions. When the heat is spread into a semi-infinite sample as show in Figure 1.12, the thermal spreading resistance is defined as:

source heatsink s T T R Q - = (1.20)
where Q is the heat transfer rate, source T is the temperature average of the heat source and where λ s is the thermal conductivity of the semi-infinite sample.

If the source is considered as an isoflux circular area, then [START_REF] Yovanovich | Thermal spreading and contact resistances[END_REF]:

2 8 1 3 s s R b π λ = (1.22)
The difference between the Equations (1.21) and (1.22) is around 7%. In the modeling of the tip-sample interaction, the ballistic constriction resistance to add here is taken however into account into the resistance of contact R contact (modeling of Nelson and

King [START_REF] Nelson | Measuring material softening with nanoscale spatial resolution using heated silicon probes[END_REF]) and into R interface ( Gotsmann et al. [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]).

More information about the modeling of the resistance of constriction can be found in ref.

[29].

Heat transfer mechanisms between SThM probes and sample

The thermal conductance at the contact G c depends on the mechanisms by which the heat is transferred from the probe to the sample. Under ambient conditions and when the probe is out of contact of the sample, the heat transfer is made through radiation and conduction through the air. Two mechanisms are added when the probe gets into contact: the solid-solid contact and the conduction through the water meniscus that is formed between the tip and the sample as shown in Figure 1.13 [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF]. In this section, we review the literature works that studied these different modes of the heat transfer or the ones that passed by some measurements that could be an indication to these mechanisms.

Heat transfer through radiation

Under ambient environment, the thermal conductance due to radiation is much smaller than the ones of other mechanisms [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] Majumdar | Scanning thermal microscopy[END_REF]. Thus, this heat mechanism is not taken into account while modeling the heat exchange under ambient conditions [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Majumdar | Scanning thermal microscopy[END_REF]. To estimate the radiation heat transfer, one must work under special conditions such as ultra-high vacuum (P ~ 10 -8 -10 -9 mbar) environment where conduction through meniscus and through the air might be suppressed.

Under UHV conditions and when the probe is out of contact of the sample, the conduction through air is suppressed and the heat is transferred to the sample only by radiation. When the separation distance between the probe and the sample becomes smaller than the peak wavelength of thermal radiation λ max (≃ 10 µm at ambient temperature), the heat transfer due to radiation may increase due to evanescent electromagnetic domains close the two surfaces.

Still we have to be careful about this hypothesis since any enhancement is dependent on the size of the two bodies [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF]. Besides, the heat conductance between the bodies depends on their nature too (dielectrics or metals [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF]) as shown by Chapuis [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF] through numerical simulations of the heat exchange between sphere of different radius and different materials.

Using the IBM type of silicon probes (part 1.3.3), Hinz et al. [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF] observed the heater temperature variations while approaching/retracting from a SiO 2 sample under high vacuum conditions (10 -5 mbar). As shown in Figure 1.14, the heater temperature decreases before the contact (from A to B) and that could be related to the heat transfer through radiation. In fact, the authors suggested this contribution or an electrostatic interaction with the sample [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF].

Working under UHV conditions (P ~ 10 -8 -10 -9 mbar), Kittel et al. [START_REF] Kittel | Near-field heat transfer in a scanning thermal microscope[END_REF] measured the heat exchange power Q between a nanotip of gold at room temperature and a cooled sample (T sample = 100 K ) of gold at different distances of separation (from 1 to 100 nm). When the separation is about 1 nm, Q is about 10 -5 W. If we suppose that the temperature difference is about 200 K, then the radiation conductance G rad is about 5 × 10 -8 W.K -1 . Narayanaswamy et al. [START_REF] Narayanaswamy | Near-field radiative heat transfer between a sphere and a substrate[END_REF] determined G rad between a silica sphere of 50 µm in diameter and a substrate of glass for a ΔT ≃ 50 K under high-vacuum conditions (P = 6.7 × 10 -5 mbar). At a separation distance of 100 nm, G rad was found around 5 × 10 -8 W.K -1 . Later, the same group showed how nearfield radiation is affected by the nature of the sample [START_REF] Shen | Surface phonon polaritons mediated energy transfer between nanoscale gaps[END_REF]. In their work, the heat transfer through radiation between SiO 2 sphere and Au sample did not show any even at a separation distance of 30 nm. That was not the case of SiO 2 sample where the heat transfer showed a significant change above the far-field limit [START_REF] Shen | Surface phonon polaritons mediated energy transfer between nanoscale gaps[END_REF].

Therefore, it can be seen that the heat transfer through radiation can be detectable in SThM where accurate measurements might be while working under special conditions (UHV conditions). However, to increase that possibility, the sample (or tips) with their materials and shapes should be appropriately selected [START_REF] Siria | A scheme for solving the plane-plane challenge in force measurements at the nanoscale[END_REF].

Heat transfer at solid-solid contact

When the tip contacts the sample, a percentage of heat is transferred through the mechanical contact. To quantify this percentage, we need first to determine the radius of contact a c . Many theories exist already to evaluate a c and the parameters that affect its value. For an elastic deformation, Hertz firstly estimates a c between a sphere of radius R sph and a flat surface as follows [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF]:

1 3 * ( ) N sph c F R a E = (1.23)
with F N the normal load and E * is the effective Young modulus given as:

2 2 * 1 1 2 1 2 1 1 4 ( ) 3 E E E ν ν - - - = + (1.24)
E 1 , E 2 are the Young modules of the sphere and the surface and ν 1 and ν 2 are their Poisson coefficients respectively. This equation does not account for the adhesion between the two bodies. Two models exist and take into account the adhesive forces between a sphere in contact with a plane surface: the JKR (Johnson, Kendall and Roberts) [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF][START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF] and DMT [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF][START_REF] Derjaguin | Effect of contact deformations on the adhesion of particles[END_REF] (Derjaguin, Muller and Toporov) models. These models are adapted to different types of pull-off forces that are defined as the required forces to separate a surface from another surface [START_REF] Dietler | Force-distance curves by atomic force microscopy[END_REF]. The JKR model suggests having non-zero contact area when no normal load is applied. Here, a c is estimated as [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF]:

1 1 2 3 3 * * [ ( )] [ ( 3 6 ( 3
) )]

sph sph c N ad ad N d sph d sph N d sph R R a F F F F w R w R F w R E E π π π = + + = + + + (1.25)
where F ad is the adhesion force and w d is the Dupré energy of adhesion.

Derjaguin, Muller and Toporov (DMT) assumed that when the sphere and the surface are separated, a c goes to zero. In the DMT model a c is determined as [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF]:

1 1 3 3 * * [ ( )] [ ( 2 )
]

sph sph c N ad N sph R R a F F F wR E E π = + = + (1.26)
The contact when it involves spheres of large radii and of large and short-range pull-off forces is well described with the JKR model [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF][START_REF] Schwarz | A generalized analytical model for the elastic deformation of an adhesive contact between a sphere and a flat surface[END_REF]. On the other hand, the DMT model explains well the contact of stiff materials and of small and long-range pull-off forces [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF][START_REF] Schwarz | A generalized analytical model for the elastic deformation of an adhesive contact between a sphere and a flat surface[END_REF]. A transition parameter exist between the two models and known as the Maugis parameter [START_REF] Butt | Force measurements with the atomic force microscope: Technique, interpretation and applications[END_REF].

Generally, in SThM, the mechanical properties of materials used in the different models are the properties at the large scales. More appropriate properties at the micro/nanoscales can be used by taking the estimation of mechanical properties of materials using the nano-indentation techniques. The Hertzian theory is included in the modeling of Majumdar for the solid-solid thermal conductance G ss [START_REF] Majumdar | Scanning thermal microscopy[END_REF]. Through an effective thermal conductivity of the contact and a conical probe of opening θ, G ss was estimated as:

tan 2 2 2 tan s p ss c eff c s p G a a λ λ θ π λ π λ λ θ = = + (1.27)
where λ p , λ s are respectively the thermal conductivity of the probe and the sample. Here, a c is the radius of the contact area linked to the mechanical contact between the probe and the sample.

Similarly, David et al. [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] estimated G ss for the Wollaston wire probe as:

2 2 s p ss c eff c s p G a a λ λ π λ π λ λ = = + (1. 28)
However, this macroscopic formula of the conductance cannot describe accurately a contact in the order of ten of nanometers or less depending on the probe size. If we suppose a c = 10 nm as determined through thermal conductivity calibration by David [START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces Application aux céramiques pour le nucléaire[END_REF], the values of G ss vary between 7 × 10 -8 W.K -1 for Silica sample (λ SiO2 ≃ 1.2 W.m -1 .K -1 at T=T a ) and 2 × 10 -6

W.K -1 for Silicon sample (λ Si ≃ 148 W.m -1 .K -1 at T=T a ). We have to note that such an approach takes an equivalent a c among the calibration samples without including the materials mechanical properties (Young modulus E and Poisson ratio ν).

Experimentally, the perfect conditions to detect the thermal conduction due to the mechanical contact are to work under vacuum conditions [START_REF] Majumdar | Scanning thermal microscopy[END_REF]. Under these conditions, there are two modes of heat transfer between the probe and the sample: the radiation and the solid-solid conductances. As mentioned before, the radiation is usually negligible in front of the other mechanisms and thus any change at the contact may be due to the solid-solid conductance.

Hinz et al. [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF] related the heater temperature change observed in Figure 1.14 to the conduction at the mechanical contact. G ss was evaluated around 1.5 × 10 -7 W.K -1 [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF].

Chapter 1 -SThM: State of the art and exposition of the issues 52 However, we can notice a signal shift while approaching and retracting the probe (lines BB' and CC' in Figure 1.14). This shift is similar to the measurements performed by Puyoo with the palladium probe [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF]. Puyoo [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF] performed simultaneous measurements of the V 3ω probe voltage and the deflection of the cantilever during the approach and retract curves at a pressure of 10 -5 mbar. As seen in Figure 1.15, a significant change of the V 3ω signal occurred at the contact. But while retracting the probe, the jump-off-contact deflection still exists and the V 3ω signal variation does not take place at the same position while approaching the probe.

The jump-off-contact deflection could be related to what remains of the water meniscus + the elastic forces or just the elastic forces. The observed effects might be also related to the nature of the sample (hydrophilic, hydrophobic) and the stiffness of the probe. If this deflection is just due to the elastic forces as the tip snaps into the sample, thus the change of the V 3ω signal is due solely to the solid-solid conduction. One needed parameter here is the probe temperature. We have to note that both of the measurements of Hinz et al. [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF] and Puyoo [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF] are not performed under UHV conditions but at a pressure around 10 -5 mbar where an eventual meniscus might exist between the probe and the sample. As we can notice, the identification of the solid-solid conductance passes too through working under appropriate experimental conditions. Working under high-vacuum conditions assures the suppression of the conduction through air but there is still a doubt whether the meniscus is totally vanished under such conditions. Another needed parameter is the probe temperature while performing the measurements that could leave or suppress the doubt about the existence or the absence of the water meniscus. An estimation of the contribution of the water meniscus to the heat transfer to the sample could help here (see later in Chapter 2).

Besides, in the whole theories and measurements cited above in this section, the contact between the probe and the sample was assumed perfect and no effect of roughness is taken into account. If the measurements are performed under UHV conditions with a rough tipsample contact, modeling the tip-sample thermal requires a better description of the contact to take into account the additive resistance due to the rough contact.

The information about the effect of the roughness on G ss conduction remains scarce in literature. Recently, Gotsmann and Lantz [START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF] identified G ss while approaching a flat-punch tip of silicon probe and taC (tetrahedral amorphous carbon) sample surface while performing measurement under high vacuum conditions (P ≃ 6 × 10 -6 mbar). As can be seen in Figure 1.18, the thermal resistance at the contact is around 2 × 10 -8 W.K -1 . The authors go beyond these measurements to determine the thermal conductance per atom-atom contacts [START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF]. The applied force between the probe and the sample was estimated and the number of atom-atom contacts is approximated [START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF]. The conductance per atom-atom contact is given as:

12 atom atom Q G G Nt - = (1.29)
where G Q is the quantum thermal conductance per phonon mode, N is the number of phonon modes and t 12 = 1 -r 12 is the transmission coefficient and r 12 is the same reflection coefficient already encountered before in Equation (1.18). G atom-atom was found around 1.2 × 10 -9 W.K -1 . A part of chapter 3 is dedicated to evaluate the solid-solid thermal conductance and to present the effect of roughness on the determination of the thermal conductivity of the samples.

Heat transfer though water meniscus

Under ambient conditions, a film of adsorbed water is formed on the sample due to the capillary condensation of the relative humidity. When the tip gets in contact with the sample, a meniscus takes shapes between the two surfaces. The heat path through water meniscus was not studied in the literature and little information exists about the meniscus thermal conductance. The reason behind is that most the properties of the water at small scales are still unknown and especially the thermal properties. Luo et al. [START_REF] Luo | Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy[END_REF] suggested that the meniscus conductance G w is the predominant among the heat mechanisms. G w was written as:

2 [ln(1 (1 cos )) 1 cos ] a w w a R G R a πλ β β ≈ + - + - (1.30)
Here, λ w and a are respectively the thermal conductivity and thickness of the meniscus and R a is the apex radius. Sin β= w/2R a where w=2r 2 is the meniscus width as shown in Figure 1.19.

Their probes were made of thermocouple junctions (Au-Ni or Au-Pt) and the tip radius around 100 nm. By heating the substrate and measuring the voltage of the probes, they found the thermoelectric voltage different to what is expected. It was related to the thermal resistance at the contact between the tip and the substrate. With a modeling of the thermal resistances between the substrate and the tip as illustrated in Figure 1.20, the contact conductance G c =1/R c was determined between 4 × 10 -6 W.K -1 and 2 × 10 -5 W.K -1 [START_REF] Luo | Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy[END_REF]. By modeling all the transfer mechanisms, it was found that the meniscus conductance is around 2 × 10 -6 W.K -1 . If the heat was transferred mainly by the meniscus, then this value is smaller to what was measured [START_REF] Luo | Sensor nanofabrication, performance, and conduction mechanisms in scanning thermal microscopy[END_REF]. This difference raises lots of questions about an eventual contribution of the other mechanisms in this heat transfer that limits the characterization of the sample. Few years later, in 2002, the same group questioned their first results based on thermal measurements with the same equipment [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF]. The authors estimated that the heat transfer through water would be rather small in front of the thermal conduction due to air, which appeared dominating the thermal interaction between the probe and the sample. They attributed this new result to the fact that the thermal conductances at the probe-water and water-sample interfaces had not been taken into account in the modeling of their previous works [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF]. Using the same configuration (heated substrate and thermocouple probe), Thiery et al. [START_REF] Thiery | Thermal contact calibration between a thermocouple probe and a microhotplate[END_REF] determined the ratio of conductances G p /G c as a function of the heater temperature T heater .

They found that G contact increased up to a limit of temperature around 90 °C and then it decreased slowly (see Figure 1.21). They just suggested that this limit is the liquid-vapor transition. Yet, many questions could be asked about the reasons behind the increasing tendency before the 90 °C and the slight decrease of G p /G c after this temperature. We note that the diameter of the thermocouple probe used in their experiment is around 1.3 µm [START_REF] Thiery | Thermal contact calibration between a thermocouple probe and a microhotplate[END_REF].

Gomès et al. [START_REF] Gomès | DC thermal microscopy: study of the thermal exchange between a probe and a sample[END_REF] suggested that the water limit is around 90 °C based on thermal signal measurements of the Wollaston probe operated at three different temperatures. Even though G w was not determined because these works were among the first ones in SThM, the thermal signal looked different at two probe temperatures 50 °C and 90°C. Then, the estimated G c and b in ref [START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF] correspond to a probe temperature ranging between 60 and 90 °C. Since the water meniscus might be affected by temperature, the water meniscus conductance as a function of temperature is required here (see Chapter 2).

As the understanding on the heat transfer through water meniscus is not clear yet, this heat path is usually avoided in the measurements. In fact, the probe is heated up to temperatures higher than 100 °C [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] or the measurements are performed under vacuum conditions [START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF][START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF] and in the two cases the water meniscus is supposed to have vanished. We should be aware that the first assumption is a point of view based on the thermodynamic laws at large scales. For the second, a doubt appears whether a high-vacuum level is sufficient to evaporate In order to answer the questions about the heat transfer through water meniscus and for a better understanding for the transfer at the contact, Chapter 2 displays in details the meniscus "issue". This study passes around the terms that affect principally the meniscus heat path:

probe shape and size, sample nature (hydrophilic and hydrophobic), capillary forces, sample roughness, probe temperature and relative humidity.

Heat transfer through air

The air conductance to the sample is dependent on the separation distance d s between the probe and the sample. When d s is much larger than air phonon mean free path Λ air (d s >> Λ air ), thus the conduction regime is diffusive and dependent directly on d and the Fourier law is applicable. When d s << Λ air , thus the air gap conductance cannot be estimated from the continuum theory and the conductance is estimated from local conductances [START_REF] Majumdar | Scanning thermal microscopy[END_REF]. Moreover, since the conduction between the tip and the sample cannot be assumed as an interaction between two plane surfaces, there is always a geometry factor δ in the conductance expressions.

In the diffusive regime, the air conductance coefficient was given by Shi et al. [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF] as follows:

( )

a diffusive s k h y d d = + (1.31)
and in the ballistic regime as follows:

3(1 2 )

air ballistic CV h f d = + (1.32)
where λ a , C and V air are the thermal conductivity, heat capacity and velocity of air molecules respectively. f is given as follows:

2(2 ) ( 1) Pr A f A γ γ - = + (1.33)
where A is the accommodation factor (0.9 for air), Pr is the Prandtl number and γ is the ratio between the heat capacities at constant pressure and at constant volume. Between the two regimes, Shi et al. [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF] introduced the slip regime for Λ < (y+d s ) < 100Λ

where h slip is given as:

( ) 2 a slip s h y d f dλ = + + Λ (1.34)
d s is taken here as the separation between the end of the tip apex and the sample and (y+d s ) is the distance between a point on the tip surface and the sample.

The conductance through air G air is then given as:

(diffusive, ballistic or slip) air G h dxdy = ∫∫ (1.35)
Lefèvre [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF] determined the equivalent conductance G eq (Equation (1.5)) after measuring the thermal signal as a function of the distance between the Wollaston probe and the sample. As shown in Figure 1.22, G eq shows 1/z dependence as a function of the distance down to distances around 300 nm. In spite of an unclear curve at the small distances because of the large steps of the used motor, the deviation from 300 nm to the contact could be a sign of the ballistic regime. Measurements with small steps of motion would let us better understand this regime. The numerical results were compared with experimental results obtained with the Wollaston probe (Figure 1.23). The numerical simulations do not fit the experimental results since the probe is assumed to a perfect torus and the roughness is neglected in the model. However, the two curves have the same tendencies with a decreasing slope at the "large distances" that is due to the diffusive regime and a limit that is due to the ballistic limit. In order to compare the different conductances of the various heat mechanisms, Chapuis [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF] measured the variation of the voltage signal measured as a function of the probe/sample distance. Figure 1.25 shows that the signal jump at the mechanical contact (from C to D) is much smaller than the drop due the heat transfer through air before the contact (from A to B).

These results show that in this case the conduction through air is the predominant mechanism in the tip-sample thermal interaction.

Supposing that the meniscus disappears at probe temperatures larger than 100 °C, Lefèvre [START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF] and David [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] estimated an equivalent G air and b air (G c ≃G air and b≃b air ) using DC calibration with the Wollaston probe (Equation (1.8)). The calibration includes various samples of well-known thermal conductivity. The obtained values of G air and b air are respectively around 5 × 10 -6 W.K -1 and 1 µm [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF]. These values are used to determine the unknown thermal conductivity of solid materials. Even though it is a fast calibration, the used procedure can generate a significant error. This due to the fact that the determined G air and b air cannot be taken constant among all the used samples. While increasing the sample thermal conductivity λ s , G air increases and b air decreases. In other terms, the heat flux lines go perpendicularly to the sample surface when λ s is large and along the surface when it is low [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF].

Chapter 3 deals with this problematic and shows a new methodology to improve the DC calibration with the Wollaston wire probe. This methodology helps decreasing the error while estimating the thermal conductivity of materials and especially materials with low thermal conductivity (between 0.1 and 3 W.m -1 .K -1 ). We have to note that while working with the silicon probes, a huge difficulty is to determine experimentally the heat conduction through air between the tip and the sample. In fact , the heat transferred through the air gap from the cantilever is greater than the one transferred from the tip though air to the substrate while working under ambient conditions [START_REF] King | Thermomechanical formation of nanoscale polymer indents with a heated silicon tip[END_REF]. The conductance of the heat path is on the order of 10 -5 W.K -1 [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. This value is much larger than the value of the thermal conductance through the tip and at the tip-sample interface (~10 -8 -10 -9 W.K -1 ). Thus, estimating the tip-air-substrate conductance needs a high sensitivity and appropriate modeling to separate this conductance from the cantilever-air-substrate thermal conductance. Analytically, Chapuis et al. [START_REF] Chapuis | Heat transfer between a nano-tip and a surface[END_REF] studied that quasi-ballistic heat transfer through between a nano-tip and a substrate. The studied nano-tip has almost the same geometry of the silicon probes. The given approach can be used to compare future experimental results where the heat through air is exclusively transferred from the tip. Through finite elements analysis FEA model, Tovee et al. [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF] studied the sensitivity of the palladium probe in air conditions. As a first approach, the model assumed a temperature continuity across the tip-sample interface [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. The apex temperature and the heater temperature change when the probe is in contact were calculated [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. The simulations showed that the probe sensitivity is limited to few W.m -1 .K -1 as shown in Figure 1.26. Later, using the same probe Bosse et al. [START_REF] Bosse | Nanothermal characterization of amorphous and crystalline phases in chalcogenide thin films with scanning thermal microscopy[END_REF] identified the thermal conductivity of amorphous and crystalline phases in GeTe and Ge 2 Sb 2 Te 5 films while working under ambient conditions. The thermal conductivity of the samples varied between 0.2 and 1.95 W.m -1 .K -1 [START_REF] Bosse | Nanothermal characterization of amorphous and crystalline phases in chalcogenide thin films with scanning thermal microscopy[END_REF]. This interval lies in the range of sensitivity determined by Tovee et al. [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF].

Juszczyk et al. [START_REF] Juszczyk | DC Experiments in Quantitative Scanning Thermal Microscopy[END_REF] performed numerical simulations in air conditions for the case of the palladium probe. The same geometry of the probe was taken into account but the probe material was varied for three different thermal conductivities [START_REF] Juszczyk | DC Experiments in Quantitative Scanning Thermal Microscopy[END_REF]. A continuity of temperature was assumed at the interface between the probe and the sample. The sensitivity when the thermal conductivity of the probe substrate is about 30 W.m -1 .K -1 (close to the case of silicon nitride) was up to few tens of the sample thermal conductivity as shown in Figure 1.27.

Conclusions

In this chapter, we have seen that the heat transfer between SThM probes and samples depends on various factors. These factors involve the shape and size of the probes, probe temperature, the sample nature, the applied load, the surfaces and the environmental conditions. An accurate thermal characterization requires a convenient modeling of the tipsample system. The modeling is accomplished first by modeling the heat transfer through the tip, the thermal interaction between the probe-sample and through the sample. Modeling the heat transfer through the tip becomes difficult when the tip dimensions decreases and become comparable to the mean free path of the heat carriers. The thermal interaction between the probe and the sample is dependent on the configuration of the probe (in contact or out of the contact with the sample), the environment (vacuum or under ambient air), the relative humidity, the sample nature (insulators or conductors) and the applied force. We add to these elements, the interaction of the probe with the environment while working under air environment.

The upcoming chapters deal with some of the questions mentioned in this chapter. They especially aim to provide answers to the following questions.

• The study and the quantification of the heat transfer through water meniscus has been,

to Can we identify parameters used to describe the heat transfer at solid-solid interfaces from measurements carried out under vacuum conditions?

• SThM calibration and measurements are often carried out under ambient conditions within the framework of the identification of the thermal conductivity of materials.

What is the rate of heat losses from the probe to its surrounding environment through thermal convection and conduction through air? In the frame of the description of the probe-sample system with a thermal conductance network, what is the dependence on the sample thermal conductivity λ s of the various thermal conductances involved?

Once this dependence clarified, what is the sensitivity to λ s of current SThM probes?

A detailed approach of all these questions is given for the Wollaston probe through the chapters 2 and 3 of this thesis. Chapter 4 addresses some of them for the KNT and doped Si DS probes.

Chapter 2: Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM

Chapter 2

Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM

As seen in the first chapter, information about the heat path through the water meniscus remains scarce. Reaching a better understanding of the interaction between the sample and the probe requires an investigation of this heat transfer mechanism. Therefore, this chapter deals with the heat transfer through water meniscus that is formed at the probe-sample contact under ambient conditions in scanning probe microscopy (SPM). First, we present two selected applications where the water meniscus (or water film) is the center of interests and how they lie within our proper interests. Later, we investigate the heat conduction through the water based on the analysis of the pull-off forces on a hydrophilic sample using the Wollaston wire probe at different probe temperatures. Through these measurements, the dimensions of the water meniscus are determined based on literature models of the meniscus. The thermal conduction of the water meniscus is quantified as a function of the probe temperature. The effect of roughness on the pull-off forces is shown on the same sample. Moreover, measurements of pull-off forces on hydrophobic sample of graphite are presented.

At the end of this chapter, results of first experiments performed in an ESEM chamber are presented and analyzed. Since the size of the MEMS/NEMS is being reduced, some tribological issues are reported.

The term "stiction" (static friction) has been used too to describe the motion between two surfaces [START_REF] Bhushan | Nanotribology and nanomechanics of MEMS/NEMS and BioMEMS/BioNEMS materials and devices[END_REF]. While working under ambient environment, stiction happens because of the condensed water layer between the surfaces [START_REF] Bhushan | Nanotribology and nanomechanics of MEMS/NEMS and BioMEMS/BioNEMS materials and devices[END_REF][START_REF] Tas | Stiction in surface micromachining[END_REF]. The corresponding capillary forces of the water film could produce damage in the performance of MEMS/NEMS and might stop their functioning [START_REF] Bhushan | Nanotribology and nanomechanics of MEMS/NEMS and BioMEMS/BioNEMS materials and devices[END_REF][START_REF] Rajagopalan | Linear high-resolution bioMEMS force sensors with large measurement range[END_REF]. Since these systems are mostly fabricated from silicon, Bhushan et al. [START_REF] Bhushan | Adhesion and friction studies of silicon and hydrophobic and low friction films and investigation of scale effects[END_REF] studied the adhesive forces between a ball of hydrophilic Si(100) against a specimen wafer of Si(100) as a function of the relative humidity and the temperature of a heated stage.

To compare his experimental results, Bhushan [START_REF] Bhushan | Adhesion and friction studies of silicon and hydrophobic and low friction films and investigation of scale effects[END_REF] identified also the adhesive forces between three hydrophobic films of DLC, Z-DOL, and HDT in contact with a ball of Si(100).

He found that the adhesive forces on Si(100) sample are larger than those of DLC, Z-DOL, and HDT, linking this observation to the hydrophilic or hydrophobic character of the sample surfaces [START_REF] Bhushan | Adhesion and friction studies of silicon and hydrophobic and low friction films and investigation of scale effects[END_REF].

Stiction forces can be reduced by changing the surface properties [START_REF] Ogwu | The effect of PECVD plasma decomposition on the wettability and dielectric constant changes in silicon modified DLC films for potential MEMS and low stiction applications[END_REF][START_REF] Puurunen | Reducing stiction in microelectromechanical systems by rough nanometer-scale films grown by atomic layer deposition[END_REF][START_REF] Hurst | Self-assembled monolayer-immobilized gold nanoparticles as durable, anti-stiction coatings for MEMS[END_REF] or by reducing the contact area [START_REF] Tas | Stiction in surface micromachining[END_REF], but the solutions to reduce the contact surface remain scarce and challenging. Based on AFM, dynamic force modulation shows a reduction in the stiction at low [START_REF] Lantz | Dynamic superlubricity and the elimination of wear on the nanoscale[END_REF] and high [START_REF] Dinelli | Ultrasound induced lubricity in microscopic contact[END_REF] frequencies. Thermally-controlled devices may be another alternative not yet explored.

In contrast to MEMS/NEMS, Dip-pen nanolithography (DPN) takes advantage of the existing water nano-layers.DPN is a scanning probe lithography technique where an AFM tip, coated with a chemical compound or mixture acting as an "ink", is used to create patterns (on scales of under 100 nanometers)directly on a substrate.

The DPN technique, introduced by Piner in 1999 [START_REF] Piner | Dip-pen" nanolithography[END_REF], consists of transporting molecules to substrate using an AFM tip. The molecules are transported through the water meniscus Weeks et al. [START_REF] Weeks | Effect of dissolution kinetics on feature size in dip-pen nanolithography[END_REF] studied the effect of humidity on the size of dots fabricated by DPN, by placing the AFM in an environmental chamber. They found high dependence of the dots diameter on the relative humidity [START_REF] Weeks | Effect of dissolution kinetics on feature size in dip-pen nanolithography[END_REF]. The DPN technique has proved to have high potential for many applications. It has been widely used to set Biomolecular Micro-and Nanoarrays on surfaces [START_REF] Ginger | The Evolution of Dip-Pen Nanolithography[END_REF]. Such microarrays are nowadays used widely in oncology to assess the Deoxyribonucleic Acid (DNA) in patient tumors [START_REF] Gonzalez-Angulo | Future of personalized medicine in oncology: a systems biology approach[END_REF]. DPN is used also to identify the cells adhesion mechanism and to study the interaction between some biomolecules for biorecognition processes [START_REF] Ginger | The Evolution of Dip-Pen Nanolithography[END_REF][START_REF] Müller | Atomic force microscopy as a multifunctional molecular toolbox in nanobiotechnology[END_REF]. Moreover, DPN is used in electrochemical mode to deposit nanofeatures of metals on substrates [START_REF] Li | Electrochemical AFM "dip-pen" nanolithography[END_REF]. Li et al. [START_REF] Li | Electrochemical AFM "dip-pen" nanolithography[END_REF] were the first to use the electrochemical dip-pen lithography to deposit small amounts of Pt on Si wafers [START_REF] Li | Electrochemical AFM "dip-pen" nanolithography[END_REF]. In this configuration, the condensed water meniscus between the tip and the substrate serves as an electrochemical cell where the metal is dissolved [START_REF] Li | Electrochemical AFM "dip-pen" nanolithography[END_REF]. Depending on the application of the DPN method, the use of a thermally-controlled AFM tip could here be a way to improve or optimize patterning.

As can be seen for both the applications cited above, the water meniscus that forms due to the capillary condensation may have a crucial role in the operating principle of systems and process. This has motivated a deep scientific research the last fifteen years for characterizing the properties of water at micro and nanoscales and investigating the parameters that affect those properties.

1 Chapter 2 -Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM 71

Characteristic dimensions and properties of the water small layers adsorbed on a surface

The characterization of the water film adsorbed on surfaces is usually performed through AFM (Atomic force microscopy) or AFM-based technique like scanning Kelvin probe force microscopy [START_REF] Hu | Scanning polarization force microscopy: A technique for imaging liquids and weakly adsorbed layers[END_REF] or STM (Scanning Tunneling Microscopy) [START_REF] He | Scanning Tunneling Microscopy Study and Nanomanipulation of Graphene-Coated Water on Mica[END_REF]. These techniques provide a better spatial resolution than optical techniques that are limited by the diffraction limit. In order to calculate the adhesive forces, the film thickness or the film physical properties [START_REF] Xu | Wetting and capillary phenomena of water on mica[END_REF][START_REF] Gao | Structured and viscous water in subnanometer gaps[END_REF][START_REF] Zitzler | Capillary forces in tapping mode atomic force microscopy[END_REF], characterization of the water film is mostly carried out with tips with small radii of curvature, due to their better force resolution [START_REF] Dietler | Force-distance curves by atomic force microscopy[END_REF]. The thickness of the water film on the surface of a sample does not exceed few nanometers as shown in Figure 2.2 [START_REF] Guo | Water adsorption behavior on metal surfaces and its influence on surface potential studied by in situ SPM[END_REF]. The factors that play the major role in determining the film thickness: the surface character (hydrophilic or hydrophobic) [START_REF] Hu | Scanning polarization force microscopy: A technique for imaging liquids and weakly adsorbed layers[END_REF][START_REF] Gil | Adsorption of water on solid surfaces studied by scanning force microscopy[END_REF] and the relative humidity [START_REF] Guo | Water adsorption behavior on metal surfaces and its influence on surface potential studied by in situ SPM[END_REF]. An additive error could be added while determining of the water thickness due to the water condensation around the tip especially while working with hydrophilic tips [START_REF] Santos | Measuring the true height of water films on surfaces[END_REF]. In fact, Mate et al. [START_REF] Mate | Atomic force microscopy of polymeric liquid films[END_REF] found a difference comparing the results of the local film thickness determined with AFM and with ellipsometry. That was linked to the water condensation around the tip [START_REF] Dietler | Force-distance curves by atomic force microscopy[END_REF]. Santos et al. [START_REF] Santos | Measuring the true height of water films on surfaces[END_REF] presented an interesting model, validating with some experiments on a BaF 2 substrate, to calculate the true height of water layers on both hydrophilic and hydrophobic surfaces. A crucial parameter is added to these parameters is the sample roughness parameter of either the tip or the sample while studying the interaction forces between the tip and the sample [START_REF] Hansson | Influence of surface topography on the interactions between nanostructured hydrophobic surfaces[END_REF][START_REF] Farshchi-Tabrizi | On the adhesion between fine particles and nanocontacts: An atomic force microscope study[END_REF][START_REF] Butt | Normal capillary forces[END_REF][START_REF] Rabinovich | Capillary forces between surfaces with nanoscale roughness[END_REF].

The dimensions of the meniscus at the tip-sample contact are estimated by using the Young-Laplace and Kelvin Equations.

Young-Laplace and Kelvin Equations

Let us consider the case of a capillary of large width in a longitudinal tube as illustrated in Due to the strong curvature of the cavity of radius r k , the pressure in the liquid is negative [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF]. The pressure in the liquid is the sum of a negative Laplace pressure P Lap and the atmospheric pressure P 0 and given by [96]:

0 cos / liquid w c k P P r γ θ = - (2.1)
where γ w is the surface tension of the liquid and θ is the contact angle (see Figure 2. Assuming the vapor as an ideal gas, we can write then [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF]:

(p ) ( ) ln(RH) (2.2) = ( ) ( ) (2
g liq sat m R T RH P P V - =
where R g is the gas constant, RH=p vap /p sat is the relative humidity and V m is the molar volume.

Due to the large magnitude of p liq, p liq -p sat is almost equal to p liq -p 0 . Therefore, using Equation (2.1), Equation (2.4) can be written [START_REF] Israelachvili | Intermolecular and surface forces[END_REF]: ln( )

w m k g V r R T RH γ = (2.5)
Equation (2.5) is the well-known Kelvin equation.

Little information exists yet in the literature on the temperature-dependence of the meniscus.

Since our SThM is an AFM-based technique with a heated probe, our study goes in the direction of fulfilling more information in that field.

2.2

Methodology developed and used to investigate the temperature-dependence of the meniscus

Experimental approach

The SThM probe to be used has to provide high sensitivity to the interaction between the probe and the sample where the heat transfer to the water meniscus could be detected.

Majumdar et al. [START_REF] Majumdar | Scanning thermal microscopy[END_REF] suggested that increasing the tip radius can result basically in increasing the meniscus diameter and its thermal conductance (see Chapter 1, section 1.6.3). So, we start our study here by using the Wollaston probe of larger radii even if thermal probes with nanoscale curvature radius are nowadays available [START_REF] Dinwiddie | Thermal conductivity contrast imaging with a scanning thermal microscope[END_REF]. The geometrical dimensions and material of this probe are given in the first section of chapter 1. Since the AC mode can induce a problem at the contact in terms of extension of the probe and thus a consequence on the stability of the water meniscus (see chapter 1), heating the probe through DC electrical current was preferred.

While working under ambient conditions, the capillary forces are the dominant forces among the pull-off forces as suggested by Cappella et al. [START_REF] Dietler | Force-distance curves by atomic force microscopy[END_REF]. Since the water meniscus might be for the sample of germanium Ge. We noticed first that the cantilever elastic constant, which is defined by the slope of the curves after contact, remains constant while varying T m . Since no hysteresis is found between the approach and withdrawal curves in contact, we can say that the deformation under the tip due to the indentation, is totally elastic [START_REF] Dietler | Force-distance curves by atomic force microscopy[END_REF][START_REF] Landman | Atomistic mechanisms and dynamics of adhesion, nanoindentation, and fracture[END_REF]. The gap δ between the withdrawal curve (in blue) and the axis of the initial cantilever deflection DFL represents the adhesive forces (see Figure 2 We have to note that the force-distance curves can be performed within different durations up to 60 seconds. This duration is the maximum time permitted by our AFM. We observed that while modifying the duration of the curves, the pull-off forces are almost constant for tipsample contact time τ ts larger than one second. That is in accordance with the measurements of the pull-off forces performed by Sirghi [100] between a classic hydrophilic silicon AFM tip (tip radius of 30 nm) in contact with a hydrophilic glass sample. Starting from a time contact of 40 ms, the pull-off forces remain constant and the meniscus reaches its thermodynamic equilibrium. In the case of our measurements, it can be said therefore that the meniscus is at its thermodynamic equilibrium. this range of T m , it can be said that the meniscus largely evaporated. Moreover, we observe a hysteresis in the measurements while increasing T m and decreasing T m . We do not have a full understanding for this hysteresis at the moment, but among the causes which could be related one should consider the hysteresis during the evaporation and the capillary condensation at the probe-sample contact. At this time, this phenomenon is not fully understood, although the reported hysteresis during evaporation and the capillary condensation at the probe-sample contact as observed by Crassous et al.

[101] is expected to be at least partly responsible.

Note that no significant changes were observed for the force magnitude and tendencies when changing the probes, even though the Wollaston wire probes that were used for the experiments are manually fabricated and then poorly reproducible.

Derived parameters: Meniscus parameters and thermal conductance of probe-sample heat transfer through water

The total thermal conductance to be considered to describe the probe-sample heat transfer through the water meniscus G total, meniscus is the inverse of the sum of the inverses of three conductances: the conductance at the probe-water interface G p-w ; the conductance at the water-sample interface G w-s ; and the conductance due to the conduction through the meniscus G water . The latter may be calculated from Equation (1.30). However, the measurements of pull-forces correspond to measurements of forces when the probe contacts the sample (a=0) [START_REF] Israelachvili | Intermolecular and surface forces[END_REF]. Consequently, R water =1/G water becomes negligible, so that G total, meniscus is only governed by G p-w and G w-s .

The thermal conductance per surface area at the interface between hydrophilic solids and water was investigated by Ge et al. The magnitudes of the pull-off forces F po at relatively high temperatures are assumed to primarily due to the elastic forces (Figure 2.7). Measurement of pull-off forces under vacuum conditions in the future could reveal more information about this assumption. To obtain the variation of the capillary forces F cap as a function of the probe mean temperature T m , the minimum value of the measured forces is then subtracted: F cap (T m )= F po (T m )-F po (T m,max ). We have considered the case of perfect wetting on the probe surface and on the sample surface, which corresponds to a single monolayer e ≃ 0.15 nm [104]. From the values of F cap at room temperature, we deduce an estimation of R a ≃ 0.35 µm. From the values of F cap at room temperature, an estimate of R a ≃ 0.35 µm was deduced. In fact, the micro-fabrication process of the Pt/Rd wire consists in drawing out the wire that leads to the appearance of many grooves at the probe surface as shown in Figure 2.9. As a consequence, this configuration is closer to a cylinder in contact with the sample surface. Thus, the radius R a in Equations (1.30) and (2.6) is almost twice that of the groove as explained on Figure 2.10 (in our case R 2 ≃ 15 µm and R 2 >>R 1 ). This value is in good agreement with the dimensions of the asperities at the Wollaston probe apex observed by David et al. [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF]. 1 2 To calculate r 2 , we make the assumption that the water meniscus confined between the probe and the sample is at the temperature T=(T m +T a )/2. It should be noted that this approximation is valid for samples of high thermal conductivity (case of germanium) where the temperature of the sample surface under the tip apex is almost at T a [105, 106] and as will be shown in Chapter 3. In the case of samples with low thermal conductivity, the sample surface has almost the same temperature of the probe apex. In this case, it is more appropriate to use the apex temperature as the temperature of the water layer. For each value of T m , a corresponding value of r 2 is then deduced from Equation (2.7) using our measurements of F cap .

a a eq R R R + = 2 1 1 2 1 1 1 1 1 eq R R R R R R >> + = 

Results

The variations of r 2 as a function of T m deduced from Equation (2.7) are shown in Figure 2.11. Once r 2 is determined, the values of G p-w and G w-s can be estimated by multiplying the areas S p- w and S w-s by H and therefore G total, meniscus can be calculated. Figure 2.12 shows the variation of G total, meniscus as function of T m .

When the meniscus is large, the values of G total, meniscus are one order of magnitude lower than the values of the thermal conductance due to the heat transfer through air (G air ≃ 4.5 µW.K - 1 [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] and G air ≃ 2.5 µW.K -1 [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF]). As a consequence, the thermal conductance through the water meniscus cannot be predominant in the thermal interaction between the probe and the sample [107]. It is striking that our findings are in accordance with the observations made with a nanoprobe [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF], even though the tackled sizes are very different. Note that the values of H=100 or 180 MW.m -2 .K -1 induce a maximum shift of 25 % of the values of G total, meniscus (see Figure 15). A variation of the probe temperature T neither modifies this conclusion. We underline that our analysis justifies SThM calibration methods that have been developed

previously with the same kind of microprobes at temperatures T m larger than 100 °C [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF].

As the meniscus is almost evaporated at these relatively high temperatures, it has a smaller impact on the heat transfer to the sample and simplifies the analysis [107]. As seen in this section, the conduction through the water meniscus is not the predominant mechanism in the thermal interaction between the probe and the sample. The water meniscus conduction is at one order of magnitude smaller than the heat conduction through air [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] (see later in chapter 3). The methodology applied here can help quantifying the heat transfer through the meniscus by measuring the pull-off forces between a probe and the sample. As observed, the meniscus that is responsible of the stiction can largely disappear at large probe temperature and thus that might be an alternative solution for the stiction reduction in the case of NEMS/MEMS devices. Moreover, the variation of the meniscus dimensions at different

probe temperatures can open an avenue for different applications such as the thermallycontrolled dip-pen nanolithography.

An important factor helped accomplishing this study is the small value of roughness of the Ge sample (arithmetic value of roughness R ar = 0.54 nm and maximum peak-to-peak roughness R p-p =5 nm measured by AFM). The next section investigates the impact of the sample roughness factor using the Wollaston probe and compares the measurements with the ones presented before on the polished germanium sample and with the results of the scientific literature for SPM studies of the roughness effect. Moreover, to conclude this part of the study, measurements of pull-off forces on a hydrophobic sample (graphite) were performed and compared with literature studies.

Effect of roughness on the pull-off forces on hydrophilic surfaces

In order to focus our study on the effect of roughness on the pull-off forces, we perform our measurements on a rough sample of germanium Ge. Its roughness parameters are R ar = 27 nm and R p-p = 277 nm measured by AFM. Figure 2.13 presents the variations of the pull-off forces F po as a function of T m on the sample of rough germanium. The measurements were performed also at different points at the sample surface for each probe temperature. As in the curves of the polished Ge sample (see Figure 2.7), the curves of F po here show a decreasing tendency up to a temperature around 120 ⁰C where the F po stays almost constant. We noticed that this value is smaller than the value observed in the case of the polished Ge. That might be related to an additional thermal resistance due to the sample roughness effect. In fact, an additional resistance at the contact induces a larger apex temperature (see chapter 1).

Therefore, the temperature of the water meniscus between the tip apex and the sample might be larger in the case of the rough Ge than that of the polished Ge.

The mean values of F po are smaller than those obtained in the case of polished germanium (see Figures 2.7 and 2.13). The surface roughness reduces basically the real contact area between the probe and the sample. This has a significant impact on the capillary forces as studied by Rabionvich et al. [START_REF] Rabinovich | Capillary forces between surfaces with nanoscale roughness[END_REF]. At relative low humidity level as in our case (RH≈ 40 %), we can expect a decrease in the mean values of the capillary forces [START_REF] Rabinovich | Capillary forces between surfaces with nanoscale roughness[END_REF]. An illustration between the contact between the probe and a surface asperity is given in Figure 2.14. In fact, there is only a thin mono-molecular film that exists at the contact and the meniscus, if it is formed, is very small and the capillary adhesion is then reduced [START_REF] Rabinovich | Capillary forces between surfaces with nanoscale roughness[END_REF]. Ata et al.

[108] studied the variation of the adhesion forces as a function of the roughness parameter R ar at the contact between a glass sphere of 10 µm in radius and a Si wafer. They found a decreasing tendency until a limit where the asperity height becomes larger than the double of the equivalent radius of curvature [108]. That is related to our case where R ar is much larger than 2×r eq and can explain the lower values of the pull-off forces measured at various positions at the sample surface. This regime is known as the "asperity regime" as defined by Halsey [109] where a single asperity is in contact with the probe. We also noticed that error bars are much larger in Figure 2.13 and the hysteresis is hardly observed in this case.

Measurement of the pull-off forces on hydrophobic surface

Due to its hydrophobic character, a graphite sample is considered in this part. At each T m , the measurements were performed at minimum of 15 points at the sample surface.

The error bars in Figure 2.15 denotes the maximum and the minimum of F po among the population of points. We can notice that the mean values F po is almost constant as a function of T m . In fact, it could be related to the non-formation of a meniscus between the tip apex and the graphite sample. Our measurements curve are similar to that obtained by He et al. [110] given in figure 2. [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. The pull-off forces F po were measured as a function of the relative humidity between hydrophobic tip (apex radius ≃ 10 -20 nm) and a hydrophilic silicon sample [110]. No variation of F po was observed as a function of RH (see Figure 19) and that was related to the meniscus that was not formed between the tip and the sample We observed in addition that at each T m , one or two points among the population have much larger values of pull-off forces than all the other points (see Figure 2.15). The experiment was repeated many times and the same behaviour was always observed. In fact, that might be related to the study of Luna et al. [111] where nanodroplets of water were observed on the graphite surface using Scanning Force Microscopy (SFM). Thus, the large values of F po observed during the measurements might be the results of the contact between the probe and adsorbed nanodroplets on the graphite sample.

As seen for the case of hydrophobic sample, the meniscus is not formed at the tip-sample contact and thus no heat conduction through meniscus exists for this type of samples.

Interestingly, a small group of measurements shows high value of pull-off forces. That might be related to nanodroplets of water that exist on the sample surface as suggested in the scientific literature.

Preliminary experiments performed in the frame of the development of a SThM-ESEM combined system

CETHIL is currently developing a SThM-ESEM combined system. This part reports the results of preliminary experiments carried out in an Environmental Scanning Electron Microscope (ESEM) with the Wollaston probe in order to evaluate the possibility to observe the meniscus under such conditions. The probe was not heated in these first experiments.

The ESEM offers a direct imaging of the meniscus in a variable-pressure environment. By decreasing the temperature of the meniscus, the curvature radius of the meniscus r k is expected to increase (see Equation (2.5)). The probe is usually brought into contact with a cooled substrate [112,113]. Figure 2.17 shows an ESEM image of the water meniscus at the contact between the Wollaston wire and a substrate of germanium Ge. The temperature of the substrate was about 2 °C and the relative humidity is around 90 %. As can be seen, the curvature radius of the meniscus r k increases while decreasing the temperature and its imagining is easier to perform. Similar experiments were performed by Van Honschoten et al. [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF] and Greiner et al. [113] and the same difference was also observed. As an example of such a difference, the radius of curvature r 1 observed at RH= 98 % in ESEM was about 200 nm where as in the Kelvin Equation the value is expected to be around 27 nm [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF]. Van Honschoten et al. [START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF] related this difference to either an additional pressure induced by the electron beam that is focused on the meniscus and/or the meniscus contact with the microscopic liquid film on the substrate. In the case of perfect wetting, the liquid film of thickness e is governed by the disjoining pressure. This pressure is the difference between the pressure of the vapor around the liquid film and the pressure of liquid [104].

In order to investigate these effects, a substrate of Germanium was placed in an ESEM chamber. The substrate was cooled through a Peltier cooler. The relative humidity was between 95 % and 100 %. The electron beam was focused on a specific zone on the substrate up to a time limit where no change is observed on the liquid film on the substrate.

Immediately after, the beam was zoomed-off and an image was collected of the liquid film that could show two zones: one where the electron beam was focused and another where the e-beam was not focused on.

Conclusion

In summary, this chapter dealt with the heat transfer through water meniscus between the Wollaston probe and samples. First, we have investigated the thermal conduction through the water meniscus at the contact between the probe and a hydrophilic sample of germanium Ge.

The values of the pull-off forces were measured whilst increasing and decreasing the probe mean temperature. The measurements show a progressive evaporation of the meniscus until a certain limit beyond which the pull-off forces remain constant. The experimental measurements were linked to the pull-off forces at elevated probe temperatures to allow the water meniscus contribution to the tip-sample heat transfer to be quantified. The largest radius of the meniscus was estimated using published models. The total thermal conductance of the meniscus was then calculated while accounting for the thermal conductance at the probemeniscus and meniscus-sample interfaces. The results show that the meniscus thermal conductance is at most one order of magnitude lower than the conductance through air for the same probe. In addition, the effect of roughness on the pull-off forces was shown on hydrophilic surfaces. The obtained results were discussed with what was observed through literature models and measurements. Moreover, the pull-off forces were measured on a hydrophobic sample of graphite as function of the probe temperature. Our results suggest the non-formation of a meniscus between the probe and the sample except for one or two points among large population of points where the meniscus is linked to a nano-droplet as suggested in the scientific literature. The same methodology is used in the case of smaller probes (KNT and DS probes) as it will be seen in Chapter 4.

In the aim of developing a SThM-ESEM combined system, the last part of this chapter reported the results of first experiments performed in an ESEM. Our results outline a first effect showing that the effect of the electron beam on SThM measurement has to be studied and to be taken into account while analyzing measurements with the SThM-ESEM system in future works.

Chapter 3: Probe-sample thermal system in DC Scanning Thermal Microscopy investigated with the Wollaston probe

Chapter 3

Probe-sample thermal system in DC Scanning Thermal Microscopy investigated with the Wollaston probe

As seen in Chapter 2, the thermal conduction through water meniscus does not play a dominant role in the thermal interaction between the SThM probe and a sample in the case of the Wollaston probe. In order to achieve a quantitative analysis of the probe-sample thermal interaction with this type of probe, the chapter is dedicated to study the probe-sample heat transfer by two other mechanisms: thermal conduction through solid-solid contact and thermal conduction through air. Studying these mechanisms requires knowledge of the probe thermal properties once it is heated to temperatures above the ambient. Once the probe thermal conductance is identified, measurements under vacuum conditions will be first used to determine the thermal conductance through the mechanical contact. Thermal boundary resistances at the tip-sample contact will be determined for different materials.

The analysis of measurements performed under ambient conditions will be then used to specify the heat losses to the environment and reach a better understanding of the probe interaction with the surrounding medium. Later, the parameters of the probe-sample thermal interaction through air (effective thermal conductance G air and thermal radius b air ) will be studied as a function of the sample thermal conductivity. The effect of roughness of samples on the measurement will be shown through a discussion of experimental results obtained with different materials. These results helped us to establish a procedure of experiments to reduce the error during the determination of the thermal conductivity of bulk samples. Based on this procedure, the thermal conductivity of samples for thermoelectric applications is determined in the last part of the chapter. If we adapt the modeling of the probe-sample thermal system as three thermal conductances in series (probe, thermal contact and sample), the variation ΔP/P c =f(T m ) can be linked to:

(1) The variation of the thermal conductance at the contact G c where the conductances of the different heat mechanisms are included or/and

(2) The variation of thermal conductance of the probe G p .

We start by identifying the variation of G p =f(T m ), as studying the thermal conduction through solid-solid contact and the thermal conduction through air requires the value of G p as it will be seen later.

Influence of the probe mean temperature T m on G p

The thermal conductance of the probe G p was used to describe the heat flux between the probe supports (in Wollaston wire) and the probe apex [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF] in the thermal conductivity calibrations and the studies of the probe-sample thermal interaction that used the Wollaston probe [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. In these calibrations, the boundaries between the Pt/Rh wire and the probe supports were supposed at ambient temperature (T support =T a ) [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. The thermal conductance of the probe G p has been given as [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]:

p p p p S G L λ = (3.1)
where λ p , S p and L p are respectively thermal conductivity, section and length of the resistive Pt/Rh wire. The thermal conductivity of the probe λ p was taken constant whatever the probe temperature (λ p = 38 W.m -1 .K -1 at T a ) and thus G p =G p1 = 3.73 × 10 -6 W.K -1 [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF].

However, the probe is heated to temperatures larger than the ambient temperature raising questions about the variations of λ p or the validity of Equation (3.1) based on the assumption taken in the thermal conductivity calibrations (T support =T a [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]). Any variation of G p should be taken into account in order to quantify accurately the heat mechanisms between the probe and the sample. Moreover, the effect of the variation of the probe temperature on the thermal interaction between the probe and the sample has not been shown yet.

In order to estimate the temperature dependence of G p , thermal measurements were carried out under vacuum conditions (P= 5 × 10 -5 mbar) where the probe is away from any contact.

Under these circumstances, probe losses to the surrounding gas may be neglected (h=0 W.m - 2 .K -1 ). Supposing that the probe supports are at ambient temperature, the apex relative temperature θ apex (θ apex =T apex -T a ) is equal to 1.5 times the probe relative mean temperature θ m (θ m =T m -T a ) [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF]. G p can be calculated as [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF]: where P p is the power dissipated in the probe.

In practice, a Wheatstone bridge was used to control T m and the variation of P p was monitored. The red circles in Figure 3. Two suggestions can be given to explain this variation: [START_REF] Raphaël | Contribution à la microscopie thermique à sonde locale en mode alternatif: Caractérisation de la réponse et de l'interaction sonde échantillon[END_REF] over our range of probe mean temperature. The variation of λ Pt/Rh in this case does not also explain the variation of G p =f(T m ).

Chapter 3 -Probe-sample thermal system in DC SThM investigated with the Wollaston probe 96 It can then be said that the difference between the two curves obtained in Figure 3.2 are related to the boundary conditions of heat sink taken at the probe support in the modeling of the approaches of Lefèvre [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF], David et al. [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF] and Raphael [START_REF] Raphaël | Contribution à la microscopie thermique à sonde locale en mode alternatif: Caractérisation de la réponse et de l'interaction sonde échantillon[END_REF].

In the following, we have considered:

Thermal conductance at the solid-solid contact G ss

The expressions of the thermal conductance for the heat transfer through solid-solid contact G ss proposed by Majumdar [START_REF] Majumdar | Scanning thermal microscopy[END_REF] and then by David [START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF] implies an increase of G ss when the thermal conductivity of the sample λ s increases (Equations (1.27) and (1.28) -Chapter 1). For samples with high thermal conductivity like silicon (λ Si ≃ 148 W.m -1 .K -1 ), the value of G ss calculated with these expressions has the same order of magnitude of the thermal conductance through air G air (for Si, G ss ≃ 2 × 10 -6 W.K -1 and G air ≃ 2.5 × 10 -6 W.K -1 [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF]) if we assume mechanical contact radius a c of 10 nm. While performing DC calibration of the Wollaston wire probe under ambient conditions (see chapter 1, section 1.3.1), G ss = f(λ s )

while assuming a c = 10 nm. Compared to the calibration curves that does not include G ss [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF], In order to distinguish between G ss and G air and to have a better estimation of G ss between the Wollaston probe and the sample in the present works, thermal measurements measured under vacuum conditions were performed. The probe and sample were mounted in a Scanning Electron Microscope (SEM) chamber. The pressure in the chamber was about 5 × 10 -5 mbar.

Under these conditions, the conduction through air and the heat losses to the environment are eliminated (h ≃ 0) [START_REF] Majumdar | Scanning thermal microscopy[END_REF][START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF]. The probe was heated up to 140 ⁰C to eliminate the rest of the meniscus that might exist between the probe and the sample as seen in Chapter 2. As explained in Chapter 1, the thermal conduction through radiation G rad may be assumed negligible in comparison to the other mechanisms. It can be supposed that under such circumstances the signal change at the contact is mainly due to the conduction at the mechanical contact. The measurements were performed on three samples of low roughness parameters (measured with AFM) and different thermal conductivities as listed in Table 3.1.

The choice of low roughness is to reduce the errors while estimating a c .

Under these circumstances, ΔP/P c can be written as:

3 4 4( ) G G P ss s P G G G G G G c p s p ss ss s ∆ = + + (3.4)
G s is the thermal conductance of the sample given as [START_REF] Prasher | Predicting the thermal resistance of nanosized constrictions[END_REF]:

1 4 s c s a G K λ = (3.5)
where K 1 is a correction factor for the diffusive thermal conductance and is about 0.694 [114,115]. Figure 3.5 shows the variation of the thermal signal measured for the different samples. The measurements were performed with two probes and repeated at an atmospheric pressure P=0.28 mbar. No significant difference was observed between the measurements at P=5 × 10 -5 mbar and P=0.28 mbar. It can be said that the heat losses to the environment and conduction through air are almost negligible at these pressures in agreement with the experiments of Chapuis [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF]. The applied force between the probe and the samples was around 500 ± 100 nN. Since we supposed the elimination of the water meniscus, a c was estimated for the three considered sample materials using the Hertzian model as shown in Table 3.1 (Equation (1.23) -Chapter 1). In this estimation, the radius of the sphere in contact with the sample was estimated around 0.35 µm as seen in Chapter 2 (Equivalent radius of a cylindrical groove). Based on these values and Equations (3.4) and (3.5), G ss can be evaluated. Figure 3.6 shows the evaluated values of G ss for the different samples. We can notice that the values of G ss increase as function of λ s . However, these values are at least one order of magnitude smaller than the values of the thermal conductance due to the heat transfer through air (G air ≃ 7 µW.K -1 ) as it will be seen in the following parts of this chapter.

Therefore, the variation of G ss =f(λ s ) (Equation (1.28)) overestimates G ss and does not explain the better fit of ΔP/P c =f(λ s ) obtained by David et al. [START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces Application aux céramiques pour le nucléaire[END_REF] (see Figure 3.3).

From the determined values of G ss , we can access to the value of an effective thermal boundary resistance R b (K.m 2 .W -1 ) at the probe-sample interface. G ss can be expressed as [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF][START_REF] Nelson | Modeling and simulation of the interface temperature between a heated silicon tip and a substrate[END_REF]117]:

2 a c G ss R b π = (3.6)
Here, G ss =1/R ss where R ss denotes the ballistic resistance at the tip-sample constriction for the case of Ge and Si samples. For these samples, the mean free path Λ of the samples is much larger than the contact radius a c as can be noticed in Table 3.1 (MICA is expected to be extremely anisotropic). As seen in this part, working under vacuum conditions does not increase the sensitivity in the calibration with the Wollaston probe for materials with high thermal conductivity like silicon and germanium. In fact, the sample thermal conductance G s is on the order 5 × 10 -6 W.K -1 while the tip-sample contact conductance G ss is on the order the 10 -7 W.K -1 . This makes it difficult to determine experimentally the thermal properties of the sample materials while working under vacuum conditions. However, experimental measurements under these conditions are of high interest for the study of the heat transport at interfaces.

The next section investigates the heat transfer to the sample through the air. The involved heat transfer mechanism appears be dominant in the tip-sample thermal interaction. We aim in this section to understand better the heat transfer to the sample and to reduce the error during the characterization of the thermal conductivity of the sample material.

Measurements under ambient conditions

Determination of the coefficient h of the heat losses to the environment

While working under ambient conditions, the thermal signal ΔP/P c is related to the coefficient h that defines the heat losses to the medium environment (see Equations (1.6) and (1.7)). To be able to determine the parameters of the tip-sample interaction through air, the coefficient h should be first estimated.

The evaluation of h by Chapuis (h≃3000 W.m -2 .K -1 [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF]) was based on supposing that the probe interaction with the environment is based on the mechanism of conduction instead of convection. Since the natural convection phenomenon is driven by the buoyancy forces, an orientation of the Wollaston probe might provide a clue to estimation the participation of the phenomenon of convection. The probe was oriented up to 70 º from its initial position as illustrated in Figure 3.8. We observed no significant change of the signal measured (see Appendix A) while performing the measurements under air conditions and at P=2 mbar as shown in Figure 3.9. This result is in accordance with what expected by the Grashof number Gr that gives the ratio of the buoyancy forces to the viscous forces [123]. In our case, for a difference of temperature of around 100 ºC between the probe and the environment, Gr is smaller than 1 and thus the convection effects are negligible in front of those of the heat conduction [124]. Therefore, the heat losses to the environment are mainly due to the heat conduction and the assumption taken by Chapuis is valid in this case.

The value of h = 3000 W.m -2 .K -1 is taken into account in our study in the following parts.

Heat conduction through air to the sample

A. Experimental study

We try in this part to study the heat transfer between the probe and the sample through air in order to quantify the air conduction parameters (b air ,G air ) and to provide more accurate determination of thermal conductivity of bulk samples. For materials with low thermal conductivity λ s , the probe provides a high level of sensitivity to λ s as seen in Figure 3.4. The sensitivity decreases with the increase of λ s . The thermal conductivity limit for the calibration with the Wollaston wire is between 20 and 30 W.m -1 .K -1 [START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. When λ s exceeds that limit, the thermal signal ΔP/P c is almost constant (see Figure 3.4). Therefore, we chose to perform our measurements with samples of thermal conductivity that is smaller than 15 W.m -1 .K -1 .

The samples used in this calibration are listed in Table 3.2. λ s is between 0.1 and 13 W.m -1 .K -1 . the thermal interaction radius b=b air and the thermal conductance at the contact G c = G air .

Figure 3.10 shows the variations of ΔP/P c as a function of λ s . The experimental curve ΔP/P c =f(λ s ) is similar to those curve observed in Figure 3.4. Table 3.3 shows the mean values identified for G air and b air while taking into account the heat losses to the environment h = 3000 W.m -2 .K -1 . Even though the values of G air (G air = 6.64 µW.K -1 ) are in the same of order of magnitude of the values identified in other studies using the same probe (G air ≃ 4.5 µW.K -1 [START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF] and G air ≃ 2.5 µW.K -1 [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF]), we can notice that the values of b air are one order of magnitude larger than expected (b air ~ µm [START_REF] Lefèvre | Nanoscale heat transfer at contact between a hot tip and a substrate[END_REF][START_REF] David | Modelling for the thermal characterization of solid materials by dc scanning thermal microscopy[END_REF][START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] Lefevre | Thermal conductivity calibration for hot wire based dc scanning thermal microscopy[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]). Unlike our experiments, we observe that the literature studies do not always account for samples of thermal conductivity smaller than 1 W.m -1 .K -1 [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. In our measurements, this range of thermal conductivity is represented by the sample of polyimide PI (λ PI = 0.17 W.m -1 .K -1 ). If we identify G air and b air while exempting the sample of PI from the calibration, we observe that the determined values of b air (b air = 4 µm) are in the same order of those estimated elsewhere [START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. In fact, when λ s decreases the heat flux lines tend to spread since the heat is not attracted efficiently by the sample anymore and then the thermal exchange radius b air increases as illustrated in Figure 3.11. Therefore, it can be said that the choice of calibration samples has a significant effect on the identification of the thermal interaction parameters between the probe and the sample. Moreover, the difference between the two cases (h= 0 and h = 3000 W.m -2 .K -1 ) in the determination of the interactions parameters is at least about 20 % as can be noticed in Table 3.3. We could adapt through empirical expressions the best fits for the variations of b air and G air as a function of λ s (see Figure 3.12). In order to apply the variations of the parameters of the conduction through air to the experimental calibration, these expressions are inserted in the formula of ΔP/P c . The corresponding curve of ΔP/P c is shown in Figure 3.10 in dotted line.

Apparently, taking into account the variations of G air and b air as a function of λ s assures a better curve rather than having constant values among all the range of the thermal conductivity (see Figure 3.10). The difference between these curves is the variation of the parameters that are linked to the conduction through air.

In the next section, we investigate the heat transfer through air to the sample with the Wollaston probe through numerical simulations in order to confirm our experimental investigation and to monitor the variations of the interactions parameters between the probe and the sample. The aim of these simulations is to shed light on the interaction between the probe and materials of low thermal conductivity and to have a precise idea of the limitation of the thermal conductivity calibration using the Wollaston wire probe.

B. Numerical study of the heat transfer through air

Numerical simulations were done using COMSOL Multiphysics. The geometry of the model is shown in Figure 3.13. Because of its symmetry, half of the probe is modeled (half-toroid)

and the symmetry plane is insulated. The solid that represents the semi-infinite sample is of 500 µm in thickness, 500 µm in width and 1 mm in length. The air surrounding the probe is represented through a solid 2 where the thermal conductivity of the air is inserted. The probe supports are represented through a cylinder of silver of 75 µm in diameter and 100 µm in length. The mesh consists of 8000 elements. The probe is at a distance of 70 nm from the sample surface. The mesh was 8 times refined at a cubic volume of 4 × 10 6 µm 3 near the probe apex. The air and the sample surfaces are at initial temperature of 300 K. A condition of continuity is assumed at the boundary condition between the probe and its support. The boundaries of the solids 1 and 2 are maintained at 300 K. The heat flux is calculated the sample surface as shown while varying the thermal conductivity of the sample λ s .

We define Ф ref as the heat power calculated with respect to a sample of large thermal conductivity. The curves reach a limit for λ s between 20 and 50 W.m -1 .K -1 where Ф ref does not vary anymore. That limit is close to the limit of the experimental curves of ΔP/P c where ΔP/P c reaches a limit at a thermal conductivity between 20 and 30 W.m -1 .K -1 [START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF][START_REF] David | Développement de la microscopie thermique à sonde locale pour la détermination de la conductivité thermique de films minces[END_REF]. Moreover, the slope of the variations of Ф ref (λ s ) is at its max at the zone where λ s is smaller than 2 W.m - 1 .K -1 . That is identical to the behaviour of the interaction parameters of the conduction through air G air and b air determined in the last section (see Figure 3.12). 3.17 are larger than those shown in Figure 3.12 but have the same order of magnitude. The results are striking here since the model does not reflect properly the probe interaction with the surrounding medium. These simulations emphasize our suggested arguments presented in the experimental part. They show that the conduction through the air depends strongly on the thermal properties of the sample. In summary, the parameters of the thermal exchange through air G air and b air are dependent on the thermal properties of the sample. In the previous models for the Wollaston probe, these values were taken constant for a set of calibration samples. In the frame of calibration for thermal conductivity measurement, we have shown that the determination of these parameters is affected by the thermal conductivity of the calibration samples. The experimental identification of G air and b air was confirmed through numerical simulations of the probesample interaction. These simulations reveal a variation of the heat received at the sample surface when λ s varies. Moreover, b air was estimated by adopting a random criterion and its variation as a function of λ s is similar to what expected through the identification based on the experimental measurements.

During the measurements and the numerical simulations, the contact between the probe and the sample was supposed perfect. The arithmetic roughness R ar of our calibration samples does not exceed 6 nm as already listed in Table 3.2. In the following section, we identify the effect of roughness of samples on the heat transfer through air using sample of different levels of roughness.

Effect of roughness of the samples on the heat transfer to the sample

We considered two samples of silicon dioxide SiO 2 and Germanium Ge. The values of the roughness parameter R a of these samples are listed in Table 3.5.

The thermal signal ΔP/P c was measured for the different samples under ambient conditions and is shown in Figure 3.18. The index (i) denotes the samples SiO 2 or Ge with the smoothest surfaces. The roughness parameters of these two samples (SiO 2 (i) and Ge(i)) can be found in tables 3.1 and 3.2. ΔP/P c is almost the same for the SiO 2 samples (see Figure 3.18). However, ΔP/P c decreases significantly while increasing the roughness parameters of the sample of germanium. These variations are due to the increase of the resistance at the contact due to the roughness parameters. As the roughness parameters are in the same order of the radius of the groove at the probe apex (R a ≃ 350 nm as seen in Chapter 2), the additive resistance at the contact might be due to the air regions between the asperities.

The difference observed for the two materials is linked to the difference between the thermal conductances G s of the two samples. Under ambient conditions, G s is about 2 × 10 -5 W.K -1 for the sample SiO 2 whereas it is about 5 × 10 -4 W.K -1 for the sample of Ge (see Equation (1.21)).

Based on a modeling of two conductances in series (G s and G contact ), the signal ΔP/P c for the Ge samples is more sensitive to the variation of G c and thus to the sample roughness than the one measured for the SiO 2 samples.

These measurements can give an idea of the effect of the roughness in reducing the conduction at the tip-sample thermal contact, especially in the case of highly conducting samples. Therefore, it is highly recommended that both the calibration samples and the sample to be characterized have polished surfaces.

Approach for a more accurate thermal conductivity calibration

The determination of the thermal conductivity of materials requires the estimation of the parameters of probe-sample interaction through air in the calibration methodology that we propose here.

As seen in the previous sections, several significant errors can be introduced and have to be avoided to determine accurately the parameters of interaction G air and b air , therefore the calibration procedure should include:

-The accurate estimation of the probe conductance G p at the considered probe mean temperature at which the measurements are performed.

-The coefficient h of the heat losses to the environment.

-Polished samples where the experiments are performed.

-The variation of G air and b air as a function of λ s through empirical expressions of G air =f(λ s ) and b air =f(λ s ) similarly to the one obtained in Figure 3.12.

Zhang et al.

[125] determined the thermal conductivity of thick films of Bi 2 Te 3 and Bi 2 Se 3 with SThM using the Wollaston probe (λ Bi2Te3 =0.36 W.m -1 .K -1 and λ Bi2Se3 =0.52 W.m -1 .K -1 ).

The film thermal conductivities were identified using two calibration samples of thermal conductivities λ s =1 W.m -1 .K -1 and λ s =1.5 W.m -1 .K -1 . In their calibration, they assumed that the interaction parameters (G air and b air ) are constant for the thermal conductivity of samples that lies between 0.1 and 2 W.m -1 .K -1 [125]. This assumption could involve a significant error while characterizing their samples. In order to calculate the error that depends on the choice of the calibration samples, we take into a similar calibration two samples made of: polyimide (λ PI = 0.17 W.m -1 .K -1 ) and silicon dioxide SiO 2 (λ SiO2 = 1.2 W.m -1 .K -1 ). The thermal interaction parameters G air and b air were determined before in this chapter (G air = 5. As a conclusion, it can be said that an attention should be paid to the evaluation of the interaction parameters of the conduction through air. In the zone of sample thermal conductivity where the probe is highly sensitive, these parameters vary largely. Assuming these parameters constant induces a significant error on the experimental characterization of the samples while working with the Wollaston probe.

Determination of the thermal conductivity of Ba 8 Si 46 clathrates

This part is dedicated to characterize the thermal conductivity of samples of barium silicon Ba 8 Si 46 clathrates [126]. They are used in thermoelectric applications and are expected to have low thermal conductivity λ s [126]. That is convenient for our measurements as the more λ s decreases the more the technique is sensitive. We followed the procedure mentioned in section 3.6 for an accurate thermal conductivity calibration. The calibration samples are the same used before and listed in Table 3.3. The obtained thermal conductivity of the of Ba 8 Si 46 clathrates is about 1.7 ± 0.17 W.m -1 .K -1 [126]. That is in agreement with the expected thermal conductivity through modeling with this material.

Conclusion

In this chapter, our study of the thermal interaction between the probe and the sample included the quantification of the heat transferred to the sample through the solid-solid contact and through air. In order to estimate accurately the contribution of these two heat mechanisms, the effective probe conductance was first determined as a function of the probe mean temperature. The probe conductance was shown to depend on the probe temperature.

This is related to the boundary conditions taken through the modeling. A more realistic model should be developed with more appropriate boundary conditions in order to estimate accurately the probe conductance.

The heat conduction through the mechanical contact was estimated while performing measurements under high vacuum conditions. The measurements were performed using three polished samples of different thermal conductivity. The contribution of this heat mechanism appears to be much smaller than that of the heat conduction through air. Through these measurements, the thermal boundary resistance was determined for the interfaces PtRh/Si (1.14-1.4 × 10 -9 m 2 .K.W -1 ) and PtRh/Ge (2.8-3.35 × 10 -9 m 2 .K.W -1 ). The obtained results appeared to be in good agreement with the values of thermal boundary resistance announced in the scientific literature for contact between a metal and a semiconductor material.

For measurements under ambient conditions, the probe voltage was measured while rotating the Wollaston probe in order to identify a variation of the convective heat transfer coefficient.

The conduction appears to be the dominant mechanism in the heat losses of the probe to its surrounding environment. Later, the variation of the interaction parameters of the conduction through air was monitored as a function of the thermal conductivity of the sample by performing measurements under ambient conditions with reference samples. The variation of these parameters appears to be significant especially in the range of samples of low thermal conductivity. As shown in this chapter, neglecting this variation might introduce a large error during the thermal characterization of the samples of interest. Based on our experimental results in chapter 2 and this chapter, the thermal signal variations for the sample of Ge as a function of the probe temperature were sketched. The variations follow the same trend of the experimental curve despite a difference between the simulated values and the ones determined experimentally. A procedure was drawn in order to perform more accurate thermal conductivity calibrations. We followed this procedure to characterize the thermal conductivity of Ba 8 Si 46 clathrates for thermoelectric applications.

Chapter 4: DC Scanning Thermal Microscopy with probes of nanoscale radius of curvature

Chapter 4

DC Scanning Thermal Microscopy with probes of nanoscale radius of curvature

In this chapter, we investigate the probe-sample thermal system for probes with tip of nanoscale curvature radius. This chapter consists mainly of two parts that are dedicated to measurements performed with (1) KNT probes (Pd/Si 3 N 4 ) and ( 2) doped silicon Si probes respectively.

The first part concerns the investigation of the different heat mechanisms that operates between the KNT probe and the sample. We study the effect of reducing the apex radius on the probe-sample thermal interaction. Similar methodologies used with the Wollaston probe are used to quantify the heat conduction through water meniscus (pull-off measurements), at the solid-solid contact (vacuum conditions) and through air (measurements using reference samples). We also analyze the possibilities of the calibration with this probe for thermal conductivity measurement.

In the second part, we studied the thermal conduction through water meniscus with the doped silicon probe. We explain the methodology and approach used in the calibration of the apex temperature. The heat conductance through meniscus is determined as a function of the apex temperature.

The last part of this chapter is dedicated to explain the idea of identifying the phonon transmission coefficient at the tip-sample interface while working with SThM probes. Based on our measurements of the boundary thermal resistance on different samples, we explain the methodology used to determine this coefficient and give some perspectives for its estimation using the different probes considered in this work.

Thermal conduction through the water meniscus

The measurements of the pull-off forces with the Wollaston probe on a hydrophilic sample, already described in Chapter 2, showed the variation of the dimensions and the thermal conductance of water meniscus as a function of the probe temperature. The pull-off forces varied between 100 and 500 nN and the variation of the largest radius r 2 was estimated between 5 and 35 nm (see Chapter 2). The heat conduction through the meniscus appeared to be at least one order of magnitude lower than the heat conduction through air.

Searching for a better resolution of the pull-off forces and meniscus and in order to monitor the effect of nano-tip on the heat conduction through water meniscus, the KNT probe is used here. The expected radius of curvature of the tip apex is expected to be around 50 nm [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF].

The temperature coefficient β of the electrical resistivity of the probe was measured using a specific oven at different levels of temperature. The obtained value is about 8 ± 0.1 × 10 -4 K -1 .

This value is close to the value obtained by Puyoo (β = 6.9 × 10 -4 [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF]). It is used in the following to determine the probe mean temperature T m . . We have to note that this value includes the information about the temperature coefficients of the resistive Pd wire β Pd at the tip extremity and the two NiCr resistors β NiCr deposited on the cantilever. To estimate the values of β Pd and β NiCr , the probe was shortcut at the resistors using an adhesion layer of silver. β NiCr was then estimated around 1.65 ± 0.05 × 10 -4 , which is also close to the value obtained by Puyoo (β NiCr = 2.4 × 10 -4 [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF]). Assuming the probe resistive elements cited above at a same temperature T m when the probe is in the oven, the value β Pd can be estimated using the following equation:

eq NiCr NiCr Pd Pd R R R β β β = + (4.1)
The estimated value is around β Pd = 1.37 ± 0.02 × 10 -3 . This value is similar to the one determined by Puyoo [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF] (β Pd = 1.2 × 10 -3 ). However, little information exists yet about the real temperature field within the cantilever when a current heats the probe. Its determination constitutes a perspective of this work.

In this work, the probe mean temperature T m was controlled using a Wheatstone bridge with a feedback circuit. We used two hydrophilic samples of two different thermal conductivities:

polyimide PI (λ s =0.17 W.m -1 .K -1 ) and germanium Ge (λ s =58.6 W.m -1 .K -1 ). Under ambient conditions (T a =30, RH=40 %), force-distance curves were performed at different levels of probe mean temperature for the two samples and the pull-off forces F po were determined. The spring constant of the probe is around 0.35 N.m -1 [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. The measurements were performed up to a maximum T m around 140 °C. The maximum working temperature of this probe is around 150 °C [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. A thermal control unit was designed and used to control the probe temperature.

More details can found in appendix A. Figure 4.1 shows the variations of F po as function of T m for the two samples of PI and Ge. For the case of PI sample, F po decreases progressively up to a certain limit of T m around 110 °C beyond which they are almost constant. Unlike the sample of PI, the curves obtained for the Ge sample shows that the evaporation of the meniscus is not the same where large amount of meniscus remains even at high T m . The difference between the variations of F po =f(T m ) for both samples can be related to the difference of thermal conductivity λ s between samples and thus to the difference between their spreading resistances. Figure 4.2 shows a description of the probe-sample thermal system using a network of thermal resistances in series. R probe describes the thermal diffusive resistance of the probe. The ballistic part of this last one is included in the contact thermal resistance R contact following a similar description of the one proposed by Gotsmann et al. [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF].

As a reminder, Δ gives the temperature at the sample surface: The spreading thermal resistance R s is inversely proportional to λ s (see Chapter 1). For low values of λ s , R s dominates R contact whereas R contact is larger than R s in the case of large values of λ s . Therefore, the temperature at the sample surface is larger in the case of the polymeric sample. The numerical simulations through finite element analysis FEA performed by Tovee et al. [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF] confirm our analysis. While working under ambient conditions, the temperature near the apex is almost the same of the heater in the case of samples of λ s in the range of our sample of polyimide [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. In the case of samples of λ s in the range of the germanium sample , the temperature near the tip is closer to the ambient temperature [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. Through the variation of the pull-off forces (see Figure 4.1), we can notice that the measurement dispersion in the case of sample of PI is larger than those obtained in the case of the sample of Ge. That might be related to the difference of roughness of the two samples.

sample surface amb s apex amb s contact T T R T T R R - ∆ = = - + (4.2)
The roughness parameters of the PI and Ge samples are respectively: R ar = 5 nm; R p-p =57 nm and R ar = 0.7 nm; R p-p =4.75 nm. A similar effect was found in the measurements with the Wollaston wire (see chapter 2). The methodology used in chapter 2 to estimate the thermal conduction through water meniscus is used again here. As a reminder, the total thermal conductance of the water meniscus can be expressed as:

1 1 1 1 total probe meniscus meniscus meniscus sample G G G G - - = + + (4.3)
The thermal conductance through meniscus tends to a large value as the probe contacts the Equations (2.6) and (2.7). We suppose that the pull-off forces at high T m in the case of PI F po (T m,max , PI) are mainly due to the elastic forces. Therefore, F cap =f(T m ) for both samples are deduced as follows:

( ) ( ) , ( , Po )
cap m po m po m max F T F T F T = - (4.4)
The radius of apex R a estimated is around 25 nm (Equation 2.6). This is close to the values estimated for this probe (~ 50 nm [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]). We note that different force-distance curves were performed using different probes at ambient temperature on a sample of Ge. The capillary forces at T a varied between 30 and 90 nN. Based on our estimation methodology using Equation (2.6), the tip apex radius R a can vary from 20 nm to 70 nm for these probes. These data are close to the estimated values (~ 50 nm [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]). values are much smaller than the values of the heat conduction through air. Moreover, our study validates the approach of Shi and Majumdar [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF] in quantifying the thermal conduction to the sample through the water meniscus. In their study, the tip radius was around 45 nm and G total, meniscus was estimated around 6.7 nW.K -1 [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF]. This is in the range of our estimated values (see Figure 4.4). However, the temperature of the heated sample in the study of Shi and

Majumdar is missing and should have been specified [START_REF] Shi | Thermal transport mechanisms at nanoscale point contacts[END_REF].

Thermal conduction through the solid-solid contact:

Working under ambient conditions makes it difficult to distinguish between the thermal conduction through air and through the mechanical contact. In order to quantify the thermal conduction through the solid-solid contact and to study the heat transport at the tip-sample interface, it is preferable to work under vacuum conditions [START_REF] Majumdar | Scanning thermal microscopy[END_REF]. Under these conditions, we assumed that the heat transfer through radiation to the sample is basically small compared to the conduction at the mechanical contact [START_REF] Majumdar | Scanning thermal microscopy[END_REF]114]. In order to quantify the thermal conductance at the solid-solid contact G ss between the KNT probe and the sample, measurements at P=0.28 mbar were performed. At this pressure, we supposed that the heat conduction through air to the sample and environment are negligible based on SThM measurements of Chapuis [START_REF] Chapuis | Contribution à l'étude des transferts thermiques à l'échelle nanométrique: interaction pointe-surface[END_REF].

Since the temperature field within the probe is yet unknown, we have supposed in a first approach that only the extremity of the probe is heated. The supports of the resistive part are assumed to be heat sinks and the Pd/Si 3 N 4 part of the probe to be thermal fins [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF]. The total electrical resistance of the probe at T a is 365 Ω. Based on discussions with the fabrication group of this probe, the electrical resistance of the Pd wire at the probe extremity is around 150 Ω. The remaining electrical resistances are linked basically to the Ni/Cr resistors and Au pads.

At first, the probe thermal conductance G p is required to quantify G ss based on experimental measurements. Through the Wheatstone bridge with its feedback and the signal measured (see Appendix A), G p can be calculated using Equation (3.2). The temperature coefficient of the electrical resistivity of the Pd wire is β Pd = 1.37 × 10 -3 as discussed in the section 4.1. Figure 4.6 shows the variation of G p as a function of the probe mean temperature T m (°C). These variations are similar to those obtained with the Wollaston probe and related to the heat sink boundary conditions taken at the supports of the Pd wire as discussed in Chapter 3.

Two samples of germanium Ge and silicon Si were used in the measurements of G ss (R ar = 0.28 nm; R p-p =1.85 nm for Si and R ar = 0.7 nm; R p-p =4.75 nm). The measurements were performed at T m ≃ 67 °C (G p = 2.65 µW.K -1 as shown in Figure 4.6) and at P=0.28 mbar.

Thermal conduction through air

In order to quantify the parameters of the thermal conduction through air G air and b air , measurements of the thermal signal ΔP/P c as a function of the sample thermal conductivity λ s were performed under ambient conditions (T a =30 °C, RH= 40 %).

Since the thermal conductance of the probe G p changes as a function of T m (see Figure 4.6), the thermal signal ΔP/P c was measured as a function of T m for a sample of germanium. Puyoo for this probe [START_REF] Puyoo | Caractérisation thermique de nanofils de silicium pour des applications à la thermoélectricité[END_REF]. The mean values of G air and b air were extracted from ΔP/P c measurements for the samples of polyimide PI (0.17 W.m -1 .K -1 ) and silicon dioxide (

W.m -1 .K -1 ). Table 4 and G air at two probe mean temperatures are respectively micrometric and around 1 µW.K -1 . ). For the KNT probe, the values of the total conductance water meniscus G total, meniscus is also much smaller than G air (see Figure 4.5).

However, the value of the solid-solid conductance G ss , which are close to those of the conduction through air, cannot be neglected while describing the heat transfer between this probe and the sample.

A 3D model is needed to map the temperature field within the probe to determine more accurately the probe-sample interaction through the different heat transfer mechanisms.

Study of the probe-sample thermal system with doped silicon (DS) probes:

Thermal conduction through the water meniscus

The details about the geometry of the DS probes are discussed in chapter 1. The DS probe here used is the AN-300 type. For this probe, the difficulty in estimating the temperature of the tip apex is due to the positioning of the heater. In fact, the heater is not positioned at the tip extremity but above the tip. Therefore, the determination of the tip temperature includes the determination or measurement of the heater temperature and afterwards the modeling of the heat transfer through the tip (diffusive and ballistic -see chapter 1) in order to determine the apex temperature.

In this part, the conduction through the water meniscus between the DS probe and sample is studied on polymeric samples. In order to determine the apex temperature, we used three references samples of polymers of well-known melting temperatures as listed in Table 4.2

(Anasys Instruments). The calibration of the apex temperature is obtained from measurements of the cantilever deflection performed on these polymers as shown in Figure 4.9. This technique is called the Local Thermal Analysis (LTA). It consists on monitoring the cantilever deflection during a controlled fast heating while the tip remains in contact with the surface. The drop of the deflection signal is a sign of the melting of the polymer (see Figure 4.10). The corresponding heating voltage at the deflection drop corresponds then to the melting temperature of the polymer. The heating is made by the NanoThermal Analyzer (NanoTA2) system of Anasys Instruments. More information can be found in appendix A.

The uncertainty linked to the calibration of the tip apex with such polymers remains very small. This is due to the fact that the spreading resistance for polymers R sample is much larger than the thermal resistance of contact R contact [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF]. Following Equation (4.2), the temperature at the tip apex is almost the same at the polymer surface. Based on our calibration, a polymeric sample of hydrophilic polyimide PI (R ar = 5 nm; R p-p =57 nm) was used to study the conduction through water meniscus between the DS probe and the sample. We followed the same procedure used to study this heat mechanism for the Wollaston probe (see chapter 2) and the KNT probe (section 4.1 in this chapter). The spring constant of the AN-300 probe is around 0.35 N.m -1 [START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF]. reported for this type of probes (G ss ≃ 0.5 µW.K -1 [START_REF] Gotsmann | Nanoscale thermal and mechanical interactions studies using heatable probes[END_REF][START_REF] Hinz | High resolution vacuum scanning thermal microscopy of HfO2 and SiO2[END_REF]).

As a conclusion, whatever the probe is, the water meniscus does not play a dominant role in the probe-sample thermal interaction.

Determination of the transmission coefficient at interfaces from SThM measurements

SThM has been recently an experimental tool used to determine the phonon transmissivity t ij at interfaces [START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF]114]. The evaluation of the transmissivity requires the determination of the thermal contact resistance at the tip-sample interface [START_REF] Gotsmann | Quantized thermal transport across contacts of rough surfaces[END_REF]114]. As seen in this chapter and in Chapter 3, it is preferable that the experiments are performed under vacuum conditions where the conduction through air is eliminated.

When the two sides are insulators or semi-conductors, the thermal boundary resistance R b is linked to the phonon interaction and R b is noted as R bp [117]. This is the case of our contact between the KNT probe (Si 3 N 4 ) and samples of germanium and silicon. If at least one side is a metal, the electrons-phonons interaction should be then taken into account in R b . In this case, the phonons in the metal transmit their energy to the electrons [127] and R b can be assumed to be the sum of the phonon-electron interaction in the metal and phonon-phonon interaction between the two materials [117, 122]: R b = R bp + R be . R be is the thermal boundary resistance linked to the phonon-electron coupling. This is the case between our Wollaston probe and samples of MICA, germanium and silicon (see Chapter 3).

There are two models that have been used widely to evaluate t ij : the acoustic mismatch model In the case of contact between the KNT probe and nonmetallic samples (R b = R bp ), t 12 can be calculated directly from our measurements. C pa and ν pa can be calculated using the phonon dispersion relation for Ge and Si found respectively in Refs.

[128] and [129]. The coefficient t 12 was found to be of 0.5 ± 0.03 for KNT/Ge contact and 0.88 ± 0.02 for KNT/Si contact.

To evaluate t 12 in the case of the contact between the Wollaston and samples, we need to distinguish first between R bp and R be . R be can be expressed as follows [127]:

1 be ep pp R G λ = (4.9)
where G ep (W.m -3 .K -1 ) and λ pp are respectively the volumetric phonon-electron coupling constant and the phonon thermal conductivity in the Pt/Rh wire. The information about λ pp is in general difficult to find for metals where the heat conduction is dominated by electrons [117]. In our case, the evaluation of t 12 requires also the phonon dispersion of Pt/Rh.

The procedure followed in this study can be applied especially for the DS and KNT probes where the phonon dispersions of the probes materials are available. The experiments can be performed on different materials of acquired phonon dispersion in order to evaluate the phonon transmission at the tip-sample interface.

Conclusion

In this chapter, the probe-sample heat transfer was studied using two different probes with tips of nanoscale curvature radius.

As for the Wollaston probe, the meniscus conductance was determined as a function of the probe temperature for the KNT probe. The thermal conduction through water meniscus appears to be negligible in front of the heat conduction through air for this probe. Our results are in good agreement with the ones of previous approaches. They consolidate and validate them.

Experimental measurements were performed under vacuum conditions on two samples of germanium and silicon. Based on these measurements, the heat conductance through the solid-solid contact and the thermal boundary resistance was determined for both the samples (1.15 ± 0.1 × 10 -8 m 2 .K.W -1 for KNT/Ge and 1.75 ± 0.25 × 10 -8 m 2 .K.W -1 for KNT/Si). The results are in agreement with the values reported in scientific literature.

Under ambient conditions, tendencies of the thermal signal as a function of the probe mean temperature similar to those seen with the Wollaston probe were observed. The curves of the thermal conductivity calibration show that the sensitivity of the KNT probe is limited to a maximum value of λ s around a few W.m -1 .K -1 . From this curve, the thermal interaction parameters of the conduction through air were evaluated. It appears that the conduction through air is the dominant heat mechanism in the heat transfer between the probe and the sample but the solid-solid heat conduction is not negligible in the heat transfer contrary to the case of the Wollaston probe. Using the doped silicon probe, the heat conduction through water meniscus to a sample of polyimide was determined. The apex temperature was calibrated using different polymers of known melting temperatures. The heat conduction through the meniscus is also negligible in front of the values of the heat conductance through solid-solid contact reported in the literature.

The last part of the chapter was dedicated to the investigation of the phonon transmission at interfaces based on the evaluation of the boundary resistance through SThM measurements.

The DMM model can be used to determine the transmission coefficient at the tip-sample interface. The transmission coefficient was determined in the case of the contact between KNT probe and our nonmetallic samples. In the case of the Wollaston wire, the electronphonon should be taken into account as a part of the thermal boundary resistance to evaluate the phonon transmission coefficient. Another parameter that should be taken in the evaluation is the roughness of both the probe and the sample. exposing the originality of the proposed technique by giving its principle, a review of the methods that involve deposited wires on the thin film to be characterized is given. Then the work of modeling carried out to design the setup for a simple reference sample in the frame of a first step of validation of the proposed method is described. The setup, the measurements and results obtained for this reference sample are discussed. To finish the chapter, some results of numerical simulations are used to estimate the potentiality of the method to determine the in-plane and cross-plane components of the thermal conductivity of anisotropic samples.

Main thermal techniques using deposited resistive heater and probe

The AC methods are the most used nowadays among the deposited wires techniques. The reason is that DC techniques give measurements with large equilibrium time and are more dependent on the thermal resistances at the substrate boundaries [130,131]. The first technique of this type for measuring the thermal properties of bulk materials was developed by Cahill [START_REF] Cahill | Thermal conductivity measurement from 30 to 750 K: the 3ω method[END_REF] and called the "3ω method". In practice, a long metallic wire of finite width In the material, the conductive diffusion of the AC heating produces a continuous temperature variation called θ DC and alternating temperature oscillations at angular frequency 2ω called

θ 2ω .
As reminder, the third harmonic voltage of the wire V 3ω can be linked to the properties of the material through the expression of the harmonic temperature θ 2ω , as:

0 2 3 cos(3 t ) 2 ac R I V ω ω β θ ω φ = + (5.1)
where R 0 is the electrical resistance of the wire at ambient temperature and β is the temperature coefficient of the electrical resistivity of the wire. where λ s , d 0 , and C are the sample thermal conductivity, density, and heat capacity respectively.

In the case of a homogeneous and isotropic bulk material that can be described as a semiinfinite substrate and for µ>>b w , the alternating temperature θ 2ω at the heater can be expressed as: is the power per unit heater length [START_REF] Cahill | Thermal conductivity measurement from 30 to 750 K: the 3ω method[END_REF]. In order to perform measurements at different levels of temperatures and to eliminate the effects of radiation, the samples were placed in specific cryostats and the measurements performed from 30 to 750 K [START_REF] Cahill | Thermal conductivity measurement from 30 to 750 K: the 3ω method[END_REF].

Later, Cahill et al. [133] extended the measured 3ω method to measure the cross-plane thermal conductivity f λ ⊥ of thin films with a thickness of a few tens of nanometers. The heater width w=2b w is much larger than the film thickness d f and the film is deposited on a substrate of high thermal conductivity. Then, the film can be described as a 1D thermal resistance. It can be shown that this thermal resistance, d f / f λ ⊥ , adds an additional second harmonic temperature to expression (5): Here, the thermal conductivity of the substrate is obtained from the slope of the curve ΔT 2ω = f(ω) and the film thermal conductivity is calculated from the temperature drop through the film (Equation (5.5)). Let us note that the obtained thermal conductivity of the film is an average along the frequency interval. In this approach, the series of resistances must be known between the heater and the substrate. Kim et al. [134] noticed a variation of the estimated value of the substrate thermal conductivity that was linked to the thermal resistance between the heater and the film.

In this method, controlling the current frequency and thus the penetration depth allows us to eliminate the thermal resistance at the lower boundary of the substrate. The method was known later as the "slope method".

Later, Borca-Tasciuc et al.

[135] developed the differential method to determine the thermal conductivity components of anisotropic substrates and very thin films.

In the case of anisotropic semi-infinite substrate:

where λ xy =λ x /λ y , x and y refer to the in-plane and cross-plane directions respectively.

The case of an anisotropic film deposited on a substrate was also developed to study an eventual lateral spreading in the film. The measurement is performed in two steps:

-First, with the heater directly deposited on the substrate (S) -Second, with the heater deposited on the film-substrate (f+S) system.

The thermal conductivity of the film can be calculated from the temperature rise [135]: For the measurements of the in-plane thermal conductivity of thin films, Zhang and

Grigoropoulos [136] were the first to develop the phase-shift method for the deposited wires.

The method, which had been widely used before but with optical heating, consists of a heater deposited on the suspended thin film of interest as shown in Figure 5.2. In such configuration, the effects of the interface between the thin film and the substrate are avoided [136]. The suspended film is settled on two substrates that act as two thermal sinks. An alternating current is passed through the heater and a sensor at distance X from the heater is used to detect the attenuation of the thermal waves. Assuming 1D lateral heat conduction, ΔT 2ω at a distance X from the sensor is [136]: -In the first step, with the suspended metallic heaters alone, -In the second step, with the suspended metallic heaters covered with the film of interest.

We have to note that the configuration of suspended structures require a high level of precision during the micro-fabrication and for some types of films it is inconvenient or impossible [141].

Another method to determine the in-plane thermal conductivity of thin films is to use different heaters of different widths and each heater is subjected to harmonic heating. Ju et al. [142] used this technique to determine the thermal conductivity of anisotropic films of polyimide.

Here, the alignment of molecular chains to the in-plane direction is the main source of anisotropy in polyimide films [142]. The cross-plane thermal conductivity can be determined in the case of a heater width that is much larger than the film thickness. The in-plane The measurements were performed at temperatures up to 300 K in a specific cryostat. As can be seen from the techniques listed above, we are able nowadays to determine the anisotropic thermal properties of thin films. One limiting factor while observing all the techniques is the heavy conditioning needed to perform measurements at different temperature levels.

Moreover, the number of measurement at temperatures above the ambient is much smaller in comparison to measurements at low temperatures. All of the studies need a specific cryostat either to eliminate effect of air convection (or conduction) or to go down in temperatures or oven to go up in temperatures. Moreover, measurements in the alternating regime require a small current to avoid an additional heating of the substrate.

In this chapter, we describe a method and a technique inspired from the studies above to determine the anisotropic thermal properties thin films above ambient temperature. This technique allows us to avoid the need for a special conditioning for our sample of interest. We start by a 1D model that allows us to determine the in-plane properties of the film. Later, we give an analytical model that allows us to determine also the cross-plane properties of the same film. The factors that affect our studies in terms of loss through convection and radiation and thermal contact resistance(s) are also investigated.

Proposed method

General principle

Our sample consists of a deposited film on a substrate as shown in A direct current i 0 of small magnitude is generated through the probing wires for the ΔR 2ω (x)

measurements. Using a lock-in amplifier, the second harmonic voltage at the probing wires can be measured:

ΔV 2ω (x) = ΔR 2ω (x) i 0 = R 0 (x).β.ΔT 2ω (x).i 0 (5.16) We have to note that in these measurements careful attention must be paid to the nature of the substrate. If the substrate is electrically conductive, then an insulating layer should be deposited between the metallic heater and the substrate in order to avoid an eventual current leakage through the substrate.

Associated thermal modeling in the alternating regime

Assuming a heater of width w in the X direction and infinite length in the Y direction, a twodimensional thermal waves propagation regime may be considered in the sample.

The sample is heated on its surface by a heat flow density Ф d modulated at the angular pulsation 2ω and homogeneous on the width w of the heating wire. The indexes 1 and 2 denote the film and the substrate respectively.

The axes x,z denote respectively the in-plane and cross-plane directions.

The heat flow density received by the sample across the surface (z=0) is:

( , z) rect ( ) ( ) ( , ) ( ) w g f x x z h T x z z φ d d = + (5.17)
where rect w denotes the rectangle function, h g is the global linearized convective and radiative sample-surrounding exchange coefficient. In the Fourier space, Equation (5.17) is as follows:

( ,z) sin ( ) ( , 0) 2 2 2 d x x g x w w f k c k h T k z φ π = + =   (5.18)
where k x is the x-component of the wave vector.

Due to the finite thickness of the sample, the temperature field in the sample may be expanded in the following plane waves spectrum:

( ) ( ) 2 1 , , 2 x ik x x x T x z T k z e dk ω π +∞ -∞ = ∫  (5.19)
The temperature field in the film can be expressed as:

( ) ( )

2 1 1 1 1 1 , , ( ) 2 2
x x ik x ik x i z i z x x x

T x z T k z e dk T e T e e dk

+ + = =   is the incident mode, 1 1 ( , 0) x T T k z - - = =  
is the reflected mode and α 1 is the film thermal diffusivity.

The boundary condition at the sample surface is given as:

( ) ( ) 1 1 , , 0 x x z T k z f k z z λ ∂ = = - ∂  (5.21)
Following Equations (5.18) and (5.21):

1 1 T A BT + - = -   (5.22)
where ( )

1 1 2 2 d x g z w wsinc k A h i φ π χ λ   -     = + (5.23) and ( ) ( ) 1 1 1 1 g z g z h i B h i χ λ χ λ - = + (5.24)
The boundary condition at Z=d (see Figure 5.4) is given as (conservation of heat flux):

( ) ( )

1 2 1 2 , , x x z z T k z T k z z z λ λ ∂ ∂ - = - ∂ ∂ (5.25)
and therefore:

1 1 1 2 1 1 1 1 1 1 2 2 2 2 2 2 i d i d i d i d z z z z i T e i T e i T e i T e χ χ χ χ χ λ χ λ χ λ χ λ - - + - + - - + = - +     (5.26)
Based on Equation (5.22):

1 2 1 2 2 2 2 2 2 1 1 1 1 1 1 i d i d i d z z z z T e T e A e T C χ χ χ χ λ χ λ χ λ χ λ - + - - - + + =    (5.27) and 1 2 2 1 1 1 1 1 1 2 2 2 2 2 2 i d i d z z z i d z A e i T C i T e T i e χ χ χ χ λ χ λ χ λ χ λ - - - + - + - = -    (5.28)
The discontinuity of the temperature between the film and the substrate at Z=d implies that: ( )

2 2 1 2 1 1 2 2 , ( ,Z ) ( , ) x i d i d i d i d th z x x T k z R T k d T k Z d T e T e T e T e z χ χ χ χ λ + -- + -- - ∂ - = = - = = + + ∂        (5.29)
with R th (K.W -1 ) is the thermal resistance at the contact between the film and the substrate. At the front surface of the sample, the average of the temperature on the width of the probing wires w for a distance x from the source (heating wire) can therefore be calculated: 5.38) and the temperature at the back-surface is:

( ) ( ) /2 /2 2 2 1 1 /2 /2 1 1 1 , 0 , 0 ( ) 2 i i x i i x w x w ik x i i x x w x w T x z T x z dx T T e dk w w ω ω π + + +∞ + - - - -∞ ∆ = = == + ∫ ∫ ∫   (
( ) ( ) 2
+ + +∞ - + - - - -∞ ∆ = = = = = + = ∫ ∫ ∫  
(5.39)

The sample temperature at any instant in the stationary regime is the sum of the two components T DC and ΔT 2ω .When an alternating current passes across the heater, this latter is expected to act as a low-pass filter. The technique consists of adjusting the frequency of the current in order to get a condition where T DC >> ΔT 2ω . Satisfying this condition allows us to determine the thermal properties of the considered materials through the variation of ΔT 2ω at the temperature T DC .

The temperature T DC is controlled by the amplitude of the alternating current generated within the heater, and the thermal properties of the sample at this temperature level is characterized through the model already discussed above.

In the next part, we adjust the analytical model for a simple reference material in order to determine the thermal properties of the sample at different levels of T DC .

Establishment of the method for a simple reference material

The reference material

Because of its well-known thermal properties versus the temperature, we used a sample of isotropic pure silicon as reference material. As the thermal expansion coefficient does not exceed 1 % at temperature below 600 K [146],

we considered it constant as a function of temperature in our work. Temperature (K) C (J.Kg -1 .K) [116] λ (W.m -1 .K -1 ) [147] α ref (×10 -5 m².s -1 ) a α exp (×10 -5 m²/s) [148] For the test and the method demonstration, a specific sample made of silicon was prepared. Its thickness, width and length were 200 μm, 7 mm and 10 mm respectively.

Design of the resistive wire array

Using COMSOL Multiphysics, numerical simulations of the thermal behavior of the Si sample were performed to determine the wires length for which the analytical 2D model is applicable and to design the electrical chip. The design includes the determination of the distances between the heating wire and the probing wires. 3D COMSOL model was first made. As the heating wire is located at the middle on the sample surface, quarter of the system was modeled. A volumetric term of the power P v is inserted in the section of the heating wire. For the coefficient of heat losses to the environment, we assumed h= 5 W.m -2 .K -1 , the emissivity ϵ equal to 1 at the sample and wires surfaces and a negligible thermal contact resistance between the wires and the sample. Let us remember that ΔT 2ω has no effect on the global sample temperature as our current frequency is much larger than the cut-off frequency of the hot wire. Figure 5.5 shows the top view of the isotherm distributions at the sample surface when a electrical current amplitude I ac =60 mA heats the sample. First, it can be said that the sample is approximately isothermal as the temperature gradient between the heater and the sample extremity does not exceed 3 K in this case. This gradient usually lies in the range of uncertainty of the thermal properties determination (see Table 5.1). Moreover, Figure 5.6 shows the variation of ΔT 2ω for different lengths as a function of the distance from the heating wire. We can notice that there is no difference for the different representative curves. That confirms that the probing wires temperatures are uniform along their length (y direction) and justifies the large length of the heater as in this configuration the heat flows perpendicularly to the probe wires. Based on these simulations, the wire array design was fabricated as will be seen in the following part. Table 5.2 gives the value of the wires section S, electrical resistivity ρ and the coefficient β.

Using Atomic Force Microscopy (AFM) in tapping mode, S was measured. Figure 5.9 shows an example of the section measurements. As noticed, the wire real section obtained after the process fabrication is uniform but not square as expected. The wires length L is of several millimeters and was measured with optical microscopy. The length of the heating wire is The mask design is shown in Figure 5.10. Actually, the probe wires (in yellow) are connected to the pads through connection wires of large section. Due to this large section, these connection wires have negligible electrical resistance compared it to that of the probe wires and have no effect on the measurements.

C) Comparison between 2D and 3D COMSOL models

Before getting to the 2D analytical model already described in section 3, 2D COMSOL model was built to compare the difference in terms of the ΔT 2ω and T DC with respect to the 3D model as illustrated in Figure 5.11. Because of its symmetry, half of the system was established. In order to be able to compare between the 2D and 3D simulations, the same boundaries conditions were maintained. Figure 5.12 presents a comparison between the variations of ΔT 2ω for the 2D and 3D models as a function of the distance from the heating wire for the same boundary conditions and I ac =60 mA. We found that whatever the boundaries conditions and the current amplitude are 2 In our technique, the temperature T DC , at which the sample thermal properties are determined through measured ΔT 2ω (x), is measured experimentally through the probing wires:

T DCexp =(1/N)Ʃ i=1 (R(x i )-R 0 (x i ))/(R 0 (x i ).β)+T a (5.40)
where N is the number of the probing wires and i is an index for a considered probing wire.

Effect of the resistance of contact between the heater and the wire

The interface thermal resistance R th between the wires and the substrate is given by: R th = R Au/Cr + R Cr + R Cr/Si (5.41) where measured R th at Al/Cu interface equal to 0.25 × 10 -9 m 2 .K.W -1 using thermo-reflectance.

Using frequency-dependent photothermal radiometry (PTR), the thermal resistance between the wires pads (supports) and the sample was measured [154]. The identified value of R th is about 3 ± 0.5 × 10 -7 m 2 .K.W -1 . We assumed that R th is the same value for the contact between the heating wire and the sample. R th was inserted in the numerical simulations in order to investigate the difference in the variation of T DC and ΔT 2ω . The variation of T DC does not exceed 2 K and is less than the uncertainty of identifying the thermal properties of the sample 5.1). Figure 5.15 presents the variation of ΔT 2ω for two cases: R th = 3 × 10 -7 m 2 .K.W -1 and for a perfect contact (R th =0). As can be observed, the difference between the values ΔT 2ω is small (less than 1%).

Way for the simplification of the thermal model: 1D approach

After pointing out the difference between the 2D and 3D COMSOL configurations, we make here a comparison between the 2D numerical simulations, the analytical model and the simplified 1D approach model (Equation 5.11). We determine the range of frequencies of the heating current and the distances between the wires where the 1D approach is valid.

For our silicon Si isotropic sample, several simplifications are applied to the analytical model already described in the section 5.

2: (1) R th =0, (2) d=0, 1 2 T T =   and λ x = λ z (α x =α z ).
As a reminder, the 1D approach model is given as: In the case of the 1D approach, the in-plane thermal diffusivity of the sample α can be determined from the exponential tendency of the amplitude ΔT 2ω and the linear tendency of the phase lag ϕ (º). Moreover, the in-plane thermal conductivity λ s is determined through the term A 1 (see Equation 5.11). For |x min | = 225 µm, the 1D approach is valid and the uncertainty of the thermal properties determination of is less than 4 %. Considering the identified frequency range, the sample thickness d tends to be smaller than μ when f decreases. A critical frequency f cr1 that should not be exceeded to maintain μ>d (f<f cr1 ). Working with frequencies larger than f cr1 induces a non-uniform sample temperature along Z and so the 1D model cannot be applicable. Another condition of the frequency is the limitation of the half-width of the sample so that 5μ<L/2 (f>f cr2 ). For example, for a sample temperature of 500 K, 15 Hz <f< 160 Hz. At this range of frequencies, the current frequency is much larger than the low cut-off frequency of the system that is estimated about 5 mHz. Let us note that the simulations presented in this part of this section concern the case of perfect contact between the wires and the substrate (R th =0 m 2 .K.W -1 ). The thermal properties of silicon (k, d and C) and their dependence on temperature were used for the calculations.

4 4 2 1 ( ) 2 2 2 X X X X i i i i s d s s s P T X A e

e e e dw

Based on the determined criterion, the 1D approach was used to determine the thermal properties of the silicon sample. We have to note that when changing the substrate material, some adjustments have to be applied in terms of the sample dimensions, range of frequency of the heating current and distances between the probing wires so that the 1D approach can be valid.

Experimental Set-up

The experimental set-up is illustrated in Figure 5.18. In this set-up, the AC generator ( 1) is connected to the current amplifier (2) to generate an alternating current I ac through the heating wire. This wire in red is deposited on the mask (3) and mounted in series with a resistance of 1 KΩ to measure the heating current (4). The resistance is linked to an ampere-meter (5) for an accurate measurement of I ac . A DC current generator (6) produces i 0 of small magnitude through the probing wires in yellow. Synchronized with the AC generator, the lock-in amplifier [START_REF] Gomès | Scanning thermal microscopy : A review[END_REF] detects the signal V 2ω and the phase lag Φ at the probing wires. In this work, these resistances are supposed negligible comparing them to those of the wires.

Moreover, using a multimeter, it was verified that the electrical resistances of the connection between the PCB 1 and the pins of the PCB 2 are of few mΩ. Taking all of this into account, it can be said that the electrical resistances measured at the PCB 1 plugs correspond to the wires of the electrical chip.

Experimental results

The measurements were performed under ambient conditions (T a =30°C, RH≃40%).

Experiments were performed for frequencies between 20 The low thermal conductivity of the sample λ z taken here is equal to 2 W.m -1 .K -1 . In the x-direction, the thermal properties are the same of Silicon Si. Numerical simulations were performed using COMSOL Multiphysics and the 2D analytical model presented in section 3.

The variation of the temperature T DC is around 5 K from 10 µm to 750 µm as distances from the heating wire as shown in

b. λ z >> λ x

In this case, λ x is taken equal to 2 W.m -1 .K -1 and the thermal properties are the same as the silicon in the z-direction. As shown in Figure 5.22, the sample is not isothermal.

Cross-plane thermal properties

The principle of the measurement of the thermal properties in the cross-plane direction is illustrated in Figure 5.4.

A probing wire might be inserted at the back surface of the substrate. The variation of ΔT 2ω can be detected as a function of the current frequency in order to determine the thermal properties of the substrate at the temperature T DC , which depends on the value of I ac .

Conclusions and Perspectives

This chapter reports the first steps of the development of a technique of characterization of the thermal properties of anisotropic solid thin films at temperatures larger than the ambient temperature.

The technique principle is based on the adaptation of the membrane method principle used in modulated regime. The modulated heating induces two temperatures in the sample: constant and alternating temperatures. The frequency of the current that should be used is larger enough than the cut-off frequency so that the alternating temperature is much smaller than the constant temperature. An array of resistive wires of different lengths has to be deposited on the front and back surfaces of the sample of interest. The longest wire heats the sample with an alternating current while the other wires detect the variation of the alternating temperature.

The measured alternating temperature helps identifying the thermal properties of the sample The identification, of the in-plane component of the thermal conductivity for an anisotropic sample is discussed. The 2D analytical model fits exactly the numerical simulations when the parameter λ xz is much larger than 1 and some uncertainties are observed when λ xz is much smaller than 1.

Moreover, the analytical model is able to describe the variation of the alternating temperature in the z-direction in order to identify the cross-plane thermal properties of the sample where the curve of 2D numerical simulations fit the curve of the analytical model.

Seeking the thermal characterization of complex materials such as multilayered materials deposited on a silicon substrate, the perspectives of this work are based on its technical adaptation for such applications. The future works include the depositing of resistive wire on both surfaces of thin films in order to experimentally determine their anisotropic thermal properties.

General conclusions

This manuscript was dedicated to the description and specification of the measurement with two thermal metrologies for the characterization of thermal conductivity of materials: the Scanning Thermal Microscopy (SThM) and the 2ω method.

Based on a global state of the art of SThM in chapter 1, we raised the questions regarding the probe-sample thermal system in order to better understanding the SThM measurement and to achieve more accurate thermal characterization of materials. The heat losses to the environment and the different heat transfer mechanisms that exist between the probe and the sample were detailed. The resistive probes used in the experiments mentioned in this manuscript were described and the tasks to be dealt with in our study were defined.

A methodology of work was initiated to study the heat transfer through water meniscus in SThM in chapter 2. This methodology was used with the Wollaston wire probe (chapter 2)

and with the KNT and doped silicon (DS) probes (chapter 4). The capillary forces were determined based on measurements of pull-off forces on hydrophilic samples as a function of the probe temperature. The dimensions of the meniscus were derived from these forces values by using literature modeling. For the three probes, the total thermal conductance of the meniscus was determined. The results have shown that this heat transfer mechanism is not predominant in the thermal interaction between the probe and the sample in SThM.

Furthermore, this contribution depends strongly on the probe temperature where the meniscus stability has been observed. This lies in the interest of different applications such as controlled dip-pen nanolithography. The meniscus almost disappears at high probe temperatures

showing that thermally-controlled system can be an alternative solution to reduce stiction at the nanoscale. On the other hand, the first results of the investigation of SThM probe-sample thermal interaction through water meniscus in an environmental scanning electron microscope (ESEM) were presented and discussed. The differences observed between the literature ESEM experimental studies and the theory (Kelvin Equation) of the meniscus dimensions were clarified. The effect of the electron beam should be taken into account while working in ESEM.

The quantification of the thermal conduction at the solid-solid contact and through air requires the comparison between SThM measurements and modeling results. The thermal conductance of the probe was shown to be a crucial modeling parameter to be determined for a more efficient approach of the measurements. In fact, the effective probe thermal conductance changes as a function of the probe temperature and cannot be assumed constant depending on the boundary conditions taken at the heater supports.

Working under vacuum conditions, the thermal conduction at the solid-solid contact was studied for the Wollaston and KNT probes. In the case of the Wollaston, the experimental results have shown that this heat transfer mechanism was previously overestimated in the probe-sample thermal interaction. The thermal conduction at the mechanical contact is small compared to heat conduction through air. However, for the case of the KNT probe, the results have shown that the conductance through solid-solid contact should be taken into account while modeling the probe-sample thermal interaction. For both these probes, the boundary resistance was determined for different polished samples and the obtained values are in good agreement with literature values. For the Wollaston probe, the thermal resistance of the phonon-electron coupling in the Pt/Rh wire is needed to estimate the transmission coefficient at the tip-sample interfaces. This coefficient can be determined directly in the case of the KNT probe (0.55 ± 0.03 for KNT/Ge and 0.88 ± 0.02 for KNT/Si).

Under ambient conditions and using different reference samples, the probe thermal signal was measured as a function of the sample thermal conductivity for the Wollaston and KNT probes.

SThM with the Wollaston probe then appears sensitive exclusively to the sample thermal conductivity λ s on a range limited to low λ s up to around 10 W.m -1 .K -1 . With the KNT probe, the sensitivity is lost for λ s larger than few W.m -1 .K -1 . The modeling parameters of the thermal conduction through air were determined using reference samples in the ranges of sensitivity of these probes. The results have pointed out that these parameters depend on the sample thermal conductivity. In the case of the Wollaston probe, numerical simulations were performed and have confirmed our analysis. Measurements on different rough samples have shown that the roughness of the sample surface increases the thermal resistance at the probe-sample contact.

It is a crucial factor and should be taken into account during the SThM probe calibration and the thermal characterization of materials. A methodology of work was set and applied to identify the thermal conductivity of Ba 8 Si 46 clathrates with the Wollaston probe. 3D modeling should be developed in order to provide a better description of the probe-sample system and more specifically of the thermal interaction through air while taking into account the real shape of the probe.

The summary of the thermal parameters of interaction of the different heat transfer mechanisms obtained using the Wollaston and KNT probe is given in the table below. We consider for each mechanism its thermal conductance G and thermal contact radius b. connector that links the unit to the measuring head. In the case of SThM probes, the blue wire links the probe to its thermal control unit (Wheatstone bridge or NanoTA2). To perform measurements under vacuum conditions (low vacuum operation (10 -3 torr)), a protective hood is mounted on the base unit with the help of a metallic frame as shown in

A.2) Thermal control unit

The thermal control units developed at CETHIL to control the temperature of the Wollaston probe and KNT probe are based on balanced Wheastone bridge circuits. La première technique est la microscopie thermique à sonde locale (SThM). La pointe d'un microscope à force atomique intègre un élément résistif. Utilisée en mode contact, cette pointe, chauffée par effet joule, permet l'excitation thermique localisée de l'échantillon. La détermination des propriétés thermiques de l'échantillon nécessite l'analyse de la réponse de cette pointe avec un modèle du système sonde-échantillon et de son environnement. Un état de l'art général des études réalisées en SThM permet de poser les questions scientifiques actuellement traitées dans le domaine ainsi que les modèles utilisés pour analyser le système pointe-échantillon. Une attention particulière est accordée à l'interaction thermique sondeéchantillon. Sa compréhension nécessite d'être approfondie. L'étude ici présentée tient compte des propriétés thermiques, de la rugosité et de la mouillabilité de la surface de différents échantillons. Une nouvelle méthodologie est établie pour la spécification du transfert de chaleur échangée par conduction thermique au travers du ménisque de l'eau formé au contact sonde-échantillon. Cette méthodologie est basée sur l'analyse de la dépendance à la température de la sonde des courbes de force-distance obtenues à l'air ambiant. Elle est appliquée à trois sondes de taille, forme et constitution différentes: la sonde Wollaston, la sonde KNT et une sonde en silicium dopé. Quels que soient la sonde et l'échantillon, la contribution du ménisque d'eau à l'interaction est montrée être inférieure à celle de l'air. Par ailleurs, pour les sondes Wollaston et KNT, les paramètres d'une modélisation, basée sur une description du système sonde-échantillon avec un réseau de conductances thermiques, sont identifiés à partir de mesures effectuées à l'ambiante et sous vide. La conductance thermique au contact solide-solide est déterminée pour différents échantillons. Cela a permis d'identifier le coefficient de transmission de phonons dans le cas de la sonde KNT et des échantillons non-métalliques. Nos résultats expérimentaux et de simulations numériques pour les mesures effectuées sous air démontrent que les paramètres utilisés pour décrire la conduction thermique via l'air dépendent fortement de la conductivité thermique de l'échantillon. Aussi des améliorations de la méthodologie actuelle d'étalonnage pour la mesure de la conductivité thermique sont proposées. Il est en outre montré que la variation de la rugosité de l'échantillon lors de l'étalonnage et de mesures doit être prise en compte. La sensibilité à la conductivité thermique pour les sondes Wollaston et KNT est part ailleurs montrée fortement réduite pour les matériaux de conductivité thermique supérieure à 10 et quelques W.m -

  AbbreviationAFM

2. 3 3 . 1 3 . 2 3 . 3 3 . 4 3 . 5 3 . 6 3 . 7 3 . 8 CHAPTER 4 : 4 . 1 4 . 2

 3313233343536373844142 Preliminary experiments performed in the frame of the development of a SThM-ESEM combined system ........................................................................................................ 2.4 Conclusion ....................................................................................................................... CHAPTER 3: PROBE-SAMPLE THERMAL SYSTEM IN DC SCANNING THERMAL MICROSCOPY INVESTIGATED WITH THE WOLLASTON PROBE . Influence of the probe temperature T m on the thermal signal ΔP/P c and on G p ....... Influence of the probe mean temperature T m on G p .................................................... Thermal conductance at the solid-solid contact G ss .................................................... Measurements under ambient conditions .................................................................. 3.4.1 Determination of the coefficient h of the heat losses to the environment .............. 3.4.2 Heat conduction through air to the sample ............................................................. 3.4.3 Effect of roughness of the samples on the heat transfer to the sample .................. Variations of ΔP/P c as a function of T m ....................................................................... Approach for a more accurate thermal conductivity calibration ............................ Determination of the thermal conductivity of Ba 8 Si 46 clathrates ............................. Conclusion ..................................................................................................................... DC SCANNING THERMAL MICROSCOPY WITH PROBES OF NANOSCALE RADIUS OF CURVATURE ..................................................................... Study of the probe-sample thermal with the KNT probe ......................................... 4.1.1 Thermal conduction through the water meniscus................................................... 4.1.2 Thermal conduction through the solid-solid contact: ............................................. 4.1.3 Thermal conduction through air ............................................................................. Study of the probe-sample thermal system with doped silicon (DS) probes: ......... 4.2.1 Thermal conduction through the water meniscus...................................................

1 .Figure 1 . 3 .Figure 1 . 6 .Figure 1 . 7 .Figure 1 . 8 .Figure 1 . 9 .

 11316171819 Figure 1.1. Description of the principle of the technique ......................................................... Figure 1.2. a) Description of the Wollaston probe b) SEM (Scanning Electron Microscope) image of the Wollaston probe. .................................................................................................. Figure 1.3. Description of the KNT probe (courtesy of Kelvin NanoTechnology). ................ Figure 1.4. SEM images of the palladium probe. ..................................................................... Figure 1.5. SEM images of the Anasys Instruments probes (Anasys instruments). ............... Figure 1.6. Schematic of the description of the Wollaston probe [22]. ...................................Figure 1.7. The heat balance on a element dx of the Pt90/Rd10 wire [22]. ............................. Figure 1.8. Description of the network of the thermal conductances of the probe-sample system. ...................................................................................................................................... Figure 1.9. The description of the KNT probe [28]. ................................................................ Figure 1.10. Modeling the tip-sample system as given by Gotsmann et al. [12]. .................... Figure 1.11. Modeling the scattering at the interface through the phonon reflection coefficient r 12 [12]. ..................................................................................................................................... Figure 1.12. Heat flow lines and isotherms during the conduction from a localized heat source [43]. .......................................................................................................................................... Figure 1.13. Schematic of the different heat mechanism that operate between the probe and the sample [13]. ........................................................................................................................ Figure 1.14. Variation the heater temperature of a silicon probe during the approach and retract curves [45] (Frequency of 1 Hz -P = 10 -5 mbar). ........................................................ Figure 1.15. The variation of the deflection and V 3ω probe voltage as a function of the probe/sample distance. The probe used here is the palladium probe and the sample is made of Silicon (P=10 -5 mbar and I probe = 308 µA [28]). ........................................................................ Figure 1.16. Variation of the thermal signal ΔP/P c as a function of the probe/sample distance between a Wollaston wire probe and a monocrystalline silicon sample at different levels of pressure[57]. ............................................................................................................................. Figure 1.17. Variation of the equivalent conductance G eq as a function of the force signal ΔI for measurements performed on a sample of Hafnium under ambient conditions [22]. .......... Figure 1.18. Approach and retraction curves between a flat-punch tip and taC sample a) Variation of the heater temperature b) Evaluation of the thermal resistance at the tip-sample

  Figure 1.1. Description of the principle of the technique ......................................................... Figure 1.2. a) Description of the Wollaston probe b) SEM (Scanning Electron Microscope) image of the Wollaston probe. .................................................................................................. Figure 1.3. Description of the KNT probe (courtesy of Kelvin NanoTechnology). ................ Figure 1.4. SEM images of the palladium probe. ..................................................................... Figure 1.5. SEM images of the Anasys Instruments probes (Anasys instruments). ............... Figure 1.6. Schematic of the description of the Wollaston probe [22]. ................................... Figure 1.7. The heat balance on a element dx of the Pt90/Rd10 wire [22]. ............................. Figure 1.8. Description of the network of the thermal conductances of the probe-sample system. ...................................................................................................................................... Figure 1.9. The description of the KNT probe [28]. ................................................................ Figure 1.10. Modeling the tip-sample system as given by Gotsmann et al. [12]. .................... Figure 1.11. Modeling the scattering at the interface through the phonon reflection coefficient r 12 [12]. ..................................................................................................................................... Figure 1.12. Heat flow lines and isotherms during the conduction from a localized heat source [43]. .......................................................................................................................................... Figure 1.13. Schematic of the different heat mechanism that operate between the probe and the sample [13]. ........................................................................................................................ Figure 1.14. Variation the heater temperature of a silicon probe during the approach and retract curves [45] (Frequency of 1 Hz -P = 10 -5 mbar). ........................................................ Figure 1.15. The variation of the deflection and V 3ω probe voltage as a function of the probe/sample distance. The probe used here is the palladium probe and the sample is made of Silicon (P=10 -5 mbar and I probe = 308 µA [28]). ........................................................................ Figure 1.16. Variation of the thermal signal ΔP/P c as a function of the probe/sample distance between a Wollaston wire probe and a monocrystalline silicon sample at different levels of pressure[57]. ............................................................................................................................. Figure 1.17. Variation of the equivalent conductance G eq as a function of the force signal ΔI for measurements performed on a sample of Hafnium under ambient conditions [22]. .......... Figure 1.18. Approach and retraction curves between a flat-punch tip and taC sample a) Variation of the heater temperature b) Evaluation of the thermal resistance at the tip-sample contact (P ≃ 6 × 10 -6 mbar) [58]. .............................................................................................. Figure 1.19. Schematic of the meniscus when the probe is in contact with the sample. ......... Figure 1.20. A model of the thermal resistances between the substrate and the tip in the experiment of Luo [59]. ........................................................................................................... Figure 1.21. Variation of the ratio G probe /G contact as a function of the heater temperature T heater in the experiment of Thiery et al. [61]. .................................................................................... Figure 1.22. Variation of the equivalent resistance R eq =1/G eq as a function of the distance between the Wollaston probe and a Silver sample [22]. ..........................................................Figure 1.23. Differential tension (V) as a function of the separation distance between the probe and the sample. The red line and the green line represent respectively the diffusive model and the complete regime [27]. .......................................................................................Figure 1.24. Variation of the probe 3ω voltage V 3ω as a function of the pressure when the probe in contact and out of contact with a Yttrium oxide sample [27]. ................................... Figure 1.25. (a) Differential tension as a function of the probe/sample distance for a sample of Niobium (λ= 59 W.m -1 .K -1 ). A zoom at the contact zone is shown in Figure (b) [27]. ............Figure 1.26. Sensitivity of the palladium probe for materials of various thermal conductivities of samples [16]. ........................................................................................................................

Figure 4 . 1 .Figure 4 . 2 .Figure 4 . 3 .Figure 4 . 6 .Figure 4 . 7 .Figure 4 . 8 .Figure 4 . 9 .Figure 4 .

 414243464748494 Figure 3.6. Estimation of the thermal conductance at the solid-solid contact G ss for the three materials (MICA, Ge and Si). ................................................................................................ 100 Figure 3.7. Thermal boundary resistance R b (m 2 .K.W -1 ) identified for the three materials (MICA, Ge and Si). ................................................................................................................ 100 Figure 3.8. Schematic of the probe in two different positions. .............................................. 102 Figure 3.9. Voltage of the probe in different conditions for different tilts of the probe. ....... 102 Figure 3.10. ΔP/P c variation as a function of the thermal conductivity λ s (T a =30°C, RH=40%). ................................................................................................................................................ 105 Figure 3.11. Schematic of the heat flux lines for two samples of different thermal conductivity λ s . ............................................................................................................................................ 105 Figure 3.12. Variation of b air and G air as a function of λ s and their corresponding fitting curves (a 1 = 1.05E-04, b 1 =-2.32, c 1 =3.42E-06, d 1 =-0.04492; a 2 =7.77E-06, b 2 =-5.18E-04 , c 2 =-5.47E-06, d 2 =-1.197). ............................................................................................................. 106 Figure 3.13. The model of the probe being at proximity of the sample. ................................ 108 Figure 3.14. Calculated relative heat power transferred to the sample as a function of its thermal conductivity λ s . .......................................................................................................... 109 Figure 3.15. Variation of the temperature on the sample surface in the direction of the major axis X. ..................................................................................................................................... 110 Figure 3.16. Variation of the temperature on the sample surface in the direction of the minor axis Y. ..................................................................................................................................... 110 Figure 3.17. Variation of the thermal interaction radii b x , b y and b' air as a function of the thermal conductivity λ s . .......................................................................................................... 111 Figure 3.18. Variation of the thermal signal ΔP/P c for the different samples of different roughness parameters (T a = 25 ⁰C, RH= 40 %). ..................................................................... 113 Figure 3.19. Variation of the modeled ΔP/P c as function of the probe mean temperature T m for the sample of germanium Ge. ................................................................................................ 114 Figure 4.1. Variation of the pull-off forces as a function of the probe mean temperature for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares). ............ 122 Figure 4.2. Network of thermal resistance in series as a description of the probe-sample thermal system. ....................................................................................................................... 122 Figure 4.3. Schematic description of the variation of R sample and Δ as a function of λ s . ......... 123 Figure 4.4. Variation of the larger radius r 2 of the meniscus as function of the probe mean temperature T m for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares). ........................................................................................................................ 124 Figure 4.5. Variation of the total thermal conductance of water meniscus G total, meniscus as function of the probe mean temperature T m for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares). ............................................................................... 125 Figure 4.6. Thermal conductance of the probe as a function of the probe mean temperature T m (P=0.28 mbar, T a =30 °C). ...................................................................................................... 127 Figure 4.7. Thermal signal ΔP/P c for the two samples of Ge and Si (P = 0.28 mbar, T a = 30 °C). ......................................................................................................................................... 127 Figure 4.8. Thermal signal ΔP/P c as function of the sample thermal conductivity λ s (T a =30 °C, RH= 40 %). ............................................................................................................................. 128 Figure 4.9. Thermal signal ΔP/P c as function of the sample thermal conductivity λ s for T m = 40 °C (blue circles) and T m = 65 °C (red circles) (T a =30 °C, RH= 40 %). .................................. 129 Figure 4.10. Monitoring the probe cantilever deflection as a function of the probe heating voltage of the three reference polymers. ................................................................................ 131 Figure 4.11. Heating Voltage (V) as a function of the apex temperature T ap (°C). ............... 131 Figure 4.12. Pull-off forces F po as a function of the apex temperature T ap for a contact between the DS probe (AN-300) and Po sample (T a =30 °C, RH= 40 %). ............................. 133

Figure 5 . 24 .

 524 Figure 5.24. Simulated Variation of the phase lag ϕ as a function of the distance from the heating wire for the numerical simulations and the 2D analytical in the case of anisotropic sample (λ x >> λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ). ........................................... Figure 5.25. Simulated variation of the (a) phase lag (b) ΔT 2ω as a function of the distance from the heating wire for the numerical simulations and the analytical model in the case of anisotropic sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ). ........................ Figure 5.26. Simulated variation of the (a) phase lag (b) ΔT 2ω as a function of the distance from the heating wire for the numerical simulations and the analytical model in the case of anisotropic sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ). ........................ Figure 5.27. Variation of the phase lag ϕ as a function of the current frequency for the numerical simulations and the analytical model for the probing wire located at the back surface of the sample (λ x << λ z , I ac = 60 mA, f = 40 Hz and ϵ=1). .......................................... Figure 5.28. Variation of ΔT 2ω as a function of the current frequency for the numerical simulations and the analytical model for the probing wire located at the back surface of the sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1). .....................................................................

Figure 1 . 1 .

 11 Figure 1.1. Description of the principle of the technique

Figure 1 . 2 .

 12 Figure 1.2. a) Description of the Wollaston probe b) SEM (Scanning Electron Microscope) image of the Wollaston probe.

Figure 1 . 3 .

 13 Figure 1.3. Description of the KNT probe (courtesy of Kelvin NanoTechnology).

Figure 1 . 4 .

 14 Figure 1.4. SEM images of the palladium probe.

Figure 1 . 5 .

 15 Figure 1.5. SEM images of the Anasys Instruments probes (Anasys instruments).

Figure 1 . 6 .

 16 Figure 1.6. Schematic of the description of the Wollaston probe [22].

Figure 1 . 7 .

 17 Figure 1.7. The heat balance on a element dx of the Pt90/Rd10 wire [22].

Chapter 1 -SThM: State of the art and exposition of the issues 39 A

 139 description of the series of thermal conductances of the probe-sample is presented in Figure1.8. R s = 1/G s is the spreading resistance in the sample where section 1.4 is dedicated to present the modeling of this resistance in its different configurations. The heat mechanisms that operate between the probe and the sample are featured in G c . Depending on the description of the heat source and on the sample (bulk, layered samples …), different expressions of G s can be used. We will present more in detail these two thermal conductances later in this chapter.

Figure 1 . 8 .

 18 Figure 1.8. Description of the network of the thermal conductances of the probe-sample system.

Figure 1 . 9 .

 19 Figure 1.9. The description of the KNT probe [28].

Chapter 1 -

 1 SThM: State of the art and exposition of the issues

Figure 1 . 10 .

 110 Figure 1.10. Modeling the tip-sample system as given by Gotsmann et al. [12].

Figure 1 . 11 .Chapter 1 -

 1111 Figure 1.11. Modeling the scattering at the interface through the phonon reflection coefficient r 12 [12].

TChapter 1 -

 1 is the temperature of a sink. Describing the area of the thermal contact between the tip and the sample as a circular one of radius b one seems the most used. b denotes the equivalent radius of the global exchange taking into account all the mechanisms of the heat transfer.If we consider the source as an isothermal circular area, thus R s is given by[START_REF] Yovanovich | Thermal spreading and contact resistances[END_REF]: SThM: State of the art and exposition of the issues[START_REF] Narayanaswamy | Near-field radiative heat transfer between a sphere and a substrate[END_REF]

Figure 1 . 12 .

 112 Figure 1.12. Heat flow lines and isotherms during the conduction from a localized heat source [43].

Figure 1 . 13 .

 113 Figure 1.13. Schematic of the different heat mechanism that operate between the probe and the sample [13].

Figure 1 . 14 .Chapter 1 -

 1141 Figure 1.14. Variation the heater temperature of a silicon probe during the approach and retract curves [45] (Frequency of 1 Hz -P = 10 -5 mbar).

Chapter 1 -

 1 SThM: State of the art and exposition of the issues 51

Figure 1 . 15 .Chapter 1 -

 1151 Figure 1.15. The variation of the deflection and V 3ω probe voltage as a function of the probe/sample distance. The probe used here is the palladium probe and the sample is made of Silicon (P=10 -5 mbar and I probe = 308 µA [28]).

Figure 1 . 16 .

 116 Figure 1.16. Variation of the thermal signal ΔP/P c as a function of the probe/sample distance between a Wollaston wire probe and a monocrystalline silicon sample at different levels of pressure[57].

Figure 1 . 17 .

 117 Figure 1.17. Variation of the equivalent conductance G eq as a function of the force signal ΔI for measurements performed on a sample of Hafnium under ambient conditions [22].

Chapter 1 -

 1 SThM: State of the art and exposition of the issues 55

Figure 1 . 18 .

 118 Figure 1.18. Approach and retraction curves between a flat-punch tip and taC sample a) Variation of the heater temperature b) Evaluation of the thermal resistance at the tip-sample contact (P ≃ 6 × 10 -6 mbar) [58].

Figure 1 . 19 .

 119 Figure 1.19. Schematic of the meniscus when the probe is in contact with the sample.

Figure 1 .

 1 Figure 1.20. A model of the thermal resistances between the substrate and the tip in the experiment of Luo [59].

Figure 1 . 21 .

 121 Figure 1.21. Variation of the ratio G probe /G contact as a function of the heater temperature T heater in the experiment of Thiery et al. [61].At a mean probe temperature of 60 °C (temperature in the DC mode T DC ), Lefèvre et al.[START_REF] Lefèvre | 3ω-scanning thermal microscope[END_REF] used the Wollaston wire probe in the 3ω mode (Equations (1.11) and (1.14)) to estimate the thermal conductance at the contact G c and the thermal interaction radius b on various samples.They found an equivalent G c about (4.7-6) × 10 -6 W.K -1 and b about 100-300 nm. As we will see later in the next section, G c is on the same of the values for the heat transfer through air but b is one order of magnitude smaller. The values of G c and b are equivalent for a range of frequency between 0 and 10 4 rad.s -1 . At low frequencies, θ 2ω ≃ θ DC and thus the mean temperature of the probe is about 60 °C. At frequencies larger than the cut-off frequency of the probe, T 2ω becomes much smaller than T DC and thus the probe temperature is around 30 °C.

Chapter 1 -

 1 SThM: State of the art and exposition of the issues 59 totally the water meniscus or the measurements must be performed under ultra-vacuum conditions with appropriate cleaning methods.

Chapter 1 -

 1 SThM: State of the art and exposition of the issues 60

Figure 1 . 22 .

 122 Figure 1.22. Variation of the equivalent resistance R eq =1/G eq as a function of the distance between the Wollaston probe and a Silver sample [22].

Figure 1 . 23 .

 123 Figure 1.23. Differential tension (V) as a function of the separation distance between the probe and the sample. The red line and the green line represent respectively the diffusive model and the complete regime [27].

Figure 1 . 24 .

 124 Figure 1.24. Variation of the probe 3ω voltage V 3ω as a function of the pressure when the probe in contact and out of contact with a Yttrium oxide sample [27].

Figure 1 .

 1 Figure 1.25. (a) Differential tension as a function of the probe/sample distance for a sample of Niobium (λ= 59 W.m -1 .K -1 ). A zoom at the contact zone is shown in Figure (b) [27].

Chapter 1 -

 1 SThM: State of the art and exposition of the issues 64

Figure 1 . 26 .

 126 Figure 1.26. Sensitivity of the palladium probe for materials of various thermal conductivities of samples [16].

Figure 1 . 27 .

 127 Figure 1.27. Variation of the probe electrical resistance when the tip contacts the sample as a function of the sample thermal conductivities. The simulations accounts for three conductivities of the probe materials [66].

1 Chapter 2 -

 2 Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM 70 formed between the tip and the substrate as shown in Figure2.1[START_REF] Piner | Dip-pen" nanolithography[END_REF]. Therefore, the writing rate depends on the water meniscus volume and the relative humidity[START_REF] Piner | Dip-pen" nanolithography[END_REF].

Figure 2 . 1 .

 21 Figure 2.1. Schematic of the principle of Dip-pen nanolithography (DPN) [76].

Figure 2 . 2 .

 22 Figure 2.2. Evolution of the water film thickness as a function of the relative humidity for three types of materials: chromium Cr, copper Cu and gold Au. The measurement were performed using a scanning Kelvin probe force microscope (SKPFM) [87].

Figure 2

 2 Figure 2.3.

Figure 2 . 3 .

 23 Figure 2.3. Schematic of a capillary confined in a longitudinal tube [95].

  3). As a simplification, the angle θ c is usually taken equal to 0 while talking about hydrophilic surfaces. Equation (2.1) represents the well-know Laplace equation. At a stable state of the capillary, the liquid and vapor have the same chemical potential

1 Chapter 2 -

 2 Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM 74 affected by temperature, measurements of the pull-off forces at different levels of T m would reveal a possible eventual change in the interaction forces between the probe and sample. We used a sample of hydrophilic germanium Ge. Figure2.4 shows the contact between a water droplet deposited on the surface of the Ge sample. The sample surface is hydrophilic as the contact angle between the droplet and the sample surface is around 27 ± 3.

Figure 2 . 4 .

 24 Figure 2.4. Image of the contact between the water droplet and the sample of Ge.

  .5.a). The pull-off forces can be determined by multiplying the distance φ c times the spring constant of the probe. The spring constant of the probe is estimated to 5 N.m -1 with an uncertainty around 20 %[START_REF]Specification Scanning Thermal Microscope Option[END_REF].

Figure 2 . 5 .

 25 Figure 2.5. Spectroscopy curves performed on Germanium (Ge) for different probe mean temperature: (a) Ambient temperature, (b) T m = 130 °C and (c) T m = 230 °C.

Figure 2 .

 2 6 presents the variation of the pull-off curves as a function of the contact time obtained by Sirghi [100].

Figure 2 . 6 .

 26 Figure 2.6. Dependence of the meniscus pull-off force on the tip-sample contact time [100].

Figure 2 .

 2 Figure 2.7 shows the variation of the pull-off forces measured as a function of T m . As can be seen, the value F po of the pull-off forces decreases when increasing the temperature until a certain limit of T m between 120 ºC and 150 ºC beyond which F po becomes almost constant. At

Figure 2 . 7 .

 27 Figure 2.7. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of polished germanium.

  [102]. The determined values lie between 100 and 180 MW. m -2 K -1 [102]. We consider an average value H = h p-w = h w-s = 140 MW.m -2 .K -1 in this study. To estimate the conductances G p-w and G w-s , the value of the largest radius associated to the meniscus r 2 is needed as shown in Figure2.8. r 2 can be estimated as a function of T m while taking into account the Kelvin Equation (see Equation2.5) and the geometrical dimensions of the meniscus between the probe apex and the sample.

Figure 2 . 8 .Figure 2 . 8 )

 2828 Figure 2.8. Schematic of the meniscus when the probe is in contact with the sample.

FromFigure 2 . 9 .

 29 Figure 2.9. Scanning Electron Microscopy SEM image of a Wollaston wire probe apex.

1

 1

Figure 2 . 10 .

 210 Figure 2.10. Schematic of a sphere and a cylinder with their radii of curvature

Figure 2 . 11 .

 211 Figure 2.11. Radius of the curvature of the meniscus r 2 as a function of the Wollaston probe mean temperature T m for the Ge sample (T a = 30 ⁰C, RH= 40 %).

Figure 2 . 12 .

 212 Figure 2.12. Total thermal conductance of the meniscus G total, meniscus as a function of the Wollaston probe mean temperature T m for the Ge sample (T a = 30 ⁰C, RH= 40 %).

Figure 2 . 13 .

 213 Figure 2.13. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of rough germanium.

Figure 2 . 14 .

 214 Figure 2.14. Schematic illustration of the meniscus of a contact between the probe and a surface asperity [94].

  Figure 2.15 presents the variation of pull-off forces F po as a function of the probe mean temperature T m .

Figure 2 . 15 .

 215 Figure 2.15. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of graphite.

  [110]. That is probably our case as the mean values F po in Figure18is close to the values of F po at large values of T m in the case of the polished Ge (see Figure 2.7) where the meniscus largely disappeared.

Figure 2 . 16 .

 216 Figure 2.16. Pull-off force as a function of RH measured between a hydrophobic tip and a flat silicon sample [110].

Figure 2 . 17 .

 217 Figure 2.17. ESEM image of the water meniscus formed at the contact between Wollaston wire and germanium sample (T substrate ≃ 2 °C, RH ≃ 90 %).

Figure 2 .

 2 [START_REF] Binnig | Ultrahigh-density atomic force microscopy data storage with erase capability[END_REF] shows an example of different images collected at different places on the substrate sample at T substrate ≃ 2 °C. The electron beam was focused around 30 minutes on the zone that appears on the left. The liquid film appears obviously to be larger in the zone where the electron beam was not focused. It can be said in this case that the electron beam factor is important and its effect on observing the dimensions of the liquid film or the meniscus is significant. In our experiment, the electron beam charges and reduces the water film observed on the substrate surface. We illustrate the contact between the probe and sample in an ESEM chamber as shown in Figure2.20. Figure2.20.a corresponds to the case where there is no effect of the electron beam and the small radius of the meniscus and the water film thickness are respectively r 1 and e. On the other hand, Figure2.20.b corresponds to the one where the electron beam is focused on the probe-sample contact and reduces the water layers as shown before. In this case, r' 1 and e' correspond respectively to the radius of the meniscus and the water film thickness. Since e' is smaller than e as shown before, we can imagine than r' 1 is larger than r 1 . That might be the reason between the difference observed while predicting the equivalent radius r k ≃r 1 through the Kelvin Equation and observing that radius in an ESEM chamber as seen in refs[START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF] 112, 113]. The future experiments in the SThM-ESEM combined system should take the effect of the e-beam while analyzing the experiments in ESEM.

Figure 2 . 18 .

 218 Figure 2.18. Meniscus height (nm) as a function of the relative humidity for a SiN cantilever tip in contact with a silicon substrate at a temperature of 5 °C .

Figure 2 . 19 .

 219 Figure 2.19. ESEM image of a liquid film on a Ge substrate showing two different zones of the liquid film (T substrate ≃ 2 °C, RH between 95 and 100 %).

Figure 2 . 20 .

 220 Figure 2.20. Schematic diagram of the cases (a) where the electron beam is not focused (b) and when it is focused in an ESEM chamber on the probe-sample contact.

Chapter 3 - 93 3. 1

 3931 Probe-sample thermal system in DC SThM investigated with the Wollaston probe Influence of the probe temperature T m on the thermal signal ΔP/P c and on G p Studying the heat transfer between the probe and the sample through the meniscus as a function of the probe mean temperature T m raised the question about an eventual effect of the variation of T m on the heat transfer between the probe and the sample. Since the Wollaston probe is used in this chapter, our experiments were dedicated to the measurements of the probe Joule power relative difference of the probe ΔP/P c (Equations (1.6) and (1.7) in Chapter 1). A sample of germanium Ge and a Wheatstone bridge to control T m were used. Figure 3.1 shows the variation of ΔP/P c as a function of T m . While increasing T m , ΔP/P c decreases but the slope of the curve ΔP/P c =f(T m ) decreases as a function of T m .

Figure 3 . 1 .

 31 Figure 3.1. Variations of the probe Joule power relative difference as a function of the probe mean temperature when the probe contacts a sample of germanium Ge (T a = 30 °C and RH= 40 %).

  2 give the variation of G p estimated from measurements with Equation (3.2) as a function of T m . The values of G p increase with the increase of T m .

Chapter 3 - 1 )/or 2 )

 312 Probe-sample thermal system in DC SThM investigated with the Wollaston probe[START_REF] Van Honschoten | Capillarity at the nanoscale[END_REF] The hypothesis of a probe support at ambient temperature in (1) is too simplifying, andThe thermal conductivity of platinum λ Pt/Rh changes with the variation of T m . The Wiedemann-Franz law gives the empirical relationship between the thermal conductivity of a metal λ (Pt/Rh in our case) and the electrical resistivity ρ at a temperature T as follows: L wf = 2.44 × 10 -8 W.Ω.K -2 .

Figure 3 . 2 .

 32 Figure 3.2. Variations of the probe conductance G p as a function of the probe mean temperature T m .

( 1 )

 1 the probe supports as heat sinks and (2) the values of G p =f(T m ), determined from Equation (3.2) and experimental data as shown in Figure 3.2, to estimate the thermal conductances through the solid-solid contact G ss and through air G air .

Figure 3 . 3 .

 33 Figure 3.3. Variation of λ Pt/Rd as a function of T m as determined by Raphael [42].

Chapter 3 -

 3 Probe-sample thermal system in DC SThM investigated with the Wollaston probe 97 the calibration of David [57] induced a better fit of the measurements of the Joule power relative difference ΔP/P c as a function of λ s as shown in Figure 3.4.However, G ss and G air are correlated in these measurements since they are performed under ambient conditions and the estimation of G ss is based on macroscopic analytical expressions.

Figure 3 . 4 .

 34 Figure 3.4. ΔP/P c Calibration curves obtained by David [56] under ambient conditions (a c = 7.5 nm, G air =4.5 × 10 -6 W.K -1 ).

Figure 3 .

 3 Figure 3.7 shows the values of R b identified through our experiments. We can notice that the values of R b are in the range of 10 -9 m 2 .K.W -1 , in agreement with the order of magnitude of the values mentioned in the literature [12, 33]. The values identified for R b by Stoner et al. through picosecond optical technique were about 5 × 10 -9 m 2 .K.W -1 for Al/Al 2 O 3 interfaces [118]. The values of R b identified by Stevens et al. [119] by the transient thermoreflectance technique between Pt/Si are in the range of 7 × 10 -9 m 2 .K.W -1 . Using the same technique, Komarov et al. [120] identified values for R b for Au/Si (metal-semiconductor) interfaces ranging from 5.5 to 7 × 10 -9 m 2 .K.W -1 . These values are in agreement with Molecular Dynamics (MD) simulations performed for the case of Au/Si interfaces by da Cruz et al. (R b = 6.3 × 10 -9 m 2 .K.W -1 [121]). These values are of the same order of magnitude of our results for the case of the MICA sample (PtRd/MICA interface) but larger than the identified values for the cases of PtRh/Si and PtRh/Ge interfaces as shown in Figure 3.7. Recently, Yang et al. [122] determined the values of R b for the case of metal-semiconductor (Al/Si) interfaces through non-equilibrium molecular dynamics (NEMD) method. The electron-phonon coupling was taken into account to evaluate R b [122]. The identified values of R b were between 2.1 and 2.7 × 10 -9 m 2 .K.W -1). These values are closer to our cases of PtRh/Si and PtRh/Ge interfaces where R b varies between 1.14 and 1.4 × 10 -9 m 2 .K.W -1 for PtRh/Si interfaces and between 2.8 and 3.35 × 10 -9 m 2 .K.W -1 for PtRh/Ge interfaces.

Figure 3 . 5 .

 35 Figure 3.5. Variation of the thermal signal for the three materials (MICA, Ge and Si) under vacuum conditions (T m = 140°C, RH=40 %, T a =30 °C).

Figure 3 . 6 .

 36 Figure 3.6. Estimation of the thermal conductance at the solid-solid contact G ss for the three materials (MICA, Ge and Si).

Figure 3 . 7 .

 37 Figure 3.7. Thermal boundary resistance R b (m 2 .K.W -1 ) identified for the three materials (MICA, Ge and Si).

Figure 3 . 8 .

 38 Figure 3.8. Schematic of the probe in two different positions.

Figure 3 . 9 .

 39 Figure 3.9. Voltage of the probe in different conditions for different tilts of the probe.

Figure 3 . 10 .

 310 Figure 3.10. ΔP/P c variation as a function of the thermal conductivity λ s (T a =30°C, RH=40%).

Figure 3 . 11 .

 311 Figure 3.11. Schematic of the heat flux lines for two samples of different thermal conductivity λ s .

Figure 3 . 12 .

 312 Figure 3.12. Variation of b air and G air as a function of λ s and their corresponding fitting curves (a 1 = 1.05E-04, b 1 =-2.32, c 1 =3.42E-06, d 1 =-0.04492; a 2 =7.77E-06, b 2 =-5.18E-04 , c 2 =-5.47E-06, d 2 =-1.197).

Figure 3 .

 3 14 presents the variations Ф ref transferred to the sample as a function of λ s . Increasing the geometrical dimensions of solid 1 and solid 2 by a factor of 2 induces a maximum deviation of Ф ref by 4%. The heat power is calculated with respect to a sample of infinite thermal conductivity and for two initial temperatures of the toroidal tip: 400 K and 500 K. The variations of Ф ref are not affected by the temperature. This shows that the heat transfer through air does not vary with the variation of the probe temperature (Figure 3.14).

Figure 3 . 13 .

 313 Figure 3.13. The model of the probe being at proximity of the sample.

Figure 3 . 14 .Figures 3 .

 3143 Figure 3.14. Calculated relative heat power transferred to the sample as a function of its thermal conductivity λ s .Based on the variations of Ф ref , it was important to determine a criterion that allows us to evaluate the variation b air =f(λ s ) in these numerical simulations. Let us remind that b air by its definition is the radius of an equivalent discoidal area on the sample surface. That area is at a uniform sample temperature called T s . The Wollaston probe is not symmetric in the two axes X and Y. In this case, b' air can be taken as the equivalent radius among the radii b x and b y[START_REF] Lefèvre | Modélisation et élaboration des métrologies de microscopie thermique à sonde locale résistive[END_REF]:

Figure 3 . 15 .

 315 Figure 3.15. Variation of the temperature on the sample surface in the direction of the major axis X.

Figure 3 . 16 .

 316 Figure 3.16. Variation of the temperature on the sample surface in the direction of the minor axis Y.

Figure 3 . 17 .

 317 Figure 3.17. Variation of the thermal interaction radii b x , b y and b' air as a function of the thermal conductivity λ s .

Figure 3 . 18 . 3 . 5

 31835 Figure 3.18. Variation of the thermal signal ΔP/P c for the different samples of different roughness parameters (T a = 25 ⁰C, RH= 40 %).

Figure 3 . 19 .

 319 Figure 3.19. Variation of the modeled ΔP/P c as function of the probe mean temperature T m for the sample of germanium Ge.

  5 W.K -1 and b air = 24.5 µm). If we suppose that the thermal conductivities of the samples of Bi 2 Te 3 and Bi 2 Se 3 are the same of what was determined [125], then we can calculate their approximate Joule power relative difference : ΔP/P c (Bi 2 Te 3 )= 0.147 and ΔP/P c (Bi 2 Se 3 )= 0.152. On the other hand, we take the SiO 2 and ZrO 2 samples into the calibration as reference samples. We already identified mean values of G air and b air between these samples (G air = 7.19 W.K -1 and b air =4.35 µm). The thermal conductivity of these two calibration samples has almost the same range of those samples taken by Zhang et al. [125]. Therefore, based on the ΔP/P c (Bi 2 Te 3 )and ΔP/P c (Bi 2 Se 3 ) values that were determined in the last step, we can estimate the "new" thermal conductivity of the Bi 2 Te 3 and Bi 2 Se 3 samples. In this parallel calibration, the thermal conductivities of these samples: λ Bi2Te3 = 0.854 and λ Bi2Se3 = 0.97 W.m -1 .K -1 . These two values are very different from the values obtained in the mentioned study[125]. It can be said then that the determination of the thermal conductivity of the materials λ s depends on the thermal conductivity of the calibration samples. Therefore, it is better that the range of the thermal conductivity of the calibration samples covers the thermal conductivity of the material to be characterized.

Chapter 4 -DC SThM with probes of nanoscale radius of curvature 120 4 . 1

 441 Study of the probe-sample thermal with the KNT probe 4.1.

Chapter 4 -

 4 DC SThM with probes of nanoscale radius of curvature 121

Figure 4 .

 4 Figure 4.3 presents a schematic description of the variation of R sample and Δ as a function of λ s .

Figure 4 . 1 .

 41 Figure 4.1. Variation of the pull-off forces as a function of the probe mean temperature for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares).

Figure 4 . 2 .Chapter 4 -Figure 4 . 3 .

 42443 Figure 4.2. Network of thermal resistance in series as a description of the probe-sample thermal system.

  sample (see Equation(1.30)). The determination of G probe-meniscus and G meniscus-sample requires the determination of the meniscus dimensions. The small radius of curvature r 1 is determined through the Kelvin Equation (see Equation (2.5)). Based on what was stated above (see Equation4.2), the temperature of the meniscus is supposed equal to T m in the case of Po and to (T m + T a )/2 in the case of Ge in this Equation. The apex radius and the meniscus largest radius r 2 =f(T m ) are respectively deduced from the equations of the capillary forces F cap :

Chapter 4 -Figure 4 .

 44 Figure 4.4 shows the variations of the meniscus radius r 2 as a function T m for the two samples of PI and Ge. It can be said that the possible resolution achieved for these variations with the KNT probe is on the nm range.

Figure 4 . 4 .

 44 Figure 4.4. Variation of the larger radius r 2 of the meniscus as function of the probe mean temperature T m for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares).

Figure 4 .

 4 Figure 4.5 shows the total thermal conductance of the water meniscus G total, meniscus as a function of T m for the two samples of Po and Ge. As it will be seen later in this chapter, these

Figure 4 . 5 .

 45 Figure 4.5. Variation of the total thermal conductance of water meniscus G total, meniscus as function of the probe mean temperature T m for the contact between KNT probe and samples of PI (red circles) and Ge (blue squares).

Figure 4 .K 1

 41 Figure 4.7 shows the measurements of the thermal signal ΔP/P c for the two samples. The model of ΔP/P c is given in chapter 3 as a function of the three conductances: G p, G ss and the sample conductance G s given as:

Figure 4 . 6 .

 46 Figure 4.6. Thermal conductance of the probe as a function of the probe mean temperature T m (P=0.28 mbar, T a =30 °C).

Figure 4 . 7 .

 47 Figure 4.7. Thermal signal ΔP/P c for the two samples of Ge and Si (P = 0.28 mbar, T a = 30 °C).

Figure 4 .

 4 7 shows the variation of ΔP/P c as a function of T m . The tendency shown by the curve ΔP/P c =f(T m ) is similar to the one obtained with the Wollaston wire as discussed before and seen in chapter 3.

Figure 4 . 8 .

 48 Figure 4.8. Thermal signal ΔP/P c as function of the sample thermal conductivity λ s (T a =30 °C, RH= 40 %).

Figure 4 .

 4 [START_REF] Williams | Photothermal imaging with sub-100-nm spatial resolution[END_REF] gives the curves ΔP/P c =f(λ s ) for T m = 40 °C and T m =65 °C. It shows that the sensitivity of the KNT probe to the sample thermal conductivity is limited to a maximum limit around a few W.m -1 .K -1 for the used experimental conditions.

Figure 4 . 9 .

 49 Figure 4.9. Thermal signal ΔP/P c as function of the sample thermal conductivity λ s for T m = 40 °C (blue circles) and T m = 65 °C (red circles) (T a =30 °C, RH= 40 %).

  .1 presents the values determined of G air and b air . At T m = 40 °C, the mean values of b air and G air are respectively 3.5 µm and 0.81 µW.K -1 whereas at T m =65 °C, b air and G air are respectively 5.2 µm and 2 µW.K -1 . As we can notice, the values determined for b air

3

 3 As expected for this small tip, the values of the effective parameters G air and b air are smaller than the values determined for the Wollaston wire probe shown in chapter 3 for the same range of λ s (b air = 25 µm and G air = 5.5 µW.K -1

Figure 4 .

 4 [START_REF] Goodson | Near-field optical thermometry[END_REF] shows the heating Chapter 4 -DC SThM with probes of nanoscale radius of curvature 131 voltage as a function of the apex temperature. For an accurate calibration, the apex temperature is calibrated between two consecutive melting temperatures.

Figure 4 . 10 .

 410 Figure 4.10. Monitoring the probe cantilever deflection as a function of the probe heating voltage of the three reference polymers.

Figure 4 . 11 .

 411 Figure 4.11. Heating Voltage (V) as a function of the apex temperature T ap (°C).

Figure 4 . 13 shows

 413 the variation of the pull-off forces F po as a function of the apex temperature. The pulloff forces decreases progressively until an apex temperature limit beyond which they become almost constant. The large error bars (specifying the dispersion of measurements) are due to the roughness of the sample as noticed before in the case of the Wollaston probe and the KNT probe.Greiner et al. [113] performed measurements of F po under ambient conditions as a function of the heater temperature for a contact between the same kind of probe and a SiO 2 sample as seen in Figure4.13. The limit reached by these measurements is close to our limit as shown in Figure4.13 (≃ 25 nN). The radius of the apex R a for the used DS probe was estimated around 45 nm. This is in good agreement with the reported values of R a of these probes (≃ 50 nm)[START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF].

Figure 4 .

 4 14 presents the variations of the largest radius of meniscus r 2 as a function of the apex temperature T ap . These values are in the range of the determined values for the KNT probe (see Figure 4.4).

Figure 4 .

 4 [START_REF] Dinwiddie | Thermal conductivity contrast imaging with a scanning thermal microscope[END_REF] shows the variation of the total conductance of the water meniscus G total, meniscus as a function of T ap . These values of are at least one order of magnitude smaller than the values of the solid-solid conductance G ss

Figure 4 . 12 .

 412 Figure 4.12. Pull-off forces F po as a function of the apex temperature T ap for a contact between the DS probe (AN-300) and Po sample (T a =30 °C, RH= 40 %).

Figure 4 . 13 .

 413 Figure 4.13. Pull-off forces F po as a function of the heater temperature for a contact between the DS probe (AN-300) and silicon dioxide SiO 2 sample [113].

Figure 4 . 14 .

 414 Figure 4.14. Variation of the larger radius r 2 of the meniscus as function of the apex temperature T ap for the contact between DS probe and sample of Po (T a =30 °C, RH= 40 %).

Figure 4 . 15 .

 415 Figure 4.15. Variation of the total thermal conductance of water meniscus G total, meniscus as function of the apex temperature T ap for the contact between DS probe and sample of Po (T a =30 °C, RH= 40 %).

5. Chapter 5 : 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient

 55 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambientChapterThis chapter describes the development of a technique that should allow in short terms the measurement of the thermal properties of anisotropic thin solid films at different levels of temperature above the ambient. The technique uses a set of deposited resistive wires. Before

  2b w and length l is deposited on the substrate surface as illustrated in Figure 5.1. The wire is heated with an alternating current I of magnitude I ac at angular frequency ω=2πf. The principle of the electrical measurement has already been described in Chapter 1.

Figure 5 . 1 .

 51 Figure 5.1. Schematic of 3ω method where a metallic wire (heating wire) is deposited on the sample to be characterized.

5 Chapter 5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 141 a and µ are respectively the thermal diffusivity of the sample and the thermal diffusion length in the sample material and given as:

5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 142

  d and w s are respectively the thickness and width of the film and P d is the power dissipated in the heater. The variation of the frequency ω allows us to determine the thermal properties of the thin film (α //f , λ //f ). In this technique, one must pay attention to the radiation and convection losses that can affect the thermal measurements.When the suspended film of interest is electrically conducting, it can be used as a heater and sensor too as studied Lu et al. [137] in the alternating regime. The thermal properties of the film are obtained solving the 1D heat conduction equation [137]. Bourgeois et al. [138] used the method of Lu et al. [137] to measure thermal conductance of silicon nanowires at low 5 Chapter 5 -Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 144 temperatures (less than 2 K). The same group studied the effect of the roughness of the nanowires surfaces on the thermal conductance of silicon wires at low temperatures [139].

Figure 5 . 2 .

 52 Figure 5.2. Schematic of the phase-shift method used by Zhang and Grigoropoulos [136].

5 Chapter 5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 145 component can be determined in the case of a narrow heater as seen in the 3ω differential technique. Using this approach, Lui et al. [143] determined the thermal conductivity of anisotropic strained Si/Ge superlattices. The same sample type was used by Jang et al. [144] to determine the in-plane thermal conductivity of encased graphene and ultrathin graphite. However, the configuration here consists of different sensors, of same length, deposited on the sample surface at different distances from the heater as shown Figure 5.3. For related applications (encased graphene layers devices), two insulating layers (ox in figure 5.3) surround the graphene of interest that have a high in-plane thermal conductivity [144]. In this case, the heat is spread laterally in the film of interest and down to the substrate [144].

Figure 5 . 3 .

 53 Figure 5.3. The configuration of the sample used by Jang et al. [144].

Figure 5 . 4 . 5 Chapter 5 -

 5455 An array of parallel and long resistive wires is deposited on the front surface and back surface of the sample. One of the resistive wires (in red in Figure5.4) is heated with an alternating current I of magnitude I ac at angular frequency ω=2πf. This wire is the thermal source for the sample.The heating produces a continuous temperature ΔT DC and alternating temperature oscillations ΔT 2ω . The other wires are the probing wires. The information about the thermal properties of the samples requires the determination of ΔT 2ω .By measuring the electrical resistance change of the probing wires ΔR 2ω (x), the surface temperature ΔT 2ω (x) is detected:ΔR 2ω (x) = R 0 (x).β.ΔT 2ω(5.15) Here, |x| is the distance from the heating wire, R 0 (x) is the electrical resistance of the probing Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 147 wire at ambient temperature T a and β is the temperature coefficient of the electrical resistivity.

Figure 5 . 4 .

 54 Figure 5.4. Schematic of method principle.

5 Chapter 5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 152 A) Determination of the probing wires length

Figure 5 . 5 .

 55 Figure 5.5. Top view of the model showing the distribution colored with isotherms while generating current amplitude I ac of 60 mA.

Figure 5 . 6 .

 56 Figure 5.6. Variation of ΔT 2ω for different lengths as a function of the distance from the heating wire.

5 Chapter 5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 154 seven mm and varies between 3 and 4 for mm for the probing wires. The justification of the wires length is given in the next section. By measuring the wire electrical resistance R by 4point probes method (4-probe sensing) at ambient temperature, ρ was determined. The chip (sample + wires) was placed in a specific oven and heated to different temperature levels. Then, β was determined from the wire resistance measurements: R = R 0 (1+β(T-T a )).

Figure 5 . 7 .

 57 Figure 5.7. Schematic of the implemented sample.

Figure 5 . 8 .

 58 Figure 5.8. Process flow for the deposition of the metallic wires array. (a) Photolithography, (b) Metal deposition, (c) Lift-off process.

Figure 5 . 9 .

 59 Figure 5.9. AFM 2D image of a gold wire.

Figure 5 . 10 . 5 Chapter 5 -

 51055 Figure 5.10. The mask design showing the probe wires (in yellow) and the heater (in red).

.

  In fact, the difference between ΔT DC (3D) and ΔT DC (2D) is related to the fact that in the 2D model, the length of the wires is assumed the same of the sample.

Figure 5 . 11 .

 511 Figure 5.11. Schematic of the 2D COMSOL model.

Figure 5 . 12 . 5 Chapter 5 -Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 157 5 . 4 Analysis of the influence parameters 5 . 4 . 1 Figure 5 .

 51255545415 Figure 5.12. Variation of ΔT 2ω in the two COMSOL configurations (2D and 3D) as a function of the distance from the heating wire for Si sample (f= 40 Hz, ϵ=1, h= 5 W.m -2 .K -1 and I ac =60 mA).

Figure 5 .

 5 Figure 5.14 presents the variations of ΔT 2ω and ϵ as a function of h and ϵ for different values of h and ϵ. As can be seen, the heat losses are not negligible and affect the temperatures T DC and ΔT 2ω .

Figure 5 . 13 .

 513 Figure 5.13. Effect of the heat losses on the temperature of the sample T DC in 2D and 3D COMSOL configurations (I ac =60 mA).

Figure 5 . 14 .

 514 Figure 5.14. Variations of ΔT 2ω as a function of the distance from heating wire for different values of h and ϵ (I ac =40 mA, f=40 Hz).

5 Chapter 5 -

 5 Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 159 (see Table

Figure 5 . 15 .

 515 Figure 5.15. Variation of ΔT 2ω (K) (calculated in 2D COMSOL configuration) as a function of the distance of the heating wire (µm) for a perfect contact (R th =0) and for R th = 3 × 10 -7 m 2 .K.W -1 .

  d and w s are respectively the thickness and width of the sample, λ s and α s are respectively the thermal conductivity and diffusivity of the Si sample and 2 f length in the sample.

Figures 5 .

 5 Figures 5.16 and 5.17 show respectively the variations of ΔT 2ω and the phase lag Ф 1 (º) as a function of the distance from the heating wire. As can be seen, the analytical model fits well the 2D numerical simulations in the two curves. On the other hand, the 1D approach curves follows the same trend of the other curves after exceeding a distance larger than 225 µm from the heating wire as indicated in Figures 5.[START_REF] Tovee | Nanoscale spatial resolution probes for Scanning Thermal Microscopy of solid state materials[END_REF] and 5.17. For distances less than the sample thickness d, the Z part of the heat flux cannot be neglected and as a result the one-dimensional conduction model is not valid anymore, in analogy with the scale analysis shown by Bejan [123].

5 Chapter 5 -Figure 5 . 16 .

 5516 Figure 5.16. Comparison of ΔT 2ω between the analytical model, 2D numerical simulations and 1D approach (I ac =60 mA, ϵ=1 and h=5 W.m -2 .K -1 ).

Figure 5 . 17 .

 517 Figure 5.17. Comparison of the phase lag (º) between the analytical model, 2D numerical simulations and 1D approach (I ac =60 mA, ϵ=1 and h=5 W.m -2 .K -1 ).

5 Chapter 5 -Figure 5 . 18 .

 5518 Figure 5.18. Schematic of the experimental set-up.

5 Chapter 5 -Figure 5 . 19 .

 55519 Figure 5.19. Electrical connections of the measurements.

Figure 5 . 20 .

 520 Figure 5.20. Experimental variations ΔT 2ω as a function of the distance from the heating wire for I ac =60 mA and for different frequencies.

Figure 5 . 21 . 5 Chapter 5 -

 52155 Figure 5.21. Experimental variations of the phase lag ϕ as a function of the distance from the heating wire for I ac =60 mA and for different frequencies.

Figure 5 . 22 .

 522 As said before, such variations are small and within the uncertainties of the determination of thermal properties of solid materials at temperatures larger than the ambient temperature. Figures 5.23 and 5.24 show the variations of the phase lag Ф and the amplitude ΔT 2ω as a function of the distance from the heating wire where I ac =60 mA for the analytical model (section 3) and the numerical simulations. As can be seen, the analytical model can describe well the variations of Ф and ΔT 2ω in order to determine the inplane thermal properties of the considered sample. The 1D model was tested in this case and the in-plane thermal properties were found again.

Figure 5 . 22 . 5 Chapter 5 -Figure 5 . 23 .

 52255523 Figure 5.22. Simulated Variation of the sample temperature T DC as a function of the distance from the heating wire for the two cases: λ x >> λ z and λ x << λ z (I ac =60 mA, ϵ=1, h=5 W.m -2 .K -1 ).

Figure 5 . 24 .

 524 Figure 5.24. Simulated Variation of the phase lag ϕ as a function of the distance from the heating wire for the numerical simulations and the 2D analytical in the case of anisotropic sample (λ x >> λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ).

Figure 5 . 25 and 5 . 5 Chapter 5 -

 525555 26 present the variations of the phase lag Ф and the amplitude ΔT 2ω as a function of the distance from the heating wire. The analytical model fits almost the numerical model despite Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient 167 some uncertainties where the curves do not follow the variation of the numerical simulations. That might be related to the significant variation of the sample temperature T DC in this case as shown in Figure 5.22. In fact, the thermal properties inserted in the analytical model correspond to the mean temperature of the variations of T DC whereas in the numerical simulations the thermal properties are as a function of the temperature.

Figure 5 . 25 .

 525 Figure 5.25. Simulated variation of the (a) phase lag (b) ΔT 2ω as a function of the distance from the heating wire for the numerical simulations and the analytical model in the case of anisotropic sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ).

Figure 5 . 26 .

 526 Figure 5.26. Simulated variation of the (a) phase lag (b) ΔT 2ω as a function of the distance from the heating wire for the numerical simulations and the analytical model in the case of anisotropic sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ).

Figures 5 .

 5 Figures 5.27 and 5.28 show respectively the variations of the Ф and ΔT 2ω as a function of the current frequency for the numerical simulations and the analytical model. Here, the thermal properties and dimensions of the sample are the same of isotropic Si used before in our experiments. As can be observed, the analytical model fits perfectly the numerical simulations in terms of the amplitude ΔT 2ω the phase lag Ф curves.

Figure 5 . 27 .

 527 Figure 5.27. Variation of the phase lag ϕ as a function of the current frequency for the numerical simulations and the analytical model for the probing wire located at the back surface of the sample (λ x << λ z , I ac = 60 mA, f = 40 Hz and ϵ=1).

Figure 5 . 28 .

 528 Figure 5.28. Variation of ΔT 2ω as a function of the current frequency for the numerical simulations and the analytical model for the probing wire located at the back surface of the sample (λ x << λ z , I ac =60 mA, f=40 Hz and ϵ=1).

  at a temperature controlled by the magnitude of the heating current. The technique does not need the determination of the heat losses to the environment since the sample temperature is measured through the probing wires. The length of the wires is justified through 3D simulations. The difference between 3D and 2D numerical configurations is noticed and explained. 2D analytical model is introduced to determine the film thermal properties in the two directions. The analytical model fits well the numerical simulations in the case of isotropic samples. The range of frequency and the distance of the probing wires are determined to ensure the validity of the 1D approach model. First set of measurement were performed on an isotropic Silicon sample. The thermal properties determined at different temperatures from ambient to 500 K are in good agreement with the values of the scientific literature.

Summary of the thermal

  parameters of the different heat transfer mechanisms based measurements with the Wollaston and KNT probes.The first steps of the development of a thermal metrology for thermal characterization of anisotropic thin films were also presented. The method, which we have called the "2ω method", should allow the identification of the thermal diffusivity and conductivity of materials at temperatures higher than the ambient. It consists of a network of resistive wires deposited on the surface of the sample to be characterized. One of the wires serves for heating the sample in the alternating regime while the others are used as temperature sensors. A 2D model that describes the thermal behavior of the implemented sample was developed. This model was based on the resolution of the heat equation in the Fourier space. Simulations with this model developed in this chapter have shown the possibility to determine the thermal properties of materials in the in-plane and cross-plane directions. The modeling was also used to derive from experimental measurements the thermal conductivity and diffusivity of a reference sample of isotropic silicon. The obtained results at different temperatures ranging from ambient to 500 K were in good agreement with literature values. They validate the method proposed by the CETHIL laboratory. The further steps include the electrical implementation and the study of thin film samples with deposited wires on their both surfaces. Image A.3 shows an image of the adjustment unit that consists of a probe holder and a

Figure A. 3 .

 3 Figure A.3. Image of the adjustment unit: 1) Probe holder and 2) connector to the measuring head (NTMDT/ User's manual).

Figure A. 4 .

 4 Figure A.4.

Figure A. 4 .

 4 Figure A.4. Image of the AFM while performing measurements under vacuum conditions: 1) protective hood and 2) its frame (NTMDT/ User's manual).

  Figure A.5 presents a schematic of the units used in the measurements. It consists of a Wheatstone bridge and a feedback loop with a ratio between the two branches of the bridge of 5.

Figure A. 5 .

 5 Figure A.5. Schematic of the Wheatstone bridge used in the measurements.

Figure A. 6 .

 6 Figure A.6. The three parts of the NanoTA2: 1) Power supply, 2) Controller and 3) CAL Box.

Figure A. 7 .

 7 Figure A.7. Illustration of the single probe mode.

Methodology developed and used to investigate the temperature-dependence of the meniscus

  Potential applications related to the water meniscus for heated AFM probes ......... 2.1.2 Characteristic dimensions and properties of the water small layers adsorbed on a surface 71 2.1.3 Young-Laplace and Kelvin Equations ..................................................................... Experimental approach ............................................................................................. 2.2.2 Derived parameters: Meniscus parameters and thermal conductance of probesample heat transfer through water ...................................................................................... 2.2.3 Results ...................................................................................................................... 2.2.4 Effect of roughness on the pull-off forces on hydrophilic surfaces ......................... 2.2.5 Measurement of the pull-off forces on hydrophobic surface ...................................

	m N Nu p R 0 RH r 1 r 12 ΔP/P c ε θ 2.1.1 2.2	m=(hp/kS) 1/2 Number of phonon modes Nusselt number Fin perimeter Electrical resistance of the probe at T a Relative humidity Smallest meniscus radius Reflection coefficient at the tip-sample interface Probe Joule power relative difference Surface emissivity Relative temperature	m Ω m K
	P r 2 θ	Pressure Largest meniscus radius Mean Relative temperature	bar m K
	P a S θ DC	Probe Joule power -Probe free in air Section Continuous relative temperature	W m 2 K
	P c r k θ 2w	Probe Joule power -Probe in contact with sample Cavity radius Second harmonic relative temperature	W m K
	P p T γ	Power dissipated in the probe Temperature C p /C v	W K
	P d T a λ max	Power dissipated in the heater (heating wire) Ambient Temperature Maximum wavelength of thermal radiation	W K µm
	P rms T m	Power per unit heater length Probe mean temperature	W.m -1 °C
	Pr t 1 λ	Prandtl number Film thickness Thermal conductivity	m W.m -1 .K -1
	Q t 12 λ pp	Heat flux Phonon transmission coefficient Phonon-phonon thermal conductivity	W W.m -1 .K -1
	R V λ w	Electrical resistance Voltage Thermal conductivity of water	Ω V W.m -1 .K -1
	Ra V air λ max	Rayleigh number Velocity of air molecules Maximum wavelength of thermal radiation	m
	R a V DC Λ	Tip apex radius Continuous voltage Phonon mean free path	m V m
	R ar V m	Arithmetic value of roughness Molar volume	m m 3 .mol -1
	R b V 3ω µ	Thermal boundary resistance Third harmonic voltage Diffusion length	m 2 .K.W -1 V m
	R bp w ν	Thermal boundary resistance of the phonon-phonon Width Poisson ratio	m 2 .K.W -1 m
	h R be w d ν pa w s ρ	Heat transfer coefficient coupling Thermal boundary resistance of the phonon-electron Dupré Energy of adhesion Velocity of acoustic phonons coupling Sample width Electrical resistivity	1 m 2 .K.W -1 N m.s -1 W.m -2 .K -1 m Ω.m -1
	h g R g χ	Global linearized convective and radiative losses Gas constant Wave vector along z	W.m -2 .K -1 J.mol -1 .K -1
	h k R s Greek letters Ф	Meniscus height Spreading thermal resistance Phase lag	m K.W -1 °
	I R th α Ф d	Electrical current Interface thermal resistance Thermal diffusivity Heat flow density	A K.W -1 m 2 .s -1 W.m -2
	K 1 k x R p-p β Ф ref L R sph R tip γ w ω Ω	Correction factor Peak-to-peak roughness Temperature coefficient of the electrical resistivity Ratio of heat power X-component of the wave vector Length Sphere radius Tip thermal resistance Water surface tension Angular frequency G eq /G p	m K -1 m m Rad.s -1 N.m -1 K.W -1
	L wf R interface Δ	Wiedemann Franz constant Thermal resistance at the tip-sample interface Temperature ratio	W.Ω.K -2 K.W -1

.................................................................................................................................. 2.2.1

  . ...............................................................................Figure 2.1. Schematic of the principle of Dip-pen nanolithography (DPN) [76]. ................... Figure 2.2. Evolution of the water film thickness as a function of the relative humidity for three types of materials: chromium Cr, copper Cu and gold Au. The measurement were performed using a scanning Kelvin probe force microscope (SKPFM) [87]. .........................Figure 2.3. Schematic of a capillary confined in a longitudinal tube [95]. ..............................Figure 2.4. Image of the contact between the water droplet and the sample of Ge. ................Figure 2.5. Spectroscopy curves performed on Germanium (Ge) for different probe mean temperature: (a) Ambient temperature, (b) T m = 130 °C and (c) T m = 230 °C. ..........................Figure 2.6. Dependence of the meniscus pull-off force on the tip-sample contact time [100]. ..................................................................................................................................................Figure 2.7. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of polished germanium. ....................................................................................Figure 2.8. Schematic of the meniscus when the probe is in contact with the sample. ...........Figure 3.6. Estimation of the thermal conductance at the solid-solid contact G ss for the three materials (MICA, Ge and Si). ................................................................................................ 100 Figure 3.7. Thermal boundary resistance R b (m 2 .K.W -1 ) identified for the three materials (MICA, Ge and Si). ................................................................................................................ 100 Figure 3.8. Schematic of the probe in two different positions. .............................................. 102 Figure 3.9. Voltage of the probe in different conditions for different tilts of the probe. ....... 102 Figure 3.10. ΔP/P c variation as a function of the thermal conductivity λ s (T a =30°C, RH=40%). ................................................................................................................................................ 105 Figure 3.11. Schematic of the heat flux lines for two samples of different thermal conductivity λ s . ............................................................................................................................................ 105 Figure 3.12. Variation of b air and G air as a function of λ s and their corresponding fitting curves (a 1 = 1.05E-04, b 1 =-2.32, c 1 =3.42E-06, d 1 =-0.04492; a 2 =7.77E-06, b 2 =-5.18E-04 , c 2 =-5.47E-06, d 2 =-1.197). ............................................................................................................. 106 Figure 3.13. The model of the probe being at proximity of the sample. ................................ 108 Figure 3.14. Calculated relative heat power transferred to the sample as a function of its thermal conductivity λ s . .......................................................................................................... 109 Figure 3.15. Variation of the temperature on the sample surface in the direction of the major

Figure 2.9. Scanning Electron Microscopy SEM image of a Wollaston wire probe apex. ......

Figure 2.10. Schematic of a sphere and a cylinder with their radii of curvature .....................

Figure 2.11. Radius of the curvature of the meniscus r 2 as a function of the Wollaston probe mean temperature T m for the Ge sample (T a = 30 ⁰C, RH= 40 %). .......................................... Figure 2.12. Total thermal conductance of the meniscus G total, meniscus as a function of the Wollaston probe mean temperature T m for the Ge sample (T a = 30 ⁰C, RH= 40 %). ................

Figure 2.13. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of rough germanium. ...................................................................................

Figure 2.14. Schematic illustration of the meniscus of a contact between the probe and a surface asperity [94]. ................................................................................................................

Figure 2.15. Variation of the pull-off forces F po as a function of the probe mean temperature T m for the sample of graphite. ..................................................................................................

Figure 2.16. Pull-off force as a function of RH measured between a hydrophobic tip and a flat silicon sample [110]. ................................................................................................................

Figure 2.17. ESEM image of the water meniscus formed at the contact between Wollaston wire and germanium sample (T substrate ≃ 2 °C, RH ≃ 90 %). ..................................................... Figure 2.18. Meniscus height (nm) as a function of the relative humidity for a SiN cantilever tip in contact with a silicon substrate at a temperature of 5 °C . ..............................................

Figure 2.19. ESEM image of a liquid film on a Ge substrate showing two different zones of the liquid film (T substrate ≃ 2 °C, RH between 95 and 100 %). ...................................................

Figure 2.20. Schematic diagram of the cases (a) where the electron beam is not focused (b) and when it is focused in an ESEM chamber on the probe-sample contact. ............................

Figure 3.1. Variations of the probe Joule power relative difference as a function of the probe mean temperature when the probe contacts a sample of germanium Ge (T a = 30 °C and RH= 40 %). .......................................................................................................................................

Figure 3.2. Variations of the probe conductance G p as a function of the probe mean temperature T m . .........................................................................................................................

Figure 3.3. Variation of λ Pt/Rd as a function of T m as determined by Raphael [42]. .................

Figure 3.4. ΔP/P c Calibration curves obtained by David [56] under ambient conditions (a c = 7.5 nm, G air =4.5 × 10 -6 W.K -1 ). ............................................................................................... Figure 3.5. Variation of the thermal signal for the three materials (MICA, Ge and Si) under vacuum conditions (T m = 140°C, RH=40 %, T a =30 °C). ........................................................ axis X. ..................................................................................................................................... 110 Figure 3.16. Variation of the temperature on the sample surface in the direction of the minor axis Y. ..................................................................................................................................... 110 Figure 3.17

  13. Pull-off forces F po as a function of the heater temperature for a contact between the DS probe (AN-300) and silicon dioxide SiO 2 sample [113]. ........................................... Figure 4.14. Variation of the larger radius r 2 of the meniscus as function of the apex temperature T ap for the contact between DS probe and sample of Po (T a =30 °C, RH= 40 %). ................................................................................................................................................Figure 4.15. Variation of the total thermal conductance of water meniscus G total, meniscus as function of the apex temperature T ap for the contact between DS probe and sample of Po (T a =30 °C, RH= 40 %). ..........................................................................................................Figure 5.1. Schematic of 3ω method where a metallic wire (heating wire) is deposited on the sample to be characterized. .................................................................................................... Figure 5.2. Schematic of the phase-shift method used by Zhang and Grigoropoulos [136]. .Figure 5.3. The configuration of the sample used by Jang et al. [144]. ................................. Figure 5.4. Schematic of method principle. ........................................................................... Figure 5.5. Top view of the model showing the distribution colored with isotherms while generating current amplitude I ac of 60 mA............................................................................. Figure 5.6. Variation of ΔT 2ω for different lengths as a function of the distance from the heating wire. .Figure 5.21. Experimental variations of the phase lag ϕ as a function of the distance from the heating wire for I ac =60 mA and for different frequencies. .....................................................Figure 5.22. Simulated Variation of the sample temperature T DC as a function of the distance from the heating wire for the two cases: λ x >> λ z and λ x << λ z (I ac =60 mA, ϵ=1, h=5 W.m -2 .K -1 ).

.......................................................................................................................... Figure 5.7. Schematic of the implemented sample. ............................................................... Figure 5.8. Process flow for the deposition of the metallic wires array. (a) Photolithography, (b) Metal deposition, (c) Lift-off process. ............................................................................ Figure 5.9. AFM 2D image of a gold wire. ............................................................................ Figure 5.10. The mask design showing the probe wires (in yellow) and the heater (in red).

Figure 5.11. Schematic of the 2D COMSOL model. .............................................................

Figure 5.12. Variation of ΔT 2ω in the two COMSOL configurations (2D and 3D) as a function of the distance from the heating wire for Si sample (f= 40 Hz, ϵ=1, h= 5 W.m -2 .K -1 and I ac =60 mA). ........................................................................................................................................ Figure 5.13. Effect of the heat losses on the temperature of the sample T DC in 2D and 3D COMSOL configurations (I ac =60 mA). ................................................................................. Figure 5.14. Variations of ΔT 2ω as a function of the distance from heating wire for different values of h and ϵ (I ac =40 mA, f=40 Hz). ................................................................................ Figure 5.15. Variation of ΔT 2ω (K) (calculated in 2D COMSOL configuration) as a function of the distance of the heating wire (µm) for a perfect contact (R th =0) and for R th = 3 × 10 -7 m 2 .K.W -1 . ................................................................................................................................ Figure 5.16. Comparison of ΔT 2ω between the analytical model, 2D numerical simulations and 1D approach (I ac =60 mA, ϵ=1 and h=5 W.m -2 .K -1 ). .............................................................. Figure 5.17. Comparison of the phase lag (º) between the analytical model, 2D numerical simulations and 1D approach (I ac =60 mA, ϵ=1 and h=5 W.m -2 .K -1 ). .................................... Figure 5.18. Schematic of the experimental set-up. ............................................................... Figure 5.19. Electrical connections of the measurements. ..................................................... Figure 5.20. Experimental variations ΔT 2ω as a function of the distance from the heating wire for I ac =60 mA and for different frequencies........................................................................... ................................................................................................................................................ Figure 5.23. Simulated Variation of ΔT 2ω as a function of the distance from the heating wire for the numerical simulations and the 2D analytical in the case of anisotropic sample (λ x >> λ z , I ac =60 mA, f=40 Hz and ϵ=1, h=5 W.m -2 .K -1 ). ......................................................................

Table 1 .

 1 1. Properties of the Wollaston wire probe. ................................................................. Table 1.2. Dimensions of different silicon nanoprobes........................................................... Table 3.1. ΔP/P c results for the samples of Mica, Germanium and Silicon under vacuum conditions (values ofλ s were given by the suppliers, Goodfellow and Neyco). ...................... Table 3.2. List of the calibration samples used in the measurements with their roughness parameters measured with AFM (values of λ s were given by the suppliers, Goodfellow and Neyco). ................................................................................................................................... Table 3.3. Identified parameters of the probe-sample thermal interaction for the two cases: h=0 and h=3000 W.m -2 .K -1 . ................................................................................................... Table 3.4. Mean values of G air and b air for different intervals of the thermal conductivity λ s . ................................................................................................................................................ Table3.5. Samples with the values of their roughness parameters (R ar and R p-p ) and corresponding identified thermal conductivity λ

s . ..................................................................

Table 4.1. Thermal interaction parameters G air and b air determined for two different temperatures. .......................................................................................................................... Table 4.2. List of the reference polymers with their corresponding melting temperatures. (reference Anasys instruments). ............................................................................................. Table 5.1. Thermal properties of pure silicon at different temperature levels. ...................... Table 5.2. Measured properties of the wires. ......................................................................... Table 5.3. Si thermal properties experimentally determined. ................................................

Chapter 1 Scanning Thermal Microscopy (SThM): State of the art and exposition of the issues

	samples and the thermal boundary resistance at the tip-sample interface is determined. Later, 1. Chapter 1: Scanning Thermal Microscopy the thermal conduction through air is determined while taking into account the sample thermal conductivity. The effect of roughness of different samples on the probe-sample thermal (SThM): State of the art and exposition of the
	interaction is presented. Several criteria are proposed in order to improve the SThM issues
	calibration for thermal conductivity measurement.
	Chapter 4 mainly focuses on measurement carried out with the KNT probe. Moreover, results
	obtained with the doped silicon DS probe are presented and discussed. Applying the
	methodology described in chapter 2 to these probes, the water meniscus conduction is studied
	on samples of different thermal conductivity. Besides, the results of measurements under
	vacuum conditions performed on different samples are presented for the KNT probe. The heat
	conduction at the tip-sample contact is investigated and the determination of the thermal
	boundary resistance for each sample is addressed. Using various reference samples and while
	working under ambient conditions, the sensitivity to the sample thermal conductivity is
	specified for the KNT probe. The last part of this chapter proposes an evaluation of the
	phonon transmission coefficient at the tip-sample contact.
	The last chapter of the manuscript presents the concept of a technique to characterize the
	thermal properties of anisotropic samples. Based on contact thermal metrologies operating in
	AC regime, the technique allows studying samples at temperatures above the ambient. The
	technique is called the "2ω method". Its principle and first results of experimental work and
	simulations for the technique validation are presented and discussed. The experimental results
	on a Si sample are compared with the literature studies. Later, numerical simulation works are
	presented in order to set a basis to identify the thermal properties of the sample in the in-plane
	and cross-plane direction of an anisotropic sample with the 2ω method.
	The thermal
	conductance at the solid-solid contact is measured under vacuum conditions on polished

Chapter 1 -SThM: State of the art and exposition of the issues 1.1 Scanning Thermal Microscopy (SThM)

	Scanning Thermal Microscopy (SThM) is a quasi non-destructive technique that allows the
	characterization of sample materials. The technique is based on Scanning Probe Microscopes
	(SPM) and allows measurements to be performed with high spatial resolution. SThM was
	used first in Scanning Tunneling Microscope (STM)

Table 1 .1. Properties of the Wollaston wire probe.

 1

Table 1 .2. Dimensions of different silicon nanoprobes.

 1

	Tip	Leg length (µm)	Leg width (µm)	Leg thickness (µm)	Tip height (µm)	Tip radius (nm)	Spring constant (N.m -1 )	Resonant Frequency (KHz)
	AN-300	300	20	2	3-6	< 30	0.5-3	15-30
	IBM [18]	50	10	0.5	2	< 20	1	200

  date, carried out simultaneously to the one of the other heat transfer mechanisms of

	the probe-sample thermal interaction: Can we propose a new methodology to specify
	this heat transfer mechanism and have a more accurate quantitative estimation of the
	thermal conductance G total,meniscus independently of the other mechanisms? What is the
	dependence of this heat transfer mechanism on the probe temperature? Does its
	contribution to the probe-sample thermal interaction depend on the size of the probe
	and sample properties? Can we avoid it to simplify the analysis of the other heat
	transfer mechanisms?

• The study of the thermal conduction at the mechanical contact between probe and sample requires working under vacuum conditions to strongly reduce the heat transfer through air: Under these conditions and assuming the radiative heat transfer negligible and the water meniscus vanished between the probe and samples, how do the characteristic parameters of the probe-sample mechanical contact influence the SThM measurement? What is the contribution of this heat mechanism to the probesample heat transfer? Is it dependent on the probe and on the sample? How and why?

1 Chapter 2 -Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM 69 2.1 Motivations 2.1.1 Potential applications related to the water meniscus for heated AFM probes

	Under ambient conditions, a water film is formed on the surfaces due to capillary
	condensation of the relative humidity. Because of several related applications or issues (some
	examples are given below), much interest has been shown in studies around the phenomena of
	capillary condensation.

  .[START_REF] Borkar | Low power design challenges for the decade[END_REF]

	vap = vap µ	µ	vap	sat P	+	RT
		µ	vap	sat P	+	m V P liq	-	sat P
	and
						ln(	)	(2.4)

Table 3 .1. ΔP/P c results for the samples of Mica, Germanium and Silicon under vacuum conditions (values of λ s were given by the suppliers, Goodfellow and Neyco).

 3

	Sample (λ s )	MICA (5 W.m -1 .K -1 )	germanium (58.6 W.m 1 .K -1 )	silicon (148 W.m 1 .K -1 )
	a c (nm) values determined with	≃12.5 ± 0.9	≃13.7 ± 1	≃ 12.8 ± 1
	Hertz's model
	Arithmetic value of roughness R ar	0.28	0.7	0.25
	Maximum peak-to-
	peak roughness	2.68	4.75	1.85
	R p-p
	Mean free path of	??	~ 200 [116]	~ 260 [116]
	phonons Λ

Table 3 .2. List of the calibration samples used in the measurements with their roughness parameters measured with AFM (values of λ s were given by the suppliers, Goodfellow and Neyco). Sample material Thermal conductivity (W.m -1 .K -1 ) Arithmetic roughness R ar (nm) Peak-to-Peak roughness R p-p (nm)

 3

	Polyimide PI	0.17	5	57
	Silicon dioxide SiO2	1.2	3.25	54.2
	Zirconium dioxide
		2.5	0.8	5
	ZrO2
	Cadium Selenide
		3.49	0.86	13.5
	CdSe
	Cadmium telluride
		6.28	3.7	44.9
	CdTe
	Zinc Telluride ZnTe	12.36	5.08	51.2
	The measurements for our thermal conductivity calibration were performed under ambient
	conditions (T a = 25 ⁰C, RH= 40 %) at a mean probe temperature T m of 140 ⁰C. The water
	meniscus has almost vanished at this temperature (see Chapter 2) and the solid-solid thermal
	conductance G

ss is at least one order of magnitude smaller than the conductance through air G air (see section 2 of this chapter). The ΔP/P c model is described in Chapter 1 (see Equations

(1.6

) and (1.7)). The thermal conductivity calibration with the measurements of ΔP/P c consists in evaluating two mean values that denote the heat transfer to the sample through air:

Table 3 .3. Identified parameters of the probe-sample thermal interaction for the two cases: h=0 and h=3000 W.m -2 .K -1 .

 3

	Heat losses coefficient	b air (µm)	G air (µW.K -1 )
	h = 3000 W.m -2 .K -1	14.7± 1.6	6.64 ± 0.16
	h = 3000 W.m -2 .K -1 (Polyimide exempted)	4 ± 0.05	7.38 ± 0.27
	h = 0 W.m -2 .K -1	11.5 ± 1.2	5.09 ± 0.13
	h = 0 W.m -2 .K -1 (Polyimide exempted)	3.12 ± 0.04	5.64 ± 0.2

Table 3 .4. Mean values of G air and b air for different intervals of the thermal conductivity λ s .

 3

	Interval of λ s	b air (µm)	G air (µW.K -1 )
	[0.17;1.2]	24.5 ± 4.5	5.5 ± 0.1
	[1.2;2.5]	4.35 ± 0.07	7.19 ± 0.25
	[2.5;3.49]	3.14 ± 0.12	7.72 ± 0.36
	[3.49;6.28]	2.87 ± 0.4	7.73 ± 0.42
	[6.28;12.36]	2.35 ± 0.77	7.77 ± 0.54

Table 3 .5. Samples with the values of their roughness parameters (R ar and R p-p ) and corresponding identified thermal conductivity λ s .

 3

	Samples	Arithmetic roughness R ar (nm)	Peak-to-Peak Roughness R p-p (nm )
	SiO 2 (1)	20 ± 2	500 ± 80
	SiO 2 (2)	58 ± 6	800 ± 100
	Ge(1)	27 ± 5	270 ± 50
	Ge(2)	67 ± 8	900 ± 100

Table 4 .1. Thermal interaction parameters G air and b air determined for two different temperatures.

 4

Table 4 .2. List of the reference polymers with their corresponding melting temperatures. (reference Anasys instruments).

 4

	Reference polymer	Melting temperature (°C)
	Polycaprolactone PCL	55 °C
	Polyethylene PE	116 °C
	Polyethylene terephthalate PET	235 °C

DC SThM with probes of nanoscale radius of curvature 135

  (AMM) and the diffuse mismatch model (DMM)[116, 117]. The AMM model assumes that scattering at interfaces is totally elastic. It means that the transmission and reflection of phonons occur at the same frequency as the incident phonon. In the DMM model, the reflection coefficient r ji in a material is equal to the transmission t ij from the other material[116]. Based on the energy conservation principle, t ij = 1 -t ji[116]. The AMM model appears to be more appropriate to be used at low temperatures while the DMM model is more used at temperatures higher than the ambient[116, 117]. Therefore, we used the DMM as a model to estimate t ij from our experimental results. C pa and ν pa are respectively the heat capacity and group velocity of acoustic phonons.The thermal boundary resistance linked to phonons R bp is expressed as [116]:

	Chapter 4 -Following the phonon dispersion of both materials in contact, the transmission coefficient
	from material (1) to material (2) t 12 is expressed in the DMM model as [116]:
	12 t	=	C	pa	,1 ν C pa pa ,1	,2 ν C pa pa ,2 ,2 ν +	pa	,2	(4.7)
	12 t C = = ,1 ,1 4 bp pa pa R ν	21 t C	,2 4 pa ν	pa	,2	(4.8)

where

Table 5 .

 5 1 gives the values of the thermal properties of silicon determined in literature at different levels of temperature. The value of d 0 was determined equal to 2329 kg.m -3 for highly pure silicon at a temperature of 298 K [145].

Table 5 .1. Thermal properties of pure silicon at different temperature levels.

 5

  Data calculated with Equation (5.2) and values of k and C from Table5.1.

	300	703	148	9.03	8.8
	400	760	105	5.93	5.33
	500	800	80	4.29	3.8
	a

Table 5 .2. Measured properties of the wires.

 5

	Section S (m²)	ρ (Ω.m) at T a Electrical resistivity	Coefficient of temperature β ( K -1 )
	1.66 ± 0.15 × 10 -12	4.15 ± 0.1 × 10 -8	8.8 ± 0.05 × 10 -4

  R Au/Cr, R Cr and R Cr/Si are the thermal resistances at Au/Cr interface, within the Cr layer and at Cr/Si interface respectively. R Cr is equivalent to d Cr /λ Cr , where d Cr is the thickness of the chromium layer and λ Cr is the chromium thermal conductivity. This resistance is about 10 - 10 m 2 .K.W -1 . R Au/Cr and R Cr/Si are dependent on the interface geometry, the materials properties, the process of deposition of the wires on the substrate, etc… However, the values of the thermal resistances found in the scientific literature for metal/semiconductor lie in the range of 10 -8 m 2 .K.W -1 . They include the determination of R th at Au/Si interface with transient thermo-reflectance [120, 150], with picosecond optical technique [118], and with simulations of molecular dynamics at Si/Au [151] and Si/Pb [152] interfaces. Besides,Gundrum [153]

  1 .K -1 respectivement. La seconde technique développée est une méthode d'analyse thermique moins locale nécessitant l'instrumentation de la surface de l'échantillon avec un réseau de sondes résistives filiformes. L'un des fils du réseau, chauffé par un courant alternatif à la fréquence f, a le rôle de source excitatrice continue et à la fréquence 2f de l'échantillon. Les autres fils sont des capteurs de la température (méthode «2ω»). Un modèle analytique 2D, basé sur le principe des ondes thermiques et développé pour identifier les propriétés thermiques d'échantillons anisotropes est présenté. Ce modèle peut être réduit à un modèle simplifié 1D dans le cas de matériaux homogènes et isotropes. Des simulations par éléments finis et avec ce modèle ont été utilisées pour dimensionner le montage expérimental et valider la méthode sur un échantillon de silicium pur. Les résultats obtenus à des températures de l'échantillon variant de l'ambiante à 500 K corroborent ceux de la littérature. Weaver Jonathan (Rapporteur), Kolosov Oleg (Rapporteur), Volz Sebastian, Trannoy Nathalie, Bourgeois Olivier, Gomès Séverine, Lefèvre Stéphane, Vaillon Rodolphe

MOTS-CLÉS

: méthodes thermiques, microscopie thermique à sonde locale, transfert de chaleur aux micro-et nano-échelles, conductance thermique, conductivité thermique, coefficient de transmission des phonons, méthode à fils résistifs déposés, ménisque d'eau Laboratoire (s) de recherche : Centre d'Energétique et de Thermique de Lyon (CETHIL) Directrice de thèse: Gomès Séverine Président de jury : Volz Sebastian Composition du jury :

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0001/these.pdf © [A. ASSY], [2015], INSA de Lyon, tous droits réservés

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0001/these.pdf © [A. ASSY], [2015], INSA de Lyon, tous droits réservés 1 Chapter 2 -Analysis of heat transfer in the water meniscus at the tip-sample contact in SThM

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0001/these.pdf © [A. ASSY], [2015], INSA de Lyon, tous droits réservés Chapter 3 -Probe-sample thermal system in DC SThM investigated with the Wollaston probe

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0001/these.pdf © [A. ASSY], [2015], INSA de Lyon, tous droits réservés Chapter 4 -DC SThM with probes of nanoscale radius of curvature

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0001/these.pdf © [A. ASSY], [2015], INSA de Lyon, tous droits réservés 5 Chapter 5 -Development of a technique for investigation of thermal properties of solid materials at temperatures above the ambient

Acknowledgements

Chapter 3 -Probe-sample thermal system in DC SThM investigated with the Wollaston probe Let α am and α Φ the values of thermal diffusivity respectively extracted through the amplitude ΔT 2ω and the phase lag Ф. Through the parameter A 1 of the exponential tendency of ΔT 2ω (see Equation (5.42)), λ can be determined directly. a Temperature experimentally determined using Equation (5.40) at the probing wires.

Case of anisotropic samples

The next part includes numerical and analytical simulations in order to determine the thermal properties of anisotropic materials.

Once the experiments are performed and the experimental results are obtained, a numerical optimization technique is needed to determine the thermal properties of the thin solid films when using the 2D analytical model. One of the techniques that can be used and that is developed in CETHIL is the Particle Swarm Optimizers (PSO). The approach has been used in various scientific communities: combustion, pure applied mathematics….

In plane thermal conductivity

Until this part, the study was made to determine of thermal properties of isotropic samples.

We extend our study in this section to include bulk samples of anisotropic thermal properties.

Here, we consider two extreme cases: a) when the thermal conductivity in the x-direction λ x is much larger than the one in the z-direction λ z (λ x >> λ z ) and b) when λ x << λ z . These two cases are not realistic and therefore determination of thermal properties of anisotropic materials will lie within these two cases.

The dimensions of the sample are taken the same of the isotropic Si substrate used in the experiments before. The head can be changed depending of the application.