Abdo Iskandar

Abdo Iskandar Phonon Heat

Technologie De

Troyes De

: Spécialité

MECANIQUE Materiaux

Maitre Y De

D E Recherche

Cnrs Président

M A Bruyant

Conferences -Hdr Examinateur

ASSOCIATE PROFESSOR Directeur de M M Kazan

M G Lérondel

Professeur Des

Universites Directeur De Thèse

M V Lysenko

M N Semmar

Universites Rapporteur

FULL PROFESSOR Directeur de thèse M. S. VOLZ DIRECTEUR DE RECHERCHE M M Tabbal

Personnalité Invitée

M L Divay

Ingenieur De

Recherche Thales

Phonon Heat Transport and Photon-phonon Interaction in Nanostructures

To my father…

My sincere gratitude goes to the Central Research Science Laboratory (CSRL) crew, particularly to Mr. Joan YOUNES and Mr. Chadi ASSAF

List of Figures

 [START_REF] Xu | Length-dependent thermal conductivity in suspended singlelayer graphene[END_REF]Schematic of a monatomic linear chain that consists of atoms of mass m, separated by a distance " a " and interacting with a harmonic potential and force constant C . .............................................................................................................. 27 m , separated by a distance " a " and interacting with a harmonic potential of force constant C . ................................................................................................. 30 (right). Dots are experimental data points; solid curves are Mie theory calculations [START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF]. (E) Experimentally measured forward (green) and backward (blue) scattering spectra and forward-to-backward ratio (orange) of a near-spherical Si nanoparticle with a diameter of about 15 nm placed on a glass substrate (a.u., arbitrary units). Insets show transmission (F) and reflection (B) dark-field microscope images and a SEM image of the nanoparticle [START_REF] Schuller | General properties of dielectric optical antennas[END_REF]; scale bar for SEM image, 500 nm. [START_REF] Ziman | Electrons and Phonons[END_REF] [START_REF] Schuller | General properties of dielectric optical antennas[END_REF] Dielectric nanoantennas. (A) Magnetic and electric near-field enhancement in a nanogap between two silicon nanocylinders (height, 15 nm; diameter, 140 nm; gap, 30 nm). Top: Schematics of the measured system; p and m indicate electric and magnetic dipole moments, respectively. Bottom: Scanning near-field optical microscopy images for the two orthogonal excitation polarizations [START_REF] Bakker | a gnetic and Electric Hotspots with Silicon Nanodimers[END_REF]. Arrows indicate polarization direction for electric (E) and magnetic (H) fields. Color varies from minimal (blue) to maximal (red) values. Dashed circles mark the positions of silicon nanocylinders. Inset shows a SEM image of the studied dimer nanoantenna. (B) Magnetic Fano resonance in alldielectric oligomers [START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by ag netic Response[END_REF]. Top: SEM images of fabricated heptamer for different diameters (d) of the central Si disk. Bottom: Theoretical and experimental demonstrations of the magnetic Fano resonances. Blue, green, red, and cyan curves cor-respond to different diameters of the central Si nanodisk: 400 nm, 380 nm, 350 nm, and 320 nm, respectively. Arrows indicate the spectral positions of the Fano resonance. (C) Nonradiating anapole mode in silicon nanodisks [START_REF] Hayashi | Evidence for surfaceenhanced Raman scattering on nonmetallic surfaces: Copper phthalocyanine molecules on GaP small particles[END_REF]: Experimental dark-field scattering spectra of an amorphous Si nanodisk (diameter, 310 nm; height, 50 nm) on a quartz substrate. A strong dip in the nanodisk scattering at the wavelength of anapole excitation is clearly observed. Inset shows experimental near-field scanning optical microscopy images of the anapole mode at three different wavelengths (640, 650, and 600 nm) near the scattering minimum. White dashed circles indicate the nanodisks' positions. (D) Nonlinear spectroscopy of an array of Si nanodisks [START_REF] Sigalas | Electric field enhancement between two Si microdisks[END_REF]. Shown is the negative logarithm of the normalized transmission spectrum of a sample marked by a gray area indicating a resonance at 1.24 nm. The third-harmonic generation (THG) spectrum of the sample normalized over the spectrum of the substrate (purple dots) is strongly enhanced within the spectral band of the resonance. The inset shows a schematic of the THG at a single silicon nanodisk. [START_REF] Ziman | Electrons and Phonons[END_REF] [START_REF] Schuller | General properties of dielectric optical antennas[END_REF] A simplified picture of the Dynacool. In green is the bucket with the liquid helium which flows to the pot in the bottom of the annulus. Around the sample chamber flows helium gas. In the bottom of the sample chamber the puck is placed in the isothermal region, under which a block heater is attached for temperature control. To achieve high vacuum, the sample chamber is exposed to a two stage cryopump. [START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF] [START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF] The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas. [ 3) Repetitions at each temperature: At each temperature point the specific heat is measured three times in order to check stability and reproducibility. The results are not averaged. 4) Relaxation measurement: The measurement time is always set to  2 . During this time data acquisition takes place. 5) Determination of initial time constant: In general determined by applying a trial heat pulse, or, if known, it can be fixed. 6) Measurement stability: A heat pulse will only be applied when the temperature is within 1% of the requested tem-perature rise. 7) Extra wait time: When measuring at high temperatures it is necessary to wait longer before the measurement starts. [START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF] [START_REF] Luk'yanchuk | ptim um Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Inde[END_REF] The heat capacity of the puck with 1 mg of grease (blue circles) and of the puck with 0.3 mg grease (yellow squares). The heat capacity of the empty puck is measured twice (red diamonds and green triangles). [START_REF] Stewart | e asurement of low-temperature specific heat[END_REF] ............................ 71 Figure II. [START_REF] Evlyukhin | ul tipole light scattering by nonspherical nanoparticles in the discrete dipole appro imation[END_REF] The average specific heat of Apiezon N Grease (solid blue line) calculated from two addenda measurements and two empty measurements (four different combinations). The uncertainty in the mean is indicated by the colored area around the average curve. The inset shows the low temperature behavior. [

List of Tables

Table III.I Detailed structural description and the obtained parameters of the crystallites size distributions in the surface layer of the Si resonators in the samples investigated. The subscripts S, , and L refer to "short", "medium", and "long", respectively ...

General Introduction

Significant enhancements in device fabrication and integration in the electronics industry have led to a rapid increase in the total power consumption in electronic circuits. Transistors, which form the basic building blocks for microprocessors, for example, have been reduced to a few nm within four decades. This has allowed manufacturers to pack more of them onto a single die. One of the beneficial aspects of this higher density packing is a reduction in signal propagation delays, permits higher clock-frequencies, and as such better performance. If we take microprocessors for example, today's processors are faster and cheaper than older ones by several orders of magnitude owing to the exponential growth in transistor density. The scaling has surpassed the growth predicted by Gordon Moore's empirical law stated in 1965 [START_REF] Oore | ramming more components onto integrated circuits[END_REF], which stated that processor transistor density would double every year.

However, in his paper, Moore also indicated that it would also be possible to cool integrated circuits efficiently due to the relatively large surface area of integrated circuits, and the limited number of components that need to be driven. Dennard scaling justifies this by indicating that to keep the electric field constant in a reduced transistor, the voltage would have to be reduced as well thus reducing the power consumption [START_REF] Dennard | Design of ion-implanted S FET's with very small physical dimensions[END_REF]. In fact, today this scaling has broken down, and we know that this is no longer true. In fact, voltage leakage which leads to heating is the most acute problem facing the advancement of current integrated circuit technology [START_REF] Mcmenamin | ast c ells are present in the choroid of the normal eye in most vertebrate classes[END_REF]. 50% of the energy consumed by a microprocessor is dissipated as heat due to its electrical resistance and/or current leakage. This leads to negative effects on the lifetime and performance of these microprocessors. In fact, the surface density power of microprocessors was approaching that of nuclear plants.

If we note the omnipresence of electronics and microprocessors in daily life, it becomes apparent that improving the efficiency and giving more thought into controlling the heat losses in these devices is a crucial need. The reduction of heat damage to electronic components would lead to increased lifetime, less waste energy, and perhaps even the transformation of heat waste into electrical energy.

Current thermal management or thermal protection methods are at a macroscopic scale and package level. Heat sinks, fans, and other devices attempt to control the temperature of the final system by coupling to the external environment. This limits the effectiveness to the efficiency of the coupling between the environment and the integrated circuit. It thus requires the use of heat spreaders, heat pastes, and other metal contacts. However, the heat itself is generated at the nanoscale or at the interfaces between the various components. Hence, in order to better improve heat management of these integrated circuits, we must deal with the issue at the microscopic/device level or at least understand the mechanisms behind heat transfer. Phonons, which are the representations of lattice vibrations in solids are the conductors of sound and heat. Understanding and controlling phonons is the first step in thermal management and impacting current technology in a way that was achieved by controlling charge and light transport.

Indeed, it is only when we are able to effectively utilize heat through phonons as an energy source, that one of the world's key questions, the dependence on fossil fuels, could also be resolved. In a paper presented to MRS bulletin [START_REF] Smalley | Future Global Energy Prosperity The Terawatt hallenge[END_REF], a possible plan to utilize solar energy was envisioned by placing solar power plants in strategic locations around the Earth. However, as the author indicates, the current state-ofthe-art technologies involving solar power do not permit a cost-effective generation of solar-electrical power to meet current energy demands. On the other hand, if we were able to develop solutions to control/stop electron-heat relaxation channels in semiconductors then production of more efficient solar cells can be achieved. In this case, the goal is to reduce the thermal conductivity rather than to increase it as in the previous cases. With this aim in mind we still must be able to understand and control the flow of heat in a semiconductor.

In this dissertation, which has been develop in the frame of a scientific collaboration between the University of Technology of Troyes (UTT) and the American University of Beirut (AUB), we tackle the issues described above and investigate the phonon heat transport and the phonon interaction with optical elementary excitations in nanostructures. The present dissertation comprises five chapters.

In the first chapter, we present an introduction to the physics of phonons and some optical elementary excitations of matter. We describe some resonances and thermal energy transport modes. We also give a brief introduction on phenomena that take place in low-dimensional systems and make the properties of the low dimensionalsystems different from their bulk counterparts.

The second chapter provides a detailed description of the samples growth and fabrication procedures as well as the various characterization techniques used. In particular, we describe in details the operation and physics of the heat pulse calorimeter. The reason is that we used this calorimetric technique to measure nanostructures properties inaccessible by existing techniques.

In the third chapter, by studying the Raman shift and line shape, we demonstrate that phonons and photons of different momenta can be confined and interact with each other within the same nanostructure.

In the fourth chapter, we present experimental evidence on the change of the phonon spectrum and vibrational properties of a bulk material through phonon hybridization mechanisms. We demonstrate that the phonon spectrum of a bulk material can be altered by hybridization mechanisms between confined phonon modes in nanostructures introduced on the surface of a bulk material and the underlying bulk phonon modes. We also show that the shape and size of the nanostructures made on the surface of the bulk substrate have strong effects on the phonon spectrum of the bulk material.

In the fifth chapter, we demonstrate that at low temperatures (below 4 K) the nanowire specific heat exhibits a clear contribution from an essentially twodimensional crystal. We also demonstrate that transition from specular to diffusive elastic transmission, and then from diffusive elastic to diffusive inelastic transmission occur at the interface between nanowires and a bulk substrate as temperature increases.

The manuscript ends with a general conclusion and perspectives with as one of the main perspectives the control of bulk material thermal properties via surface nanostructuring.

: Introduction to phonon heat transport and Chapter I elementary optical excitations in nanosized materials

I.1. Heat transfer and phonons

The interest and study of the confinement and transfer of energy at the atomic scale has experienced a massive expansion with the emergence of nanotechnologies in the 1990s. In this introduction we will briefly explain how this area of research has evolved by showcasing its essential discoveries in the history of nanoscience.

At the beginning of the twentieth century, several key scientific discoveries have altered our understanding of matter. The discoveries of electrons and nuclei by Thomson and Ruther ford respectively, helped to understand the structure of atoms. In addition, around that time, Quantum Mechanics was born.

The latter sparked a debate, particularly because of the notion of probability that it introduces to explain physical phenomena. These theoretical revolutions helped to understand the behavior of matter and its basic components (electrons and nuclei). The energy levels quantization was one of the first hypotheses of Quantum Mechanics that helped explaining different physical phenomena such as: the blackbody radiation and the presence of discrete lines in the spectrum of the atoms. The particle/wave duality, introduced by De Broglie, revealed that all of the physical entities (bosons and fermions) can behave either like waves or like particles. A relevant example of this concept is the interference phenomenon from a particle beam consisting of molecules of fullerenes, for example, which are molecules of carbon in the form of a hollow sphere, ellipsoid, tube and many other shapes. Many other non-intuitive hypotheses and results were discovered afterwards such as the Heisenberg Uncertainty Principle, quantum states, tunnel effect.

In 1959, a famous lecture was given by Richard Feynman, at an American Physical Society meeting at Caltech, in which he stated "There is plenty of room at the bottom", underlining the possibility of direct manipulation of individual atoms and therefore the potential might of nanoscience in the future. Many technologies to probe the infinitesimally small materials were developed, such as the Scanning Tunneling Microscopy and the Atomic-Resolution Electron Microscopy, which made the manipulation of individual atoms possible.

These technological breakthroughs, along with the development of new methods of synthesis smaller and complex materials, have spread out to the industrial world, and paved the way for high-tech fields such as microelectronics and more recently nanotechnology.

The continuous miniaturization of electronic components has allowed even more integration of transistors, having a size as low as 10 nm.

Nevertheless, several technological limits have recently emerged. The first one is regarding the small size of the transistors, giving rise to undesirable quantum effects (such as tunnel effect). The second obstacle is that the improvement of the performance of the microelectronic systems was accompanied by a continuous increase of the clock frequency which reached a limit of 3 GHz, in 2004. The reason behind this originates from the fact that the amount of heat flow generated by the electronic components becomes too large to be properly dissipated, which results in an irreversible damage.

The problems described above led to the foundation of the study of energy transfer in sub-micrometric systems. At this scale, the volume/bulk effects become negligible in front of the surface effect, which greatly affects the properties of the nanosized materials. Throughout this thesis, many applications related to the thermal and phonon phenomena at the nanoscale which have emerged in various fields such as optics, electronics, and biological, will be presented.

On the other hand, nanophotonics emerged as a frontier research to exploit the interaction of nanoscale phenomena with electromagnetic waves and the nanoscale confinement and guidance of these waves. Extensive research has been conducted to exploit in particular the confinement and guidance of electromagnetic energy at the surface of a material. The confinement and guidance of electromagnetic energy on the surface of metals in the form of surface plasmon waves has attracted wide interest, but unfortunately, losses due to plasmon waves scattering are significant, making light confinement and guidance at the surface somehow problematic.

In a polar material, the frequencies of the optical phonon modes at the zone center, are split into longitudinal and transverse optical components by the internal macroscopic electric field. This macroscopic electric field serves to stiffen the force constant of the phonon and thereby raise the frequency of the longitudinal optical

  

Re is negative. This has the consequence that, within the Reststrahlen band, electromagnetic plane waves acquire an evanescent character giving rise to modes confined at the surface. These surface modes are known as the surface phonon-polariton (SPhP) modes. Hence, the SPhP can be regarded as surface electromagnetic waves arising from the coupling of electromagnetic modes (photons) with lattice vibration modes (optical phonons) in polar dielectric materials. Relevant research work suggests that the lifetime of SPhP is orders of magnitude larger than that of surface plasmon. This makes SPhP a potential candidate for enhancing the lifetime of electromagnetic energy confined at the surface, contributing to many technological applications in the field of nanophotonics. As such, it has been demonstrated that the SPhP modes are the dominant energy carriers in the near-field and their diffraction by gratings introduced on the material surface produces coherent infrared emission in the farfield.

This introductory chapter aims at presenting the essential concepts for a better understanding of the study of the energy confinement and energy transfer where the quantum of lattice vibration (phonon) plays a key role.

I.1.1. Heat transfer in matter

The understanding of the fundamentals of heat transfer concerned the scientists since the dawn of time. In antiquity, the Greeks were trying to explain the world by referring to the four basic elements: water, Earth, fire and air. The fire being the element responsible for the heat. Later in history, the Phlogiston theory was the standard theory to explain the combustion until the eighteenth century. In this model, the combustible material supposedly contained a substance called phlogiston, which was released during the combustion resulting in a loss of the material's mass. Lavoisier questioned this theory in the second half of the 18 th century, particularly with the discovery of oxygen which is an essential element in the phenomena of combustion. This period witnessed the birth of the field of thermodynamics, first formalized by Sadi Carnot in 1824 when he established a link between the work supplied by a heat engine and the heat transport. Other wellknow scientists also contributed to the field of thermodynamics such as Lord Kelvin, Clausius, and Joule.

The empirical law introduced by Joseph Fourier, which bears his name, linked the heat current density with the difference of temperature across system. In addition, Boltzmann formalized the kinetic theory of gases by describing the macroscopic properties of gas molecules using the equation that also bears his name. Furthermore, it was not until the end of the nineteenth century that thermal radiation started to be understood. Slovenian physicist, Josef Stefan, discovered that the power radiated by a body is proportional to its temperature to the power 4.

Wien and Rayleigh determined the spectral distribution of the radiated energy as a function of the material's temperature. At the beginning of the twentieth century, Planck made the assumption that the blackbody's energy levels are discrete, which allowed to establish the law of the radiated power by a blackbody as a function of the wavelength.

This historical summary gives a global overview regarding the development of the heat transfer laws as well as the evolution of the scientific reasoning over the centuries. We are going now to present these macroscopic transfer laws along with their limitations, particularly when the system has a characteristic size of the order of a few tens of nanometers.

I.1.1.1.The macroscopic laws of energy transfer and their limits

Heat transfer can be divided into three different modes: thermal conduction, convection and radiation. We will now overview each one of these modes and discuss their validity at the Sub-micrometer scale [START_REF] Volz | Microscale and nanoscale heat transfer[END_REF].

A. Energy transfer by conduction

The thermal conductivity corresponds to an exchange of energy without having any macroscopic motion of the system. Fourier stated the empirical law that relates the density of the heat flux j  to the temperature gradient that is present in the system as:

T j       (I.1)
Where is the thermal conductivity of the material. By combining the Fourier law with the conservation of energy equation, we can infer that the temporal evolution of temperature can be written as:

T c t T p       (I.2)
With  being the density of the material and p c its heat capacity at constant pressure. The thermal diffusivity D is defined by the ratio p c   . The latter is related to the ability of the material to diffuse energy by conduction.

Fourier's law assumes instantaneity between the e citation (temperature difference) and the response of the system (heat flux). This is valid when the characteristic time of observation is very long in comparison with the system's characteristic response time. In order to confirm that, it is important to know the physical origin of this conductive heat flux. The energy is transported from the hot source to the cold one by means of phonons and free electrons (in the case of a metal) in the context of heat conduction.

These energy carriers are transferred from the hot source to the cool one by undergoing collisions. The latter corresponds to various types of interaction that would alter the transport of energy in the system. The relaxation time is defined as the average time that elapses between two collisions that an energy carrier would undergo. The relaxation time is in general in the order of a picosecond for the phonons and tens of femtoseconds for the electrons.

If the characteristic time of the phenomena studied is close to the relaxation time, then Fourier's law can no longer be applied and Boltzmann's equation must be considered instead. The latter takes into account the time evolution of the energy carrier's distribution. That same analogy is valid if the system's size is of the same order of magnitude as the mean free path of energy carriers, corresponding to the average distance that they would travel between two collisions. Figure I.1 shows the two possible transport regimes. When the mean free path  is small in comparison with the system's size L , the energy carriers undergo many collisions, and the transport regime is said to be diffusive, and hence validating Fourier's law. However, when

L  

, the edge and interface effects limit the transport of energy carriers, and therefore we are under a regime of reduced collisions. This type of transport is named as ballistic regime. Fourier's law loses its validity at these small scales.

In addition, when the dimensions are small, the thermal conductivity is no longer an intrinsic characteristic of the material but rather strongly depends on its geometry. This is illustrated in Figure I.2 which shows the variation of the thermal conductivity of a suspended monolayer of graphene as a function of the material size [START_REF] Xu | Length-dependent thermal conductivity in suspended singlelayer graphene[END_REF]. The mean free path in graphene is very large in comparison with the lengths considered in this study, which explains the logarithmic growth of thermal conductivity as a function of L . Indeed, the phonons undergo very few collisions in this regime and propagate directly from the hot source to the cool one. This has also been observed on more conventional materials such as Silicon thin films. These examples conclude the presentation of transport by conduction. We will now address the transport by convection.

B. Energy Transfer by Convection

Unlike transfer by conduction, energy transfer by convection is associated with a macroscopic movement within medium. It takes place in fluids in which the molecules can freely move. Only the intermolecular collisions can alter their mean free path.

There are different types of convection. In the case of natural convection, the forces of gravity are responsible for the dynamics of the fluid due to the density gradients in the fluid, whereas in the case of forced convection, a collective movement of ensembles of molecules is imposed on the fluid, which greatly increases the thermal transfer performance between the surface and the volume of the fluid. If the dimension of the fluid system becomes insignificant in comparison with the mean free path of the molecules, then the regime is said to be ballistic. Since energy transfer in fluid is beyond the scope of this manuscript, it will not be discussed in future details.

C. Energy Transfer by Radiation

Any body that is characterized by a non-zero temperature emits and absorbs electromagnetic radiation. In fact, each material consists of positive nuclei charged positively and of electrons which are charged negatively. Therefore, there exist dipoles which oscillate under the effect of temperature in the system and allow the absorption or emission of radiation. Contrary to conductive and convective transfers, radiative transfer being carried by an electromagnetic waves does not require the presence of a material medium to propagate.

The radiative energy flux through a surface dS in the direction of u  , in a solid angle  d , at a temperature T and within the frequency interval  

   d  , is defined by:          d d dS T r u L d   cos , ,    (I.3) Here,   T r u L    , ,
 is defined as the luminance and it corresponds to the radiative energy that crosses a surface dS at the point r  in the direction u  at a frequency  multiplied by the speed of light c . We will now discuss the blackbody concept, which is a theoretical object that absorbs all of the incident radiation and reemit it totaly. Once thermodynamic equilibrium is reached at a temperatureT , and then the luminance of a blackbody is written as:

  1 1 2 , , 3 2 0   T k h B e c c h h T r u L        (I.4)
Luminance is therefore defined as the product of the energy of a photon with the density of states in a cavity, the speed of light, and the Bose-Einstein distribution.

It is possible to infer from the luminance expression, the surface flux radiated by a blackbody   T  , which is known as the Stephan-Boltzmann law:

      4 0 0 cos , T d d dS r u L T             (I.5)
Here  is the Stephan-Boltzmann constant. As can be noticed, this quantity depends only on the temperature of the blackbody. However, when the system cannot be labeled as a blackbody, then the emissivity of the material  

u   
, which depends on both the direction of the radiation and the frequency must be considered. It corresponds to the percentage of energy that will be re-emitted by the material at equilibrium in comparison to that of a blackbody:

      T L T u L u e e     ,    (I.6)
The same analogy is valid for the absorbance of the material

  u    . Kirchhoff's
law relates these two quantities by the equality    

u u        .
Radiation is characterized by different characteristic lengths. The main one is its wavelength  . If  is very large in comparison with the dimension of the system L, then the delay effects can be neglected and the problem becomes essentially an electrostatics problem. However, when  is smaller than the characteristic size of the system, the electromagnetic wave cannot be considered as a plane wave and the electrostatic description of the thermal radiation breaks down. Generally speaking, the far field approximation serves only when the characteristic lengths are larger than the wavelength.

Very close to the surface, the equations of Maxwell predict non-propagative waves, which are known as evanescent waves. The radiative transfer of the energy carried by these waves is known as the near field. It is worth noting that the evanescent waves carry energy densities that are much greater than those of propagative waves [START_REF] Joulain | Surface Electromagnetic Waves Thermally Excited: Radiative Heat Transfer, oherence Properties and asim ir Forces Revisited in the Near Field[END_REF], as is shown in Figure I.3. This figure represents the energy density over a flat surface, separating a semi-infinite medium of glass at 300 K from vacuum at 0 K at different distances from the surface. When that distance is 100 µm, the spectrum is similar to that of a blackbody, therefore this corresponds to the far field approximation. However, if the distance is only 3 μm from the surface, an energy peak shows up at around THz 100   corresponding to a surface mode called the surface phonon-polaritons SPhPs. At a distance of 100 nm from the surface, the SPhPs modes dominate the heat transfer and carry the majority of the energy. This is due to its evanescent behavior which decreases exponentially as we move away from the surface. These near field modes introduce new methods of energy transport in a radiative way with high energy densities.

In conclusion, where the size of the systems is comparable to the characteristic lengths of energy carriers, the macroscopic laws of heat transfer are no longer valid and therefore new theories must be introduced.

I.1.2. Introduction to the physics of the phonons

Since this thesis deals with energy confinement and transfer through processes involving phonons, we will therefore introduce in this section the phonons, their properties and the mechanisms that govern their transport.

I.1.2.1. Phonons

The exact concept of phonon is present only in crystalline materials which exhibit a periodicity in their structure. If the translational symmetry is broken, as in the case of amorphous solids, soft matter, liquids, most of the properties of the phonon are no longer applicable and therefore new formalisms are needed to study the conduction phenomena in such materials. Before we focus on the characteristics of the phonon, it is essential to start with an introduction to the crystal structures and the properties that arise from the presence of a periodicity in a crystalline material [START_REF] Kittel | Introduction to solid state physics[END_REF]- [START_REF] Ziman | Electrons and Phonons[END_REF].

A. Periodicity of Crystal Structures and Brillouin Zone

A crystal lattice is characterized by at least translational periodicity in its structure. In perfect three-dimensional massive materials, the three directions of space exhibit a periodicity.

The Bravais lattice establishes the periodicity in the structure from a set of vectors. We note that the Bravais lattice is however not uniquely defined, as shown in The lattice nodes positions are therefore written as:

   3 1 i i i a n R   (I.7)
Where i n represent the integers. Thus, the potential energy associated with the interactions between the individual atoms of the system   r U  is invariant by a translation of vector R 

. Therefore, we can develop the potential energy in a Fourier series:

      k k r k i c r U      . exp (I.8)
The invariance by a translation of vector R 

implies the following relationship n R k  2 .   
, where n is an integer. Thus, it is possible to show that the set of allowed wave vectors is:

   3 1 * i i i a m k   (I.9)
Where * i a  represents the reciprocal lattice primitive vectors and are defined as:

cell cell cell V a a a V a a a V a a a 2 1 * 3 1 3 * 2 3 2 * 1 2 , 2 , 2                   (I.10)
Where cell V is the volume of a unit cell. In a way similar to the direct space, a translation by vector k  leaves the reciprocal lattice invariant. Thus, the first Brillouin zone, defined by the set of points of the reciprocal space which are the closest to a node, contains all of the system's phonons characteristics. For e ample, for a one-dimensional lattice of period a with a Bravais lattice We will now present the main characteristics of the phonons in the case of a onedimensional linear chain.

B. Phonons in a one dimensional model

The phonons are the eigen-modes of a crystal lattice vibration, and can be considered as waves of vibration characterized by an angular frequency  , a wave- vector k  and a polarization. In this part, we will focus on the calculation of the dispersion relations of phonons in the case of a simple system, in order to explain the associated physics without using a heavy formalism in notation.

Consider a monoatomic linear chain made up of one type of atoms only, separated by a distance " a " from each other.

              t u t u C t u t u C t t u m n n n n n          1 1 2 2 (I.12)
The phonons are considered as progressive planes waves and expressed as:  where it attains its maximum.

      kna i u t u n    exp 0 (I.
Typically, the frequencies of the thermal phonon vary between 1 THz and 100 THz. Thus, unlike the free electrons or photons, phonons have a wide range of frequencies, which renders the understanding of the thermal phenomena difficult to grasp. The phonons are characterized by a wave packet structure, which allows the definition of their group velocity g  as the derivative of the angular frequency with respect to the wave vector: m , respectively, are alternated. A distance " a " always separates every neighbor. The interactions between atoms remain identical to the previous case. Bravais lattice has now a period of a 2 , whereas the first Brillouin zone, which is inversely proportional to the periodicity in the direct space, is therefore   By analogy to the case of the monatomic linear chain, we can write

    k k k g       (I.
                            t u t u C t u t u C t dt u d m t u t u C t u t u C t dt u d m n n n n n n n n n n 1 2 2 2 2 1 2 2 1 2 2 1 2 1 2 1 2 2 2 2 2 1                 (I.17)
The displacement for each atom of the primitive cell can be written as progressive planar waves:

                  a n k t i k u t u na k t i k u t u n n 1 2 exp 2 exp 2 1 2 1 2         (I.18)
It is important to note here that the amplitudes of vibration are different for each atom of the unit cell. For each vibration mode, the two atoms do not vibrate in the same manner. Therefore, we are left with a system of two equations with two unknowns. . The first branch is called the acoustic branch. It is characterized, as in the case of the monatomic chain, by a zero frequency for a zero wave vector. The number of acoustic branches is equal to the dimensionality of the system. Again, in the case of long-wavelengths approximation, the speed of sound is found in the chain, as indicated in Figure I.9.b. The group velocity goes to zero again near the edge/boundary of the Brillouin zone, and therefore it corresponds to a standing wave. The highest branch is called the optical branch. The number of optical branches is normally equal to the total number of degrees of freedom in the primitive cell minus the number of acoustic branches. For these optical branches, the frequency is never zero, even for a zero wave vector. The group velocity goes to zero at the center and near the boundary of the Brillouin zone, and therefore its group velocity is negative, with an absolute value smaller than that of the acoustic branch. shows the amplitudes of vibration of the two atoms of the unit cell, for the acoustic branch. The amplitudes conserving the same sign signifies that the atoms vibrate in phase. The center of gravity of the unit cell undergoes a major shift enabling an efficient transport of energy through the system. We note that when we get closer to the edge of the first Brillouin, the light atom's contribution becomes less and less dominant, and eventually drops to zero near the edge of the Brillouin zone. This means that the energy associated with this mode cannot flow in the crystal.

In the case of the optical branch (Figure I.10.b.), the amplitudes of the two atoms have opposite signs: they vibrate out of phase. Thus, these modes allow a large variations of the dipole moment, which leads to some optically active modes. This time, it is the light atom that contributes the most to the optical modes. Now that we have introduced the fundamental properties of the phonons, we move on to the calculation of the dispersion relations in the general case of an arbitrary crystal.

I.1.2.2.Lattice Dynamics

The dynamics lattice is a numerical technique is used for the calculation of the dispersion relations of phonons in 3D crystals. The general idea of this method relies on the variation of the potential energy due to the displacement of atoms in the vicinity of their equilibrium position. We consider an arbitrary perfect crystal for which the positions of the unit cell l at equilibrium are noted l R 

  t R b 
is given by:

    t R R t R b l b l      (I.19)
The displacement of this atom is noted as  

t u b l  . The atoms of the crystal interact via a potential       t u U b l 
This potential energy can be expanded in a Taylor series as a function of the atomic displacement with respect to the atoms equilibrium position. The expansion will involve the first 2 orders only, which means that the interactions are only harmonic:

      t u t u u u U t u u U E E b l b l b l b l b l b l b l b l b l tot                                           0 , , , , 2 0 , , 0 2 1
(I.20)
Where  and  are the directions in direct space (x, y or z) and the 0 index refers to the values at equilibrium. The first term of the expansion is zero because at equilibrium the forces are zero. We write down again the atomic displacements in the form of progressive planar waves:

      l b k b b l R k t i u m t u      . exp 1    (I.21)
Therefore, the dynamics equation for this atom is:

        b l b l b l b l b l b l b u u u U u U t u m                         0 , , 2   (I.22) Where    b l b l b l u u u U                0 2
corresponds to the element of the force constants matrix

                      , , , 2 . exp b l l l b l b b b l lb b k R R k i u m m C u     (I.23)
It is possible to simplify this equation by considering the translational symmetry of crystal. In fact, the force constant depends only on the relative distance of the l and l unit cell:

    b l l b b l lb C C       ,
l l l    
we get the following:

                   , , , 0 2 . exp b l l b l b b b l b b k R k i u m m C u    (I.25)
Now introducing an essential quantity in Lattice Dynamics, the dynamic matrix

  k D b b    ,
, which is the spatial Fourier transform of the force constants matrix:

              l l b b b l b b b R k i m m C k D    . exp , 0 ,   (I.26)
The system of m N 3 equations of Lattice Dynamics that should be solved is therefore:

    0 , , , 2 ,               b k b b b b b u k D   (I.27) With 1 , ,      b b if    and b b   , otherwise it is Zero.
Therefore, it is enough to diagonalize the dynamic matrix for each value of the vector wave k  in order to get the m N 3 eigenvalues corresponding to the square of the angular frequency of each mode. Due to the symmetry of the dynamic matrix, the eigenvalues can be real, while the eigenvectors which correspond to the amplitude of vibration of each atom for a given mode can be orthonormal. Having two atoms in its elementary unit cell due to its diamond structure, 6 branches are present: three acoustic branches and three optical. Unlike the one-dimensional case in the previous section, the phonons have a polarization with respect to the wave-vector that dictates the direction of propagation of the wave. Thus, the phonons can be longitudinal if the atoms are moving, for a given mode, in the same direction as k  , otherwise, transverse phonons. For the acoustic branches, the longitudinal branch has always a higher frequency than that of the transverse branches.

I.1.2.3.Phonons Anharmonicity

We explained in the previous section, the lattice dynamics approach. This technique takes into account the harmonic part of the potential solely. Phonons were therefore considered as independent of each other, meaning that they do not interact with each other, which gave them an infinite lifetime. Thus, the thermal conductivity of a purely harmonic material is infinite because the slightest disturbance to the system that would excite certain modes cannot be relaxed. In reality, phonons do not have an infinite lifetime; this time is determined by phonon/phonon, phonon/electron, and phonon/defect collisions. Each of these processes affect the phonons in different ways. A complete overview of these phenomena is given in the reference [START_REF] Srivastava | The physics of phonons[END_REF]. We consider, in this case, a system without imperfection, and we take into account the processes of diffusion among phonons, solely. To take into consideration the phonon/phonon interactions, then the terms of order three or higher, in the Taylor expansion of the potential energy, must be examined (equation (I.20)). For example, the observed inelastic processes, at the third order, are outlined in Figure I.12.

k      ,  (process a), or a phonon   k  ,  can be divided into two phonons   k   ,  and   k      ,  (process b).
These processes of diffusion, formulated by Peierls [START_REF] Peierls | ur kinetischen Theorie der Wärmeleitung in Kristallen[END_REF], must respect the principles of conservation of energy and those of pseudo-pulse, the wave-vector of a phonon being always defined with respect to a vector of the reciprocal lattice.

These processes can be summarized as following:

       ' G k k k           (I.28) Where G  is a vector of the reciprocal lattice. If 0    G
, the process is called normal. It creates a phonon, which its wave-vector approaches the edge of the Brillouin zone. In the opposite case, the sum of two wave-vectors surpasses the edge of the Brillouin zone, and the process is called Umklapp process. They have different impacts on the thermal conductivity. The life time of the phonons is an essential quantity in order to calculate the intrinsic thermal properties. This time, when multiplied by the group velocity of the phonon, it led to the mode's mean free path, in other words to the average distance that it can cover before undergoing a collision.

I.2.Optically Resonant Dielectric Nanostructures

The study of light scattering by nanoparticles has been a crucial part of the optical science field, with plasmonic nanoparticles consisting the corner stone in this study. These nanoparticles, in particular those that are made of gold or silver, supply the resonant optical modes with an ability to manipulate light at the nanoscale, which led to the development of a diverse set of applications such as: biosensors, thermotherapy, solar cells and information storage [START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF]. However, not all of these plasmonic applications were successfully realized in practice, due to high losses of metals at visible frequencies, in addition to their incompatibility with complementary metal oxide semiconductor (CMOS) fabrication processes.

Alternatively, the optical resonances of high-index dielectric and semiconductor nanostructures can also provide light manipulation below the free-space diffraction limit. These dielectric structures minimize the optical losses, while providing a wealth of distinct optical resonances, and the ability to manipulate the resonances by electrical doping. Furthermore, these dielectrics and semiconductors are formed of materials that are compatible with semiconductor device technology. For these reasons, we will analyze in the next how such high-index dielectric structures can be used as building blocks for real meta-devices and novel structures, by reviewing the basic optical properties of these nanostructures that are made of resonant highindex nanoparticles.

I.2.1. Mie Resonances in Sub-wavelength Particles

In order to showcase the fundamental properties of light scattering by nanoparticles, the study will be conducted on a spherical particle illuminated by a plane wave.

Based on the Mie theory, and as seen in Figure I.13.A, both metallic and dielectric spherical particles can possess strong scattering resonances. For a fixed size parameter q , which is proportional to the ratio between the nanoparticle radius and the wavelength of light  

   R q 2 
, the main difference between metallic and dielectric particles is in the sign of the dielectric permittivity, which is negative for metals and positive for dielectrics. As seen in Figure I.13.A, the magnetic response of small metallic spheres  

1  q
remains almost negligible because the field vanishes inside the sphere, while they produce only localized surface plasmon resonances of an electric type-dipole, quadrupole etc. Noting that for a magnetic response to be generated from metallic structures, the particle's geometry should be altered.

Furthermore, Figure I.13.A shows both electric and dielectric-type responses of comparable strengths for dielectric particles, with the resonant magnetic dipole response results from a coupling of incoming light to the circular displacement currents of the electric field, owing to the field penetration and phase retardation inside the particle. The latter phenomenon occurs when the wavelength inside the particle becomes comparable to the particle's diameter shows the field structure of the four major resonant modes in highindex dielectric particles: magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole. According to the Mie theory, the maximum achievable scattering efficiency for a specific multipolar excitation of a subwavelength particle depends only on the resonance frequency and not the type of material, which suggests that various plasmonic effects observed for the scattering of light by metallic nanoparticles can be realized with high-index dielectric nanoparticles [START_REF] Schuller | General properties of dielectric optical antennas[END_REF]. , all main multipoles are well defined, and their spectral positions correspond to a fixed ratio of the wavelength inside the particle, n  , to its diameter, R 2 . The scattering efficiency of all multipoles also increases with increasing n [17]- [19].

Achieving optically induced magnetic dipole resonances in high-index dielectric nanoparticles is not limited to spheres but can also be detected for spheroids [START_REF] Fu | Directional visible light scattering by silicon nanoparticles[END_REF], [START_REF] Luk'yanchuk | ptim um Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Inde[END_REF], disks and cylinders [START_REF] Evlyukhin | ul tipole light scattering by nonspherical nanoparticles in the discrete dipole appro imation[END_REF], rings [START_REF] Van De Haar | ontrolling magnetic and electric dipole modes in hollow silicon nanocylinders[END_REF], and many other geometries [START_REF] Heludev | Near-infrared trapped mode magnetic resonance in an all-dielectric metamaterial[END_REF]. This diversity provides opportunities for designing a variety of all-dielectric nanostructures with desirable spectral positions of the resonances. In addition, the spectral positions of both electric and magnetic dipole resonances can be tuned independently by changing the geometrical parameters of the particles [START_REF] Luk'yanchuk | ptim um Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Inde[END_REF], [START_REF] Evlyukhin | ul tipole light scattering by nonspherical nanoparticles in the discrete dipole appro imation[END_REF], [START_REF] Staude | Tailoring Directional Scattering through agnetic and Electric Resonances in Subwavelength Silicon Nanodisks[END_REF].

I.2.2. Observation of Optical Magnetic Resonances in Dielectric Nanoparticles

In this section, we report an example of experimental study of magnetic resonances in dielectric particles at optical frequencies (mid-infrared IR) for silicon carbide microrods [START_REF] Schuller | Dielectric Metamaterials Based on Electric and Magnetic Resonances of Silicon arbide Particles[END_REF].

Furthermore, that same experimental observation was done on dielectric and semiconductor mircrorods and nanorods, which exhibited scattering resonances in the visible and near-IR spectral range [START_REF] Fan | Tuning the ol or of Silicon Nanostructures[END_REF], [START_REF] Seo | ul ticolored Vertical Silicon Nanowires[END_REF].

It was reported in various works [START_REF] Evlyukhin | ptica l response features of Si-nanoparticle arrays[END_REF], [START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF] that silicon (Si) nanospheres with sizes ranging from 100 to 300 nm support strong magnetic and electric dipole resonances in the visible and near-IR spectral range. Silicon's absorption in the visible spectrum is much lower than that of metals, making it a good material for the study of Mie resonances, despite that it is not a totally loss-free material. Figure I.14.(A to C) shows an experimental data of optical magnetic and electric responses at visible wavelengths for resonant silicon nanospheres. As it can be noticed, different colors are observed in the dark-field microscope images, which correspond to magnetic dipole resonances of almost perfect spherical Si nanoparticles with sizes ranging from 100 to 200 nm.

In addition Figure I.14.D displays a newly developed technique that is based on laser-induced transfer method, which allows the generation of arrays of resonant nanoparticles with almost perfect spherical shape [START_REF] Kuznetsov | Laser-induced backward transfer of gold nanodroplets[END_REF], [START_REF] Evlyukhin | Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses[END_REF].

However, Figure I.14.E proves that conventional electron beam lithography and photolithography provide much better reproducibility and control over the structural parameters [START_REF] Staude | Tailoring Directional Scattering through agnetic and Electric Resonances in Subwavelength Silicon Nanodisks[END_REF], [START_REF] Fan | Tuning the ol or of Silicon Nanostructures[END_REF]. Moreover, chemical synthesis as seen in Figure I.14.F, in addition to soft-imprint lithography [START_REF] Spinelli | Broadband omnidirectional antireflection coating based on subwavelength surface i e resonators[END_REF], and nanosphere lithography [START_REF] Moitra | Large-Scale All-Dielectric Metamaterial Perfect Reflectors[END_REF], can also be used for generating silicon nanostructures with the required parameters to display strong scattering resonances in the visible and near-IR spectrum.

I.2.3. Directional Scattering of Light

It is already known that any particle which is much smaller than the wavelength of

light     R
, acts as an electric dipole, and therefore scattering light symmetrically in the plane transverse to the dipole axis. However, to achieve an asymmetry in the light scattering, interference effects of a few or several different modes can be exploited. In the case of plasmonic particles, the arising Fano resonance from interference of a broad electrical dipole and narrow quadrupole modes, can cause a scattering asymmetry [START_REF] Luk'yanchuk | The Fano resonance in plasmonic nanostructures and metamaterials[END_REF], [START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF]. Nevertheless, as mentioned previously, there is a drawback to these typical plasmonic metals since they display a large amount of loss.

Moving on to high-index dielectric nanospheres, where strong directional scattering of light results from the interference of magnetic and electric dipole responses excited one word in the nanoparticle with comparable strength. b a  , while other higher-order modes are considered to be negligible, then the backward scattering vanishes [START_REF] Luk'yanchuk | ptim um Forward Light Scattering by Spherical and Spheroidal Dielectric Nanoparticles with High Refractive Inde[END_REF], [START_REF] Nieto-Vesperinas | Angle-suppressed scattering and optical forces on submicrometer dielectric particles[END_REF], [START_REF] Gómez-Edina | Electric and magnetic dipolar response of germanium nanospheres interference effects, scattering anisotropy, and optical forces[END_REF]. [START_REF] Geffrin | agnetic and electric coherence in forward-and backscattered electromagnetic waves by a single dielectric subwavelength sphere[END_REF]. (E) Experimentally measured forward (green) and backward (blue) scattering spectra and forward-to-backward ratio (orange) of a near-spherical Si nanoparticle with a diameter of about 15 nm placed on a glass substrate (a.u., arbitrary units). Insets show transmission (F) and reflection (B) dark-field microscope images and a SEM image of the nanoparticle [START_REF] Fu | Directional visible light scattering by silicon nanoparticles[END_REF]; scale bar for SEM image, 500 nm [START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF]. This observation is conceptually similar to the condition for the zero-backward scattering derived by Kerker et al. in 1983 for a particle having similar electric and magnetic properties,    , often called the first Kerker condition [40].

As reported in Figure I.15.D, (showing a direct measurements of angular scattering patterns via microwave experiments), an experimental verification was carried out to validate the Kerker-type asymmetry in the scattering of electromagnetic waves by high-index dielectric particles in the microwave regime for spherical ceramic particles with diameter of 18 mm and refractive index of 4. Figure I.15.E shows a demonstration of the scattering asymmetry by Si nanoparticles at visible frequencies, resulting in different colors of the same nanoparticles when observed in the dark-field microscope in transmission or reflection mode [START_REF] Fu | Directional visible light scattering by silicon nanoparticles[END_REF].

I.2.4. Dielectric Nano-antennas

The field of nanoantennas is a rapidly growing area of research, and various geometries have been demonstrated successfully for non-classical light emission, fluorescence enhancement, high-harmonic generation, nanoscale photodetectors, and single-molecule detection [START_REF] Novotny | Antennas for light[END_REF].

The interference effects, discussed previously, between the electric and the magnetic dipole resonances of dielectric nanoparticles, have various applications that include efficient nanoantenna structures. However, typical optical elements are not applicable at the subwavelength scale, which leads to the emergence of a conceptually new approach for the design of nanoscale photonic devices, such as optical nanoantennas.

Plamonics nanoantennas are still far from attaining the same good characteristics as conventional antennas at radio frequencies, due to the high loss in metallic elements. For that matter, dielectric nanoantennas have emerged as an alternative to plasmonic structures, which led to many advantages such: low loss, small size, light weight, high radiation efficiency and reasonable bandwidth.

In the case of a single-particle nanoantenna illuminated by a localized light source, the directional scattering conditions differ from plane-wave excitation. These conditions depend not only on the frequency but also on the relative distance between the source dipole and the nanoparticle [START_REF] Rolly | Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles[END_REF]. Therefore, a single dielectric nanoparticle can either reflect or collect the light emitted by a photon source [START_REF] Schmidt | Dielectric antennas -a suitable platform for controlling magnetic dipolar emission errata[END_REF]. In spite of that, this directivity can be adjusted by controlling the distance between the light source and the nanoparticle at the scale of n 6

 [START_REF] Rolly | Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles[END_REF].

Furthermore, the overall directivity of optical nanoantennas can be improved by using an array of nanoparticles. (D) Nonlinear spectroscopy of an array of Si nanodisks [START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by ag netic Response[END_REF]. Shown is the negative logarithm of the normalized transmission spectrum of a sample marked by a gray area indicating a resonance at 1.24 nm. The third-harmonic generation (THG) spectrum of the sample normalized over the spectrum of the substrate (purple dots) is strongly enhanced within the spectral band of the resonance. The inset shows a schematic of the THG at a single silicon nanodisk [START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF].

Another important property of nanoantennas is their ability to concentrate electromagnetic energy at the nanoscale. For instance, resonant dielectric nanostructures not only can enhance electric near-fields [START_REF] Hayashi | Evidence for surfaceenhanced Raman scattering on nonmetallic surfaces: Copper phthalocyanine molecules on GaP small particles[END_REF], [START_REF] Sigalas | Electric field enhancement between two Si microdisks[END_REF], but can also behave like magnetic near-field concentrators. The latter has been predicted theoretically [START_REF] Albella | Low-Loss Electric and Magnetic Field-Enhanced Spectroscopy with Subwavelength Silicon Dimers[END_REF], and then proven experimentally for both: microwaves and visible wavelengths [START_REF] Bakker | a gnetic and Electric Hotspots with Silicon Nanodimers[END_REF], [START_REF] Boudarham | Enhancing the magnetic field intensity with a dielectric gap antenna[END_REF]. As seen in Figure I.16.A, magnetic and electric near-field intensity enhancement occurs simultaneously. It should be noted that both, the magnetic hotspots and the electric hotspots, result in an increase of the local optical density of states, and therefore leading to a modification of the magnetic transition rates of molecules or quantum emitters with an enhancement of the Purcell factor [START_REF] Schmidt | Dielectric antennas -a suitable platform for controlling magnetic dipolar emission errata[END_REF], [START_REF] Albella | Low-Loss Electric and Magnetic Field-Enhanced Spectroscopy with Subwavelength Silicon Dimers[END_REF], [START_REF] Rolly | Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances[END_REF].

As we have seen so far, many factors can contribute to the overall scattering efficiency, but the most significant contributor is having an ideal/appropriate arrangement of nanoparticles. It has been recently demonstrated that certain arrangements of nanoparticles, such as oligomers, can lead to resonant suppression of the total scattering efficiency, a phenomenon associated with Fano resonances [START_REF] Luk'yanchuk | The Fano resonance in plasmonic nanostructures and metamaterials[END_REF], [START_REF] Miroshnichenko | Fano resonances in nanoscale structures[END_REF]. Noting that the Fano resonance arises from the interference of broad and narrow spectral lines. As seen in Figure I.16.B, achieving Fano resonances in nanoparticle oligomers can be made possible through the use of electric and magnetic resonances in a low loss, high-index dielectric nanoparticles [START_REF] Chong | bs ervation of Fano Resonances in All-Dielectric Nanoparticle lig omers[END_REF], [START_REF] Kivshar | Fano Resonances in All-Dielectric Oligomers[END_REF].

The observed Fano resonance in Figure I.16.B results from the interference of the optically induced magnetic dipole mode of the central particle with the collective mode of the nanoparticle structure [START_REF] Kivshar | Fano Resonances in All-Dielectric Oligomers[END_REF].

The low-loss and strong magnetic response properties of the resonant dielectric nanoparticles, make them a unique attractive playground for the demonstration of new nanophotonics effects. One of those effects is the existence of a nonradiating anapole (i.e., poleless) mode recently observed in silicon nanodisks, which originates in the dielectric nanodisks from mere interference.

The anapole mode was recently demonstrated within an individual dielectric nanoparticle in the visible spectrum [START_REF] Miroshnichenko | Nonradiating anapole modes in dielectric nanoparticles[END_REF]. Figure I.16.C shows a nonradiating anapole mode in silicon nanodisks, with a height of 50 nm and a diameter of 310 nm. A strong dip in the nanodisk scattering at the wavelength of anapole excitation is clearly observed in the far-field scattering and a simultaneous near-field enhancement inside and around the disk [START_REF] Miroshnichenko | Nonradiating anapole modes in dielectric nanoparticles[END_REF].

Anapole modes provide new perspective for investigations into electromagnetic properties of various charge-current distributions.

I.2.5. Nonlinear Optics with Resonant Dielectric Nanostructures

Novel nonlinear effects can be achieved from the near-field enhancement of electric and magnetic response in all-dielectric nanostructures. Among those effects we distinguish the following: second and third-harmonic generation (SHG and THG), self-action of light, and Raman scattering are affected by strong confinement resulting from geometrical resonances [START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF].

Figure I.16.D shows an enhanced THG from Si nanodisks exhibiting both electric and magnetic dipole resonances. This enhancement was observed experimentally through third-harmonic microscopy and spectroscopy, with the TH signal being enhanced in the vicinity of the magnetic dipole resonance [START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by ag netic Response[END_REF].

Furthermore, strong THG has also been reported from Si-based metasurfaces [START_REF] Yang | Nonlinear Fano-Resonant Dielectric eta surfaces[END_REF] Apart from the harmonic generation, efficient tuning of optical properties of highindex nanoparticles near magnetic Mie-type resonances by means of femtosecond laser irradiation has also been demonstrated [START_REF] Makarov | Tuning of agnetic ptic al Response in a Dielectric Nanoparticle by Ultrafast Photoexcitation of Dense Electron-Hole Plasma[END_REF]. The effect is based on ultrafast

  fs 100 
photoinjection of dense electron-hole plasma within such nanoparticles, drastically changing their transient dielectric permittivity.

A similar experiment has shown, ultrafast all-optical switching in subwavelength nonlinear dielectric nanostructures exhibiting localized magnetic resonances [START_REF] Shcherbakov | Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures[END_REF].

These experimental demonstrations indicate great promise for the rapidly developing field of nonlinear and quantum nanophotonics. The main advantage of resonant dielectric nanostructures over plasmonics is the localization and strong enhancement of optical fields inside the particles, which promotes strong volumetric nonlinearities and allows for large conversion efficiencies.

: Experiments: Growth Techniques and Chapter II Characterization Methods

This chapter consists of a description of the experimental aspects of the work. We will describe how the samples namely nanostructured silicon surfaces and ZnO nanowires, were synthesized. Then we present structural characterization of such samples by scanning electron microscopy (SEM) and X-Ray Diffraction (XRD).

In the second part of this chapter, we will introduce in details the two techniques that were used to characterize the sample, namely Raman spectroscopy and the physical properties measurement system.

II.1. Samples Preparation and Characterization

II.1.1. Nanostructured Si

II.1.1.1. Preparation

In this part, we report on the synthesis of resonators in the form of pillars and truncated cones, which were fabricated in Si by reaction ion etching (RIE) using a ZnO hard mask, while interferential lithography using a 355 nm laser interference was employed to define the mask patterns [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF].

Nowadays, the growing need of material structuring at the micro and nanoscale level, has led to the development of various etching methods such as the reactiveion etching (RIE). This is due to the fact that many applications require welldefined and ordered structures. In the structuration of any material, the choice of the mask plays a crucial role in the etching process, since the mask should ensure a high etching selectivity along with reliable deposition and structuring. Typically, hard mask materials are usually employed for deep etching. However, the utilization of hard masks involves implementation of additional steps in the fabrication process, and at times the material used as a mask cannot be completely removed from the structures after etching. Such problems were circumvented by the use of ZnO as a mask for creating structured silicon (UTT patented technology) [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. Typically, 10 min of Si etching using SF 6 /O 2 gas mixture was conducted in a Plassys MU400 reactor, performed with RIE power of 90 W, the depth of the features is about 2µm, with the etching rate for silicon was 240 nm/min. The ZnO mask still remains on top of the structures and its thickness remains comparable to the initial one. Noting that under this condition the etching rate of silicon is about 20 times higher than that of ZnO. The height of the resonator could be controlled by varying the etching time.

As it can be seen in

The etching time, the power of the laser beam, and the temperature of substrate were varied leading to structures having different heights and shapes as will be described below.

These samples are fabricated at the UTT by the PhD student A. GWIAZDA.

II.1.1.2. Characterization

Figure II.2 shows the effect of etching temperature on the shape of the Si nanostructures. The SEM images correspond to nanostructured surface etched at 5, 10, 15, 20 and 30 C, at a fixed pressure of 4 mTorr, a laser power of 90 W and an etching time of 5 minutes. In all case, the obtained structures appear as truncated cones, with the diameter of the base of the cone varying little with substrate temperature (300±30 nm). The etch depth and therefore cone height also vary a little (typically 995±20 nm) except for the 30°C sample where cone heights are lower ( 720 nm). Finally, XRD measurements performed on the bare Si substrate and on the substrate with Si nanostructures are very similar, indicating that the etching process does not lead to the formation of any crystalline phase other than of pure Si.

II.1.2. Zinc Oxide Nanowires

II.1.2.1. Preparation

The synthesis of the ZnO nanowires (NW) consists of two steps, namely the preparation of seed ZnO layer followed by the growth of the nanowires.

ZnO seed layer is a very thin layer (15-20 nm) consisting of ZnO nanoparticles that act as a nucleation layer for the growth of ZnO NWs. This seed layer was prepared in the following manner: 2.0197 g of Zinc Acetate (Zn(CH 3 COO 2 ) 2 ,2H 2 O) was dissolved in 20 ml of ethanol by stirring constantly for 24 hrs. This solution was spin-coated onto a clean silicon substrate p-Si (100). After spin-coating the seed layer was annealed at C on a hot plate for a time period of 10 mins. The spin coating process was repeated twice, followed by annealing after each spin coating step. Subsequently, for the growth of ZnO NWs, the chemical bath deposition (CBD) method was used. 0.025 M Zinc Acetate was dissolved in 250 ml of deionized water. This solution was stirred for a few minutes, following which 0.3 ml of ammonium hydroxide (concentration 28%) was added to the solution maintained under constant stirring. This solution was then heated to C in a three-neck flask (show Figure II.5). Once the temperature is attained, the Si substrate with the ZnO seed layer was introduced into the solution for NW growth for a time period of 15 minute or more depending on the length of NWs needed. As a last step, the sample was thoroughly washed with deionized water and dried in air.

Process parameters such as the concentration of zinc acetate, the dipping time can be varied leading to the synthesis of NW with different length, density an even orientation as discussed below.

These samples are fabricated at the UTT by Mrs. A. GOKARNA.

II.1.2.2. Characterization

First, two different durations of dip-coating processes were tested at a fi ed concentration of inc Acetate . It was also shown that the growth of well aligned ZnO nanowires can be controlled by varying the concentration of the zinc acetate solution. Indeed, 3 different concentrations varying between 0.023 and 0.03 M were tested, and each one was accompanied by a different duration of the dip-coating process, while fixing the temperature. It was found that the largest concentration of Zinc Acetate 0.03 M yielded the nanowires quality, with a high degree of vertical alignment and the most uniform size and shape within the same sample (Figure II.8). The average diameter size was found to increase from 25 to 95 nm when increasing the Zinc Acetate concentration from 0.023 to 0.03 M.

The crystalline quality of the ZnO nanowires was studied by XRD. The first figure displayed below, Figure II.9 of XRD corresponds to that of the Silicon substrate, is well indexed to the p-type (100) surface oriented Si wafers structure. There are no additional diffraction peaks observed that might correspond to impurities. They correspond to ZnO nanowires grown on the silicon substrate by varying the temperature of the heated mixed solution, the Zinc Acetate concentration and the duration of the dip-coating process, respectively. All of the peaks that were obtained in the XRD patterns can be indexed to the hexagonal ZnO wurtzite structure.

Regarding the parameters by which the growth process of the ZnO is manipulated, we noticed that when increasing the Zinc Acetate concentration from 0.023 M to 0.025 M, the intensity of the characteristic peaks almost doubled up, while fi ing the temperature at and decreasing the dip-coating process duration to half. Then, when fixing the concentration of Zinc Acetate and the temperature, while increasing the dip-coating process from 15 min to 90 min, the intensity of the characteristic peaks increased dramatically.

Finally, when keeping the concentration of inc Acetate constant, while reducing the temperature, the intensity of the peaks remained almost stable from and C.

II.2.Characterization Techniques

II.2.1. Raman Spectroscopy

Raman Spectroscopy is a powerful optical spectroscopic technique that can be used to examine the vibrational energy levels of molecules in various fields, and it is based on the inelastic scattering of light during light-matter interactions. Of all the various systems that can be investigated by Raman Spectroscopy, we distinguish the following: chemical identification, reactivity, diffusion, charge states, phenomenology at interfaces, in addition to the mechanical and electrical stress and the influence of electric fields on molecular properties. Raman spectroscopy is also a noninvasive nondestructive imaging technique and requires minimal sample preparation, if any. (002)

II.2.1.1. Basic Theory of Raman Spectroscopy

The ground breaking work of Sir Chandrasekhara Venkata Raman (C.V. Raman) in the field of light scattering, started out in 1922 when he published his work on the " ol ecular Diffraction of Light" the first part of a series of studies with his collaborators which eventually led to his discovery, in 1928 [START_REF] Raman | A new class of spectra due to secondary radiation. Part I[END_REF], of the inelastic scattering of light and its effects which led him to the 1930 Nobel Prize in Physics.

The process starts with an electromagnetic radiation striking on a molecule and interacting with the cloud of electrons around the nucleus of the molecule causing them to distort/polarize and to form a transient and unstable energy state called the virtual energy level. During this transient light-matter interaction, a very tiny fraction of photons collides with the heavier nucleus of the molecule which is initially in either the ground rovibronic state (lowest rotational and vibrational energy level of the ground state) or in an excited rovibronic state. This results in an energy exchange between the photon and the molecule, causing a change in the energy and in the frequency of the scattered light, and therefore resulting in an inelastically scattered photon, hence the Raman process.

In Raman Spectroscopy, the vibrational frequency   However, if the molecule is in a higher excited energy level, it will transfer its energy to the scattered photon and falls back to the ground level, and the output is then characterized by a blue shifted frequency, The total distribution of the molecules in any energy level is given by the Maxwell-Boltzmann equation:

  kT E E m n m n m n e g g N N    (II.1)
Where:

 k is the Boltzmann's constant (1. 9 *1 -23 JK -1 )  T is the given temperature. On the other hand, from a classical view point, the electric field   E of the electromagnetic wave (laser beam) fluctuates with time   t as will be described here:

  t E E 0 0 2 cos   (II.2)

57

Where:

 0 E Amplitude.  0  Frequency of the laser.
When this laser light is incident on a molecule, then an electric dipole moment P is induced:

  t E E P 0 0 2 cos      (II.3)
Where  is a proportionality constant, ''the polarizability of the material'' which depends on the molecular structure and the nature of the bonds.

If the molecule is vibrating with a frequency m  , the physical displacement of the atom about their equilibrium position is expressed as:

  t q q m  2 cos 0  (II.4)
Where 0 q is the vibrational amplitude.

For a small amplitude of vibration, the polarizability  is a linear function of q , and can be approximated using a Taylor series expansion: ....

0              q q q    (II.5)
Where  0  is the polarizability at the equilibrium position.

 q q         



Rate of change of  with respect to the change in q evaluated at equilibrium position.

Combining equations (II.3), (II.4) and (II.5), we obtain:

  t E P 0 0 2 cos        t qE q t E P 0 0 0 0 0 0 2 cos 2 cos                       t t E q q t E P m      2 cos 2 cos 2 cos 0 0 0 0 0 0 0             (II.6)             t t E q q t E P m m                         0 0 0 0 0 0 0 0 2 cos 2 cos 2 1 2 cos
The first term of this equation represents an oscillating dipole that radiates light of frequency 0  (Rayleigh scattering). The second term corresponds to the Raman Scattering of frequency

m    0 (Anti-Stokes), and m    0 (Stokes). If 0 0            q 
, then the vibration is not Raman-active.

II.2.1.2. Experimental aspects

The Raman measurements were performed in the backscattering configuration. Very narrow slit widths, extremely low power and long integration time were adopted to avoid spectrum distortion by sample heating and phenomena related to interaction of light with electrons. An 850 mm focal length spectrometer with 1800 groves/mm holographic grating was used in the measurements.

II.2.2. Physical Property Measurement System (PPMS)

Physical Property Measurement System (PPMS) is a versatile instrument offering the option of measuring large variety of physical properties, such as heat capacity, resistivity and thermal conductivity. In our case, the study is solely focused on the measurements of the heat capacity of our samples, which can provide considerable information about the lattice, the electronic and the magnetic properties of the material.

An important characteristic of heat capacity measurements, when performed at temperatures well below the Debye temperature, is the ability to directly delve into and analyze the electronic and magnetic energy levels of the material studied, hence, allowing comparisons between the theory and the experiment this is not the case for resistivity measurements, where the connection between experiment and theory is not always as clear.

In addition, knowledge of the heat capacity of the material is necessary to any successful thermal design, since such thermal devices like refrigerators, cryostats, and so on, require a thermal characterization. The specific heat measurements were carried out by using the Physical Property Measurement System (PPMS) Dynacool developed by Quantum Design. This automated system can measure several physical properties such as resistivity, magnetization and heat capacity. This system is equipped with a 9 T superconducting magnet and the temperature can cover range from 400 doun to 1.8 K.

II.2.2.1. Specific Heat Measurement Methods

Different measuring methods can be used to determine the specific heat of a sample, but a trade-off needs to be made between accuracy and resolution. The adiabatic method, is a classical method that uses the definition of the heat capacity:

dT dQ C p 
, and it revolves around applying a heat pulse   dQ to the sample while recording the change in temperature   dT , so that the specific heat is determined experimentally. However, this method has some disadvantages such as a required large sample size (a minimum mass ∼10 g). The reason behind this particular sample size, is to minimize the amount of stray heat leaks and the need of a mechanical or superconducting switch to establish thermal contact between the sample and the environment (often a liquid He bath) [START_REF] Stewart | e asurement of low-temperature specific heat[END_REF]. Moreover, the sensitivity may not be good enough to detect minor changes in the heat capacity [START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF].

In order to find a way around these problems, Sullivan and Siedel developed a new method, namely, the AC-method, which allows the measurement of the specific heat of small samples (1-100 mg) [START_REF] Sullivan | Steady-State, ac-Temperature alorimetry[END_REF].

Furthermore, the sample and the heat reservoir are thermally coupled, while a heat pulse with a known frequency is applied to the sample. The measured temperature response is then inversely proportional to the specific heat of the sample. Although this method is better than the adiabatic method in measuring small changes in the

specific heat   % 04 . 0  C
, but it does not offer greater accuracy when measuring the absolute specific heat  

% 8  C
compared to the adiabatic method [START_REF] Stewart | e asurement of low-temperature specific heat[END_REF].

Moving on to a different technique, also designed to measure the heat capacity of small samples, is the relaxation technique developed by Bachmann et al. [START_REF] Bachmann | Heat apacity ea surements on Small Samples at Low Temperatures[END_REF]. The relaxation method for the heat capacity measurements combines the best measurement accuracy with powerful analysis techniques. It consists of applying a heat pulse to the sample, and then to track the corresponding temperature response. By solving the heat flow equations with the corresponding boundary conditions C is found to be proportional to the relaxation constant of the temperature response curve. The relaxation technique utilized by our PPMS, is an extension to the method of Bachmann et al. and takes into account the heat flow between the sample and the platform, which is defined as the Two-Tau effect   2  .

In our work, two separate measurements known as "addenda run" and "sample run", are carried out for the quantitative determination of heat capacity. The addenda measurement consists of measuring the heat capacity of the sample holder (platform) plus some grease applied, prior to mounting the sample, for the purpose of separating the sample's heat capacity contribution from the total heat capacity. Whereas in the Two-Tau model also a heat flow between the sample and the sample holder (platform) is considered, but different conditions are taken into account in this case, see Figure II.13 than however, we need to present aware thermal model before further description of the addenda measurement

II.2.2.2. Thermal model

 Simple Model

The heat capacity in addenda run is determined based on the "Simple model" which assumes that the sample and sample platform are in good thermal contact and are at the same temperature during the measurement. The temperature T of the platform as a function of time t, obeys the following equation that is based on Fourier's law of heat conduction and the law of conservation of energy: The process starts with a heat flow supply generated the heater, and directed to the sample/platform. The power of the heater has a constant magnitude P 0 during heating and is zero during the cooling part. Then a second heat flow from the sample/platform is directed towards the latter's environment. This is determined by the thermal conductance of the wires and the temperature difference between the sample/platform and the bath formulated in the second term of equation. The solution of the differential equation which can be solved for   t T for the heating part and for the cooling part of the temperature response curve, is the following:

    0 T T dt dT C t P pl w pl total     (II.7)     t P T T dt dT C pl w pl total     0  Where:
                           0 0 0 1 1 T C
      2 0 , ,    i i i w tot T t T T C S  (II.9)
 The Two-Tau Model

In the case when the thermal contact between the sample and sample platform is poor, the Heat Capacity software uses a more sophisticated Two-Tau model to measure the heat capacity. In other words, this model is applied when there is a temperature difference between the sample and sample platform , as shown in Figure II.15.

The Two-Tau model is expressed by the following equations: Equation (II.10a) depicts the heat flow from and to the platform. The two incoming heat flows to the platform are: (a) from the heater, with magnitude P when the heater is on, and (b) a flow from the sample to the platform, where the magnitude is determined by g  . The last term describes the flow from the platform to the bath, which is governed by w  .

            t T t T t P T t T dt dT C pl s g pl w pl pl         0       t T t T dt dT C p s s s s     0 (II.10a) (II.
Equation (II.10b) shows the heat flow from and to the sample. When the heater is on there will be a flow from the platform to the sample and when the heater is off there will be a flow from the sample to the platform. Equation (II.10) solution is not straightforward and therefore cannot be solved directly as in the simple model. A clear derivation is given in a paper of Shang Hwang et al. [66], where it is demonstrated that the two differential equations can be converted into three equations with four unknown parameters: w

 , g  , s C , pl C
, where pl C is known from the addenda measurement which is explained in section II.2.2.7. By fitting the three parameters to the three solutions of the coupled differential equations, one can determine the heat capacity of the sample.

II.2.2.3. Description of the PPMS Dynacool system

A schematic of the Dynacool PPMS is depicted in Figure II.16. The system makes use of a two stage pulse tube cooler, without the need to transfer liquid cryogens. Instead, the pulse tube cooler provides a small amount of liquid helium in the bucket and cools the superconducting magnet to 4 K. The variable temperature insert can be controlled between 1.8 and 400 K, by combining a block heater below the sample chamber with a mixture of liquid helium and gaseous helium that flows from the bucket through the cooling annulus surrounding the sample chamber. The sample platform (holder) and the sample chamber form the "calorimeter puck", which contains as well a platform heater attached to the bottom side of the sample platform, and a puck thermometer that is buried within the puck. It should be noted that for heat capacity measurements, the bottom of the sample chamber (referred to as the isothermal region) is made of copper in order to provide a uniform temperature region. The block thermometer is connected to the bottom of the isothermal region below the puck interface and measures the temperature of the puck.

Figure II.16

A simplified picture of the Dynacool. In green is the bucket with the liquid helium which flows to the pot in the bottom of the annulus. Around the sample chamber flows helium gas. In the bottom of the sample chamber the puck is placed in the isothermal region, under which a block heater is attached for temperature control. To achieve high vacuum, the sample chamber is exposed to a two stage cryopump [START_REF] Design | Physical Property easurement System, Torque a gnetometer ption Use r ' s anu al[END_REF].

For measuring the heat capacity, it is necessary that there is a minimal heat transfer from the sample to the environment and vice versa. Therefore, the sample is sealed with a cap that acts as a radiation shield, and placed inside the chamber under vacuum (˂1 -3 mbar), protecting the sample and the sample platform from unwanted exterior heating created by warmer surfaces in the sample chamber. The PPMS is equipped with a cryopump, providing a sufficient vacuum (˂1 -6 Torr) so that the thermal conductance between the sample platform and the thermal bath is totally dominated by the conductance of the wires. The cryopump consists of two stages: The first stage that operates at 70 K, while the second stage is at 4 K. Regarding the second stage, the cryopump contains charcoal which, once at 4 K, will serve as an absorbent of any remaining gas from the sample chamber. This high vacuum option is fully integrated in the system. The sample chamber is automatically evacuated to less than 5 Torr, whenever starting a heat capacity measurement, in conjunction with an external scroll chamber pump. To obtain a higher vacuum the chamber isolation valve is closed and the high vacuum valve is opened in order to expose the sample chamber to the cryostat. The pressure will drop within a few seconds to a high vacuum. . Four wires are for the 4-point probe method of the heater and four wires for the 4-point probe method of the thermometer. In each 4-point probe method, two wires are for determining the voltage and two for the current. These eight wires act as a thermal link to the bath. The current supplied to the thermometer is ∼1 nA and to the heater is ∼10 nA. The contacts between the heater and thermometer to the wires are provided by a gold layer evaporated on the sapphire platform.

II.2.2.4. Calibration of the Puck

Calibrating the thermometer and the heater of the puck is a crucial and a necessary step before attempting to measure the specific heat of the sample. The calibration consists of two parts, Pass 1 and Pass 2. In Pass 1, the aim is to relate the resistance to the temperature. For that matter, the resistance of the platform thermometer must be determined, noting that the reference thermometer for calibration is the block thermometer in the bottom of the PPMS. In Pass 2, the resistance of the heater is measured, as well as the thermal conductivity of the wires connected to the platform. When making the transition from Pass 1 to 2, the chamber needs to be opened in order to place the baffle assembly, as shown in Figure II. [START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF]. The charcoal holder is placed at the end of the baffle assembly, with a small amount of charcoal inside. As mentioned previously, in order to prevent adsorption on the sample platform, the charcoal main role is the adsorption of Helium gas from the sample chamber. Furthermore, during Pass 2 the heat capacity of the puck is determined; therefore a high vacuum is necessary. The results of the calibration are shown [START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF] The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas [START_REF] Design | Physical Property easurement System, Torque a gnetometer ption Use r ' s anu al[END_REF].

II.2.2.5. Sample Mounting

The experiments were performed on samples of Silicon Si nanocones, and ZnO nanowires, of different thickness and diameter, that were prepared at the Laboratory of Nanotechnology, Instrumentation and Optics (LNIO), at the University of Technology of Troyes (UTT) as previously by described. The PPMS measurements were done at the Central Research Science Laboratory (CRSL), at the American University of Bierut (AUB). These samples have been cut into one or two small cubes that have an approximate size of 3x3 mm 2 , by using a Talentool diamond cutter. These synthesized samples are ideal for the specific heat measurement, in particular due to their thin and flat surface, which ensure that the heat is evenly distributed in the sample in a reasonable period of time. The samples have a mass that ranges from 5 to 25 mg.

The puck is placed in a sample mounting station and fixed in position with the interlock arm. In order to stabilize the platform and ensure that no force is exerted on the fragile wires during mounting, a small vacuum pump is connected to the mounting station. A small amount of Apiezon N Grease is applied on the platform, right after fixing the puck firmly. Then, the sample can be pressed into the grease.

To perform the addenda measurement which consists of measuring the specific heat of the sample holder (platform) plus that of the applied grease, prior to mounting the sample, for the purpose of separating the sample's specific heat contribution from the total specific heat. Quantum Design recommends Apiezon N Grease when measuring below 200 K, due to anomalies in its specific heat above 200 K. It is worth mentioning that Apiezon H Grease is recommended for measurements above 200 K, because below 200 K, H Grease can pop off the platform, whereas, for covering the whole temperature range of 1.8-400 K, Apiezon N Grease is recommended. When doing so, it is recommended that above 200 K a temperature spacing of ∼5 K between the data points is used in order to resolve any anomalies. These were not detected here.

II.2.2.6. Heat capacity measurement procedure

When the heat capacity is activated, a dialogue box will open, as seen in Figure II.20, where some parameters can be modified. Before starting any measurement, some settings need to be adjusted to get a more stable and precise measurement of determined the heat capacity over the whole temperature range. For instance, the temperature start-and end-points and the spacing need to be modified. Then, another important setting that needs adjustment is 'Extra wait time at new temperature' under the tab 'Advanced'. The measurement will start immediately, with each new temperature point, when the temperature has settled within 1% of the requested temperature rise. Measurement stability: A heat pulse will only be applied when the temperature is within 1% of the requested tem-perature rise. 7) Extra wait time: When measuring at high temperatures it is necessary to wait longer before the measurement starts [START_REF] Design | Physical Property easurement System, Torque a gnetometer ption Use r ' s anu al[END_REF].

Once the steps mentioned previously are adjusted, the measurement can start.Data acquisition only takes place during the first  2 after applying a heat pulse. The time constant  must be estimated first, by applying a trial heat pulse, especially that the measurement time is in units of  . The magnitude of the heat pulse is determined by reading the calibration file with the values for the thermal conductivity of the wires in order to get the requested temperature rise which is the ratio of the power and the thermal conductivity of the wires. After determining an initial time constant, the temperature will stabilize again to the base temperature and a heat pulse is applied. Once the requested temperature is reached at time , the heater is turned off and the sample will cool down. The temperature response curve is shown in Figure II.14. After  2 seconds the data acquisition stops, but the temperature of the sample still decreases to the base temperature. This marks the end of the curve shown in Figure II.14. The temperature response curve obtained in this way will be fitted to the simple model and the two tau model discussed in section II.2.2.2. The same procedure is repeated by applying a second heat pulse, if the temperature is not varying within 1% of the requested temperature rise after every  seconds. At each temperature point, the heat pulses are repeated three times and with that three values for the heat capacity are obtained. The base temperature can now be increased to the next temperature point using the block heater below the sample chamber. At the next temperature point, the obtained time constant of the previous measurement is used as the initial time constant.

II.2.2.7. The addenda measurement

As mentioned in a previous section, the addenda measurement consists of measuring the heat capacity of the sample holder (platform) plus some grease applied, prior to mounting the sample, for the purpose of separating the sample's heat capacity contribution from the total heat capacity. Afterwards, the addenda measurement is subtracted from the total heat capacity in order to obtain the sample heat capacity. The addenda measurements are of significant importance, in particular because the samples under consideration are fairly small, with a mass of only a few mg.

The specific heat of Apiezon N Grease

Knowing the specific heat of the grease helps determining the exact contribution to the specific heat of one mg extra of grease, and with that, we can make an estimate of the error in the determination of the specific heat of each sample. Therefore, the specific heat of the empty puck and that of the puck with a known mass of grease are measured. The results are displayed in

II.2.2.8. Difference between p C and v C

The specific heat is usually measured at constant pressure. However, it should be clarified that the theoretical relations for the specific heat are all derived at constant volume. Therefore, the contribution of the volume expansion to v C needs to be determined. The difference between the heat capacity at constant volume and constant pressure is given by [START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF]:

T V C C T v p   2   (II.11)
Where:

 V is the molar volume of the sample.    p T V V    1  Is the volumetric expansion coefficient at constant pressure.    T T T V V     1  is the isothermal compressibility.
These parameters were estimated from previous works and experiments conducted on similar materials [START_REF] Gasparini | Thesis Quantum criticality in correlated metals, a gruneisen parameter study[END_REF].

: Interaction between confined phonons and Chapter III photons in periodic silicon resonators

In this chapter, we demonstrate that phonons and photons of different momenta can be confined and interact with each other within the same nanostructure. The interaction between confined phonons and confined photons in silicon resonators arrays is observed by means of Raman spectroscopy. The Raman spectra from large arrays of dielectric silicon resonators exhibited Raman enhancement accompanied with a downshift and broadening. The analysis of the Raman intensity and line shape using finite-difference time-domain simulations and a spatial correlation model demonstrated an interaction between photons confined in the resonators and phonons confined in highly defective regions prompted by the structuring process. It was shown that the Raman enhancement is due to collective lattice resonance inducing field confinement in the resonators, while the spectra downshift and broadening are signatures of the relaxation of the phonon wavevector due to phonon confinement in defective regions located in the surface layer of the Si resonators. We found that as the resonators increase in height and their shape becomes cylindrical, the amplitude of their coherent oscillation increases and hence their ability to confine the incoming electric field increases.

III.1. Introduction

Due to the need for low cost, clean, and sustainable energy sources to replace shrinking supply of fossil fuels, considerable efforts have been made in pursuit of highly efficient photovoltaic effect in silicon (Si), which is a material known for its ease of fabrication and being compatible with large scale processes. Indeed, this is not an easy struggle because of the high reflectivity and indirect electronic band structure of Si. The high reflectance of the incoming radiations from the Si surface drastically reduces the number of photons that contribute to the photovoltaic generation of electricity and the Si indirect electronic band structure disfavors the electron photoexcitation from the maximum of the valence band to the minimum of the conduction band. In fact, the electron excitation from the maximum of the valence band to the minimum of the conduction band in crystalline Si requires the assistance of phonons to make up for the momentum mismatch between the maximum of the valence band and the minimum of the conduction band. Therefore, the significant enhancement in the efficiency of the photovoltaic effect in Si requires reduction in lossy light processes and coupling between trapped photons and phonons confined in low dimensional crystalline structures where the crystal translational symmetry is broken. In such short spatial correlation crystals, the total momentum needs not be conserved in order to excite electrons from the maximum of the valence band to the minimum of the conduction band [START_REF] Arora | Raman spectroscopy of optical phonon confinement in nanostructured materials[END_REF].

The surface structuring has shown potential in reducing the reflectance and enhancing the interaction between light and matter. As such, it was reported on significant reflectance reduction over a wide spectral range through surface structuring in the form of periodic or random arrays of micron-sized pyramids or rods [START_REF] Yablonovitch | Statistical ray optics[END_REF]- [START_REF] Koynov | Black nonreflecting silicon surfaces for solar cells[END_REF]. On the other hand, Raman light scattering measurements have revealed evidence of phonon confinement in nanosized crystalline zones in highly defective crystals whereby the phonon selection rule is relaxed [START_REF] Shuker | Raman-Scattering Selection-Rule Breaking and the Density of States in Amorphous a terials[END_REF], [START_REF] Tiong | Effects of As + ion implantation on the Raman spectra of GaAs '"Spatial correlation"' interpretation[END_REF]. Thus, micron-sized structures are required to trap incident photons, whereas nanosized crystallites are needed to relax the requirement of total momentum conservation and enhance the probability of the electron transition from the maximum of the valence band to the minimum of the conduction band. Consequently, in order to enhance the interaction between light and confined phonons, photons should be confined in micron-sized structures in which the spatial correlation length does not exceed a few nanometers.

It is well known now that plasmonic nanoresonators forward scatter normally incident light towards the substrate on which they are deposited [START_REF] Atwater | Plasmonics for improved photovoltaic devices[END_REF]. Although this process enhances the coupling between light and matter, it is much too lossy to confine photons. However, when identical plasmonic nanoresonators are arranged into a two-dimensional lattice such that the lattice period is comparable to the wavelength of the normally incident light, the fields scattered by the individual nanoresonators combine together to give rise to a field in the plane of the nanoresonators lattice [START_REF] Carron | Resonances of two-dimensional particle gratings in surface-enhanced Raman scattering[END_REF], [START_REF] Zou | Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes[END_REF]. If that field is spectrally close to the resonance frequency of the individual plasmonic nanoresonators, doubly degenerate hybrid light-plasmon collective modes (known as plasmonic lattice resonance modes) occur, inducing highly coherent plasmonic oscillations at all the lattice points [START_REF] Chatz | Silver nanoparticle array structures that produce giant enhancements in electromagnetic fields[END_REF]. This makes all the nanoresonators oscillate in a coherent fashion. This coherent oscillation of the nanoresonators makes the field scattered by one nanoresonator absorbed by the others [START_REF] Zou | Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes[END_REF], and consequently the electric field confined in the nanoresonators. Similar to localized plasmon resonances in plasmonic nanoresonators, localized Mie resonances in dielectric resonators induce forward scattering and coupling of normally incident light into the substrate beneath the resonators [START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF], [START_REF] Spinelli | Broadband omnidirectional antireflection coating based on subwavelength surface i e resonators[END_REF], [START_REF] Bezares | i e resonance-enhanced light absorption in periodic silicon nanopillar arrays[END_REF], and localized Mie resonances at two-dimensional lattice points induce highly coherent oscillations in the dielectric resonators, which leads to strong field confinement in the dielectric resonators [START_REF] Evlyukhin | ptica l response features of Si-nanoparticle arrays[END_REF], [START_REF] Tsoi | E perimental demonstration of the optical lattice resonance in arrays of Si nanoresonators[END_REF]. On the other hand, the structuring of Si surface in the form of periodic micron-sized resonators over a large area triggers the development of nanosized crystallites embedded in regions with high structural defect densities in the surface layers of the resonators.

In these short spatial correlation zones, the phonon selection rule may be broken due to phonon confinement [START_REF] Cloutier | Enhancement of Radiative Recombination in Silicon via Phonon Localization and Selection-Rule Breaking[END_REF], [START_REF] Guico | Phonon localization in periodic unia ially nanostructured silicon[END_REF]. The tips of tiny Si nanowires also present potential short spatial correlation zones for breaking the phonon selection rules [START_REF] Hofmann | Size effects on phonon localization and Raman enhancement in silicon nanotips[END_REF], [START_REF] Banerjee | Phonon processes in vertically aligned silicon nanowire arrays produced by low-cost all-solution galvanic displacement method[END_REF]. Thus, dielectric resonators periodically arranged over a large area have the potential to simultaneously confine photons and generate confined phonons. In other words, dielectric resonators at lattice points can ensure the interaction between long living confined photons and nanosized crystallites in which the phonon selection rule is broken.

The important question, which arises now, is how the size and shape of the Si resonators influence the light confinement strength and its interaction with the nanosized crystallites in the resonators. In this contribution, we address this question and present a thorough experimental study of the effect of size and shape of Si resonators periodically arranged over a large area on the strength of the confined electric field and its interaction with nanosized crystallites in the surface layers of the resonators. We demonstrate the Si surface structuring that best confines the incoming electric field and ensures its interaction with nanosized crystallites for high efficiency Si-based optical devices. Raman spectroscopy has demonstrated a high potential for the evaluation of the confinement of the electric field [START_REF] Bezares | i e resonance-enhanced light absorption in periodic silicon nanopillar arrays[END_REF], [START_REF] Tsoi | E perimental demonstration of the optical lattice resonance in arrays of Si nanoresonators[END_REF]. In addition, the analysis of Raman line shape can be carried out to yield quantitative determination of the spatial correlation length in the measured material [START_REF] Arora | Raman spectroscopy of optical phonon confinement in nanostructured materials[END_REF], [START_REF] Shuker | Raman-Scattering Selection-Rule Breaking and the Density of States in Amorphous a terials[END_REF], [START_REF] Tiong | Effects of As + ion implantation on the Raman spectra of GaAs '"Spatial correlation"' interpretation[END_REF], [START_REF] Bassi | Raman spectroscopy characterization of titania nanoparticles produced by flame pyrolysis: The influence of size and stoichiometry[END_REF]- [START_REF] Silva | odi fied Phonon onfinement odel a nd Its Application to dSe/ dS Core-Shell Magic-Sized Quantum Dots Synthesized in Aqueous Solution by a New Route[END_REF]. Therefore, we use Raman spectroscopy to evaluate the confined electric field and measure the crystallites sizes in the surface layers of the Si resonators.

III.2.Experiments

Resonators in the form of pillars and truncated cones were fabricated in Si by Reactive Ion Etching (RIE) using a hard mask. A special technique called MUZ using ZnO as a hard mask was employed [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. While ZnO can be easily dissolved chemically, it is difficult to dry etch. To define the mask patterns, interferential lithography using a 355 nm laser source was used. A square lattice with circular patterns of a period of 660 nm was created by a double exposure of the sample, whereby the diameter of the features could be controlled by changing the exposure time [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. Thereafter, Si etching was conducted in a Plassys MU400 reactor using a SF 6 /O 2 gas mixture. A selectivity of about 100 was demonstrated with Si using ZnO as a hard mask. The height of the resonators could be controlled by varying the etching time.

Three sets of samples were obtained. The first set consists of two samples (samples S S,1 and S S,2 ) in which the resonators have almost the same height. However, the resonators of the first sample are cone-like, whereas the resonators of the second sample are pillar-like. The second set consists of three samples (samples S M,1 , S M,2 , and S M,3 ). The height of the resonators in this set is greater than that of the resonators in the first set. The samples in the second set, like in the first set, differ only in that the shapes of their resonators are different. The third set consists of three samples (samples S L,1 , S L,2 , and S L,3 ). The resonators in this set are the highest, and again the only difference between the samples is the shape of their resonators. Thus, within the same set, the only difference between the samples is the ratio of the diameter at the top of the resonators to the diameter at the bottom of the resonators. For the sake of clarity, the samples sets will be hereafter indexed according to their height namely S, M and L for short, medium and long respectively. With the present arrangement of the samples, we will be able to investigate the effect of the resonators height and shape on the confinement of the electric field and its interaction with phonons confined in nanosized crystallites.

In In Table III.I., we present a detailed structural description of all the samples investigated. Raman measurements were performed in the backscattering configuration using a 532 nm laser line from an Ar-ion laser. Very narrow slit widths, extremely low power and long integration time were adopted to avoid spectrum distortion by sample heating and phenomena related to interaction of light with electrons. An 850 mm focal length spectrometer with 1800 groves/mm holographic grating was used in the measurements. Table III.I Detailed structural description and the obtained parameters of the crystallites size distributions in the surface layer of the Si resonators in the samples investigated. The subscripts S, , and L refer to "short", "medium", and "long", respectively

Set

III.3. Results And Discussion

In infinite crystalline materials, the Raman shift and line shape are determined by the energy and lifetime of the optical phonon at the center of the Brillouin zone. Impurities and extended elastic fields (caused by internal or external stress), however, alter the energy and lifetime of the zone center optical phonon, and consequently modify the Raman shift and line shape [START_REF] Klemens | Effect of point defects on the decay of the longitudinal optical mode[END_REF]- [START_REF] Paillard | Improved one-phonon confinement model for an accurate size determination of silicon nanocrystals[END_REF]. In addition, the interaction between light and matter may trigger several mechanisms that can further distort the Raman spectrum. For instance, high laser powers can induce high concentrations of free carrier, and hence the induced free carriers continuum gas can couple with the discrete phonon modes to give rise to the so-called Fano resonance [START_REF] Lee | Phonon populations by nanosecond-pulsed Raman scattering in Si[END_REF], which usually results in an asymmetric Raman line shape [START_REF] Beserman | Fano-type interference in the Raman spectrum of photoe cited Si[END_REF], [START_REF] Nickel | Raman spectroscopy of heavily doped polycrystalline silicon thin films[END_REF]. Moreover, high laser powers can cause a local temperature rise, which reduces the lifetime of the excited phonons and weakens their energies. This temperature-related phenomenon further broadens the Raman peak and shifts it toward lower frequencies [102]. In nanosized crystallites, additional size effects combine with the effects described above, distorting further the Raman spectrum [START_REF] Nagata | Evaluation of phonon confinement in ultrathin-film silicon-on-insulator by Raman spectroscopy[END_REF]. In fact, in nanosized crystallites, the Heisenberg uncertainty principle imposes the relaxation of the conservation of the total momentum, and consequently the relaxation of the requirement that only the zone center optical phonons participate in the Raman scattering processes. Thus, phonons away from the Brillouin zone center contribute to the response of the nanosized crystallite to an optical excitation. According to Heisenberg uncertainty principle, all the phonons whose wavevectors are roughly equal to or smaller than the inverse of the crystallite size contribute to the response of the crystallite to an optical excitation [START_REF] Richter | The one phonon Raman spectrum in microcrystalline silicon[END_REF]- [START_REF] Ampbell | Raman spectroscopy of low-dimensional semiconductors[END_REF]. This results in a further downshift and broadening of the Raman peak. It follows from the discussion above that the Raman shift and line shape may be determined by many effects combined together [START_REF] Piscanec | Raman spectroscopy of silicon nanowires[END_REF]. The samples investigated in this work consist of Si resonators periodically arranged in a two-dimensional lattice on a Si substrate, and the surface structuring of Si was made at low temperature to prevent any impurity diffusion. Hence, we do not expect any stress-or impurity-induced elastic fields contributing to the measured Raman spectra. In order to rule out the effect of laser heating and laser-induced Fano resonance, we have decreased the excitation laser power successively until both Raman peak position and broadening became independent of laser power. It is worth noting here that we found effects of Fano resonance and laser heating for a laser power as low as 0.2 mW/µm 2 , which is five times lower than the reported threshold power for the effects of laser heating and Fano resonance [START_REF] Faraci | Quantum confinement and thermal effects on the Raman spectra of Si nanocrystals[END_REF], [START_REF] Gupta | Laser-Induced Fano Resonance Scattering in Silicon Nanowires[END_REF]].

We address the reason behind this observation at the end of this section.

In In other words, we attribute the observed softening and broadening of the Raman resonance to nanosized crystallites embedded in highly defective Si in the surface layers of the resonators induced by the structuring process. This hypothesis is supported by previously reported Raman and thermal measurements demonstrating the presence of defect-induced confined phonons in large arrays of Si resonators [START_REF] Cloutier | Enhancement of Radiative Recombination in Silicon via Phonon Localization and Selection-Rule Breaking[END_REF], [START_REF] Guico | Phonon localization in periodic unia ially nanostructured silicon[END_REF], [START_REF] Iskandar | odi fication of the phonon spectrum of bulk Si through surface nanostructuring[END_REF].

Besides D if we consider a phonon dispersion in a three- dimensional Brillouin zone. However, on the other hand, the Raman scattering efficiency is proportional to the interaction volume, which is roughly equal to 3 D .

Therefore, we believe that the observed Raman enhancement is not due to phonon confinement effect. It is also known that isolated nanosized crystallites, or nanosized crystallites embedded in an amorphous medium with a very different index of refraction n can show Raman enhancement due to electromagnetic resonances induced by local field effect and surface phonon modes [START_REF] Hayashi | Raman study of gas-evaporated germanium microcrystals[END_REF]- [START_REF] Hayashi | Amorphous-like Raman spectra of semiconductor microcrystals[END_REF]. However, since crystalline and amorphous Si have similar refraction indices, the Raman enhancement observed in our experiments cannot also be attributed to local field inducing electromagnetic resonances and surface modes contributing to the measured Raman spectra. Nonetheless, it has been demonstrated theoretically and experimentally that, similar to localized plasmon resonances in plasmonic nanoresonators, localized Mie resonances in dielectric resonators at the points of a two-dimensional lattice induce highly coherent oscillations in the dielectric resonators, which leads to strong field confinement in the dielectric resonators [START_REF] Evlyukhin | ptica l response features of Si-nanoparticle arrays[END_REF], [START_REF] Tsoi | E perimental demonstration of the optical lattice resonance in arrays of Si nanoresonators[END_REF]. We believe that this is the only reasonable explanation of the Raman enhancement observed in our experiments. Therefore, we can conclude from the Raman measurements illustrated in The FDTD simulations clearly show that the electric field is weakly confined in the cone-like resonators. Therefore, the corresponding Raman spectrum does not exhibit neither an appreciable Raman enhancement nor a downshift and broadening. However, a stronger field confinement can be noticed in the pillar-like resonators. Therefore, the corresponding Raman spectrum exhibits Raman enhancement (a signature of field confinement in the resonators) and a downshift and broadening (a signature of phonon confinement in the resonators). Thus, in the first samples set, interaction between confined photons and confined phonons is observed only in pillar-like resonators.

Several models can be used to quantify the size of the crystallites confining phonons in the pillar-like resonators from the downshift and broadening of the Raman spectrum. The most popular ones are the microscopic force model [START_REF] Cheng | alculations on the size effects of Raman intensities of silicon quantum dots[END_REF], the bond polarization model [START_REF] Xie | omparison of models for Raman spectra of Si nanocrystals[END_REF]- [START_REF] Thonhauser | Predicted Raman spectra of Si[111] nanowires[END_REF], and the spatial correlation models [START_REF] Fauchet | The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors[END_REF], [START_REF] Parayanthal | Raman Scattering in Alloy Semiconductors: "Spatial Correlation" Model[END_REF]. The spatial correlation models, however, have the advantage that they are less affected by some strong assumptions and approximations such as the continuum medium approximation. Therefore, we adopted a spatial correlation model to deduce from the downshift and broadening of the measured Raman spectrum the crystallites size distribution in the surface layers of the resonators. In the Appendix 1, we detail the spatial correlation model used in this work. The spatial correlation model was fitted to the measured Raman spectra with adjusting the parameters that determine the lognormal function describing the crystallites size distribution (i.e., the most probable size and the skewness of the distribution). We recall here that the height of the resonators in this set is greater than that of the resonators in the first set (see Table III Both Raman spectra and FDTD simulations demonstrate that the resonators of the samples in the third set are remarkably more efficient in electric field confinement than the resonators of the samples of the first and second set. However, similar to the first and second set, the field confinement gets stronger as the resonators become more pillar-like. The measured Raman spectra of the samples of the third set were analyzed using the spatial correlation model, and the crystallites sizes in the measured samples were obtained by fitting the spatial correlation model to the measured Raman spectra. The calculated Raman spectra are plotted in Figure III.5 using solid lines, and the obtained crystallites size in each of the measured samples is indicated in Table III.I. We notice that among all the measured samples, sample S L,3, which is characterized by the highest pillar-like resonators, best ensures the desirable interaction between confined photons and confined phonons.

III.3.4. Effect of resonator size and shape on the interaction between confined photons and confined phonons

The Raman spectra and FDTD simulations carried out on the samples of the three samples sets described in Table III.I. revealed important information about the effect of the resonators shape and size on the confinement of photons and phonons. is greater than that of the resonators of sample S L,3 , the confinement of the electric field in sample S L,2 is weaker than that in sample S L,3 . These results suggest the following. While the volume of the resonators does not present an appreciable effect on the electric field confinement in the resonators, the height and shape of the resonators are key parameters determining the strength of the field confinement in the dielectric resonators. The confinement of the electric field gets stronger as the resonators increase in height and their shape becomes cylindrical. We believe that this is because the highest cylindrical resonators can attain the greatest oscillation amplitude, which is the prime determinant of field confinement.

As the resonators height decreases and their shape deviates from the ideal cylindrical shape, the resonators lose their ability to vibrate, and consequently lose their efficiency in confining the incident electric field.

Phonon confinement in crystallites of comparable average sizes was observed in all the measured samples. This suggests that the resonators size and shape do not have an appreciable effect on the spatial correlation length in the resonators. However, by comparing the FDTD simulations in III.I, we notice that the skewness of the size distribution increases as the confined electric field spreads in the resonators. This can be understood if we realize that the damage in the resonator surface layers induced by the structuring process is not uniform over the resonator height. Hence, when the electric field spreads in the resonators it interacts with phonons confined in crystallites of different sizes.

As a result, long cylindrical resonators arranged on two-dimensional lattice points over a large area present great potential for confining an important amount of the incident electric field and promoting the interaction between the confined electric field and phonons confined in nanosized crystallites in the surface layers of the resonators. Such an interaction could significantly enhance the photovoltaic processes in Si because it strongly amplifies the number of photons interacting with Si and does not require total momentum conservation.

III.3.5. Electric field confinement and heating in the resonators

It is now well known that high laser powers can accumulate an important amount of energy in nanosized crystallites (because of their low thermal properties), and hence distort their Raman spectrum by strengthening the effects of laser heating and Fano resonance. It was found that the threshold power for the effects of laser heating and Fano resonance in nanosized Si crystallites is about 1mW/µm 2 [START_REF] Faraci | Quantum confinement and thermal effects on the Raman spectra of Si nanocrystals[END_REF], [START_REF] Gupta | Laser-Induced Fano Resonance Scattering in Silicon Nanowires[END_REF]. As mentioned previously, in order to rule out the effect of laser heating and laser-induced Fano resonance, we have decreased the excitation laser power successively until the Raman shift and line shape became independent of laser power.

In Figure III.6 we plot the measured Raman shift and spectrum broadening versus the incident laser power. As can be noticed, the Raman shift and spectrum broadening become independent of laser power only for laser powers lower than approximately 0.2 mW/µm 2 . This implies that the effects of Fano resonance and laser heating on the measured Raman shift and line shape vanish completely only for a laser power lower than 0.2 mW/µm 2 , which is five times lower than the reported threshold power for the effects of laser heating and Fano resonance in nanosized Si crystallites. Based on the discussion above, we attribute the existence of effects of Fano resonance and laser heating at such low laser powers to the fact that the electric field confinement in the resonators increases significantly the temperature of the crystallites embedded in the resonators. In Figure III.7, we show the Raman spectra of samples S M,3 and S L,3 (the samples with pillar-like resonators, which showed the strongest field confinement) measured with a laser power of 1mW/µm 2 (the reported threshold power for the effects of laser heating and Fano resonance) and a laser power of 0.1 mW/µm 2 . The spectra are normalized for the sake of comparison.

The spectra measured with a laser power of 1mW/µm 2 are heavily distorted. A downshift and asymmetric broadening relative to the spectra measured with a laser power of 0.1 mW/µm 2 can be clearly noticed. This further highlights the strength of the electric field that can be confined in arrays of dielectric resonators.

III.4. Conclusion

Raman spectra from large arrays of dielectric Si resonators attached to Si substrates showed an appreciable Raman enhancement accompanied by a downshift and broadening relative to the Raman spectrum from a bare Si substrate. FDTD simulations demonstrated that the collective lattice resonance induces an efficient field confinement in the resonators, which results in an appreciable Raman enhancement. A spatial correlation model demonstrated that the spectrum downshift and broadening is the result of the relaxation of the phonon wavevector, which is due to phonon confinement in nanosized Si crystallites in the surface layers of the resonators. The effect of resonators size and shape on the confinement of photons and phonons was investigated. It was demonstrated that, as the resonators increase in height and their shape becomes cylindrical, the amplitude of their coherent oscillation increases and hence their ability to confine the incoming electric field increases. However, it was found that the resonators shape and size do not have a remarkable effect on phonon confinement. The results presented in this work show that large arrays of long cylindrical dielectric Si resonators drastically suppress the light scattering losses by field confinement and promote the interaction of the confined field with confined phonons. The results highlighted in

: Modification of the phonon spectrum of bulk Chapter IV Si through surface nanostructuring

In this chapter, we present experimental evidence on the change of the phonon spectrum and vibrational properties of a bulk material through phonon hybridization mechanisms. The phonon spectrum in a finite material is strongly affected by the presence of free surfaces, which is the addition of a contribution from an essentially two-dimensional crystal. The phonon spectrum of a bulk material can hence be altered by a hybridization mechanism between confined phonon modes in nanostructures introduced on the surface of a bulk material and the underlying bulk phonon modes. We measured the heat capacities of bare and surface-structured silicon substrates originating from the same silicon wafer. Then, we deduced important features of the phonon spectra of the samples investigated through a rigorous analysis of the measured heat capacity curves. The results clearly showed that the shape and size of the nanostructures made on the surface of the bulk substrate have a strong effect on the phonon spectrum of the bulk material.

IV.1. Introduction

Efficient thermal management is often considered as a key step towards a successful technological system. The fast removal of excess heat from electronic systems exposed to temperature extremes improves the reliability and prevents the premature failure of these systems. On the other hand, maintaining a high temperature difference between the two sides of a thermoelectric module can significantly enhance the high temperature energy harvesting through the direct recovery of waste heat and its conversion into useful electrical energy. Nowadays, the usual approaches to evacuate heat and maintain the system at a desired temperature consist in using a heat sink or a complex fan speed control system that relies on continuous temperature measurement [START_REF] Kandlikar | High Flu Heat Removal with i crochannels -A Roadmap of hallenges and ppor tunities[END_REF], [START_REF] Acikalin | Purdue e-Pubs Experimental Investigation of the Thermal Performance of Piezoelectric Fans[END_REF]. However, with the continuous reduction in the size of the devices and their time scales the applicability of these approaches to ensure adequate system cooling becomes questionable. In thermoelectricity, the current trend is nanosizing and nanoengineering semiconductor materials in order to scatter the phonons (heat carriers) without intensely affecting the motion of electrons [126]- [START_REF] Tian | Heat Transfer in Thermoelectric ate rials and Devices[END_REF]. Although this approach has demonstrated potential in creating nanoengineered materials that are somewhat efficient in thermal insulation while conserving their high electrical conduction, it did not lead to the development of solid-state based alternative energy technologies, this is because the phonon scattering centers in these materials may also scatter electrons and increase the ohmic heating.

One way to circumvent this problem would be to find means to modify the material phonon spectrum. With such a technology in hand, one can enhance the material thermal conductivity for fast heat removal or rationally reduce the thermal conductivity of a thermoelectric material without the inclusion of phonon scattering centers and boundaries that may undesirably cause obstruction to electron transport and reduce the thermoelectric conversion efficiency. It was first shown by Weyl that the asymptotic distribution of normal mode frequencies is independent from the body shape provided that the size of the body is sufficiently large [131]. According to Weyl, the error terms will always be of an order involving the ratio of the number of atoms in the surface to the total number in the solid and hence go to zero as the size of the solid increases without limit. The distribution of the eigenvalues of the differential equation describing the material vibrations is thus independent from the shape of the boundary of the region to which the eigenvalues correspond. Consequently, thermodynamic properties and phonon spectrum of solids are independent of sample boundary shape. Therefore, at a first glance, it seems impossible to considerably alter the phonon spectrum of a bulk material by surface processing. However, stimulated by the theoretical work of Davis and Hussein [START_REF] Davis | Nanophononic etamaterial Thermal onductivity Reduction by Local Resonance[END_REF], we demonstrate here experimentally that the phonon spectrum of a crystalline substrate can be altered by a hybridization mechanism between confined phonon modes in nanostructures made on the surface and the underlying bulk lattice dispersion. The phonon hybridization mechanism has been recently intensively studied in nanosized materials [START_REF] Yang | E treme Low Thermal onductivity in Nanoscale D Si Phononic r ystal with Spherical Pores[END_REF]- [START_REF] Honarvar | Spectral energy analysis of locally resonant nanophononic metamaterials by molecular simulations[END_REF]. We demonstrate in this work that confined phonon modes may also hybridize with the phonon dispersion of an underlying bulk material, causing important modifications in the phonon spectrum and thermodynamic properties.

IV.2. Experiments

Phosphorus-doped (2.28*10 17 cm -3 < N d < 4.48*10 -3 ) silicon nanocones were fabricated by reactive-ion etching (RIE) using a hard mask. Laser interference lithography (laser wavelength used: 355 nm) was employed to define the mask of ZnO. A square lattice with circular patterns of period 700 nm was created by a double exosure of the samples whereby the diameter of the features was varied from 280 nm to 460 nm by changing the exposure time. Thereafter, silicon etching was conducted in a Plassys MU400 reactor using SF 6 /O 2 gas mixture [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. Structures with a height varying between 400 nm to 1 µm were obtained by increasing the etching time. Many different techniques for measuring the heat capacity are optimized for different sample sizes and accuracy requirements [START_REF] Stewart | e asurement of low-temperature specific heat[END_REF]. The measurements of the heat capacity of the samples investigated in this work were performed by using the Quantum Design Physical Properties Measurement System that uses a relaxation technique combining the best heat capacity measurement accuracy with robust analysis technique. After each measurement cycle, which is a heating period followed by a cooling period, the system fits the entire temperature response of the sample platform to a model that accounts for both the thermal relaxation of the sample platform to the bath temperature and the relaxation between the sample platform and the sample itself [66]. In the case when the thermal connection shared by the sample and platform is poor, the system models the effect of the relaxation between the platform and the sample and reports the correct heat capacity values despite poor contact.

The system first assumes that the sample and sample platform are in good thermal contact with each other and are at the same temperature during the measurement period. In that simple case, the temperature T of the platform as a function of time t obeys the equation: The system generally uses equation (IV.1) to measure most samples as well as the addenda (the sample holder and grease). However, when the thermal contact between the sample and sample platform is poor, the system automatically simulates the effect of heat flowing between the sample platform and sample by using a two-time-constant model, and fits the entire temperature response accordingly. In the two-time-constant model, the temperature of the platform p T and the temperature of the sample s T obey the equations: We set the measurement time way longer than the amount of time it takes for the sample to reach internal thermal equilibrium to avoid the effect of the thermal path and achieve accurate heat capacity measurement. The reproducibility of the measurements was checked by measuring each sample several times. All the recorded data for the same sample were identical within the nonlinear least-square fitting standard error (which is determined from the sensitivity of the fit deviation to small variations in the fitting parameters).

    t P T T dt dT C b w total      (IV.1)
            t T t T T t T t P dt dT C p s g b p w p platform        (IV.2) and       t T t T dt dT C p s g s sam ple     (IV.

IV.3. Results And Discussion

We measured the constant pressure heat capacity of bare silicon (Si) substrate and eight Si structured substrates with truncated cone-like Si structures over a 3-300 K temperature range. All the measured samples were obtained from the same Si wafer, and the surfaces structuration was made at low temperatures and with appropriate care to prevent any impurity diffusion or defect formation in the Si structures. The measured curves are presented in The measurement uncertainty is indicated by the size of the symbols used to plot the data. As can be noticed, the measured data for the surface-structured samples differ from those of the bare Si substrate. This difference in heat capacity is certainly due to differences in the samples harmonic vibration properties or differences in the properties of the electron gas in the samples. In the following discussion of the measurements we refer to constant volume specific heat, which is a direct measurement of the constant volume heat capacity times the mass of the measured sample, for convenience. Since constant volume is hard to arrange experimentally, we measure the heat capacity at constant pressure. However, one can readily show that at room temperature and below, the ratio of the solid constant pressure specific heat to its constant volume specific heat is very close to one.

At a few degrees Kelvin, where the crystal chemical potential coincides with the Fermi energy, the crystal constant volume specific heat is given by:

94 3 2 2 2 5 4 3 2                    D B F B v t v T rNk T n k T E C     (IV.4)
Where t E is the total energy of the sample, B k the Boltzmann constant, n the concentration of free carriers, F  the Fermi energy, r the number of atoms per unit cell, N the number of unit cells in the sample, and D  the Debye temperature. The first and second term on the right-hand side in equation (IV.4) represent the contribution of the electron gas and the contribution of the lattice harmonic dynamics to the total specific heat, respectively. The lattice dynamics contribution completely dominates the specific heat at high temperatures. However, well below room temperature the lattice dynamics contribution falls off as the cube of the temperature, and at very low temperatures it drops and may become comparable to the electronic contribution, which only decreases linearly with temperature. In order to separate out these two contributions, we write Eq.4 as Table IV.I The characteristics of the samples investigated and the parameters of their phonon spectra obtained from the analysis of the specific heat measurements.

The obtained small values of  clearly demonstrate the absence of observable contribution of the electron gas to the specific heat in all the measured samples. Therefore, we can reasonably consider that the lattice harmonic dynamics of the samples are the only determinants of their specific heats. This allows writing the total constant volume specific heat as an additive function of the normal modes frequencies

  k j  in the form:                                j k B j B j B v v T k k T k k k T E C , 2 2 2 sinh 2     (IV.5)
Where the summation runs over all the phonon states

       k and polarizations          j
. This has the consequence that the specific heat can be expressed as average over the phonon spectrum    g as:

                    L d g T k h T k rNk C B B B v      0 2 2 2 csc 2 3   (IV.6)
Where L  is the frequency that determines the upper limit of the phonon band. In the following, we make use of the heat capacity measurements and equation (IV.6) to estimate the effect of the confined phonon modes on some important features of    g .

IV.3.1. Effect of phonon hybridization mechanisms on the low frequency end of the phonon spectrum

At low temperatures the expression for the specific heat,equation (IV.6) takes the form:

                          L d g T k n n T k rNk T C n B B L B v      0 1 2 exp 3   (IV.7)
Since the exponential factor in equation (IV.7) will cut out contributions from all except small values of  and T , we need to consider only the low frequency end of the phonon spectrum. Furthermore, for small values of , the phonon spectrum of a three-dimensional crystal has the expansion:

  ... 4 4 2 2       a a g (IV.8)
When this expansion is substituted into equation (IV.7), and the upper limit on the integral is removed to infinity, we find that:

                         ... 21
             T x x D B v D e dx e x T rNk C   0 2 4 3 1 3 (IV.11) Where T k x B   
is a dimensionless parameter. In the limit where the upper limit on the integral in equation (IV.11) can be removed to infinity, the expression of the specific heat reduces to:

  3 2 0 5 4 3          D B v T rNk C   (IV.12)
Which is the contribution of the lattice harmonic dynamics to the low temperature specific heat in equation (IV.4). Moreover, we see from equation (IV.11) that the specific heat of any solid in the Debye approximation is characterized by only one parameter T D  . Hence, in order to obtain accurate description of the solid specific heat, it is necessary to use a different value of D  at each temperature [140]- [142]. The determination of the temperature dependence of D  can be done by equating the measured values of the specific heat to the Debye expression for the specific heat, equation (IV.11), and consider the resulting equation as an equation for D  . The temperature-dependent D  of the surface-structured samples and that of the bare Si substrate are plotted together in Figure IV.4 by using symbols. Now, comparing equation (IV.9) and equation (IV.12) we see that:

  2 3 3 3 3 0 1 a k B D    (IV.13) Where   0 D 
is the Debye temperature at extremely low temperatures. On the other hand, equation (IV.12) gives the rigorous low temperature specific heat of a crystal. However, in view of equation (IV.9) if this is the case, D  must be a function of temperature whose value at any temperature is determined by equating equation (IV.12) and (IV.6). When this is done it is found that at low temperatures: IV.I. According to equation (IV.8), the coefficients 2 a and 4 a can be used to plot the low frequency end of the phonon spectra of the measured samples.

                     ...
In order to test the accuracy of the approach outlined above in determining the low frequency end of the phonon spectrum from a specific heat measurement, we have used it to determine the low frequency end of the phonon spectrum of the bare Si substrate and compared the results to first principle calculations of the phonon density of states of Si [START_REF] Ka Zan | First principles calculation of the thermal conductance of GaN/Si and GaN/Si interfaces as functions of the interface conditions[END_REF], [START_REF] Ka Zan | Interpolation Between the Acoustic ism atch ode l and the Diffuse Mismatch Model for the Interface Thermal Conductance: Application to InN/GaN Superlattice[END_REF]. The obtained curves are plotted together in The good agreement between the derived and calculated phonon spectra at low frequencies clearly demonstrates the adequacy of the used experimental approach in determining the low frequency end of the phonon spectrum. The low frequency phonon spectra of the surface-structured samples as derived by using the approach described above are presented and compared to the bare Si substrate phonon spectrum in Figure IV.6. It can be noticed that truncated cone-like structures of small height and broad apex (S4 and S5) do not alter the phonon spectrum at low frequencies. This can be expected because these structures can be regarded as a highly correlated surface roughness. However, periodically arranged cone-like structures presenting sharp apexes (S1 and S3) enhance the phonon frequency, and periodically arranged pillarlike structures (S2) and tall truncated cone-like structures (S6, S7, and S8) weaken the phonon spectrum at low frequencies. The enhancement of the phonon spectrum at low frequencies is certainly due to flatness in the acoustic phonon branches. This can be attributed to hybridization mechanisms between local resonances and bulk phonon modes. The reduction in the phonon spectrum at low frequencies, on the other hand, can be due to enhancement of phonon modes frequencies. This may be attributed to a hybridization mechanism between acoustic phonon modes of folded Brillouin zone and bulk phonon modes. Thus, cone-like structures of sharp apexes appear to flatten the acoustic phonon branches and reduce the acoustic phonon group velocities due to the hybridization mechanism between local resonances and bulk phonon modes, whereas pillar-like structures and truncated cone-like structures of height exceeding one micrometer appear to have the effect of increasing the slopes of the acoustic phonon branches, and consequently increasing the acoustic phonon group velocities due to a hybridization mechanism between phonon modes of a folded Brillouin zone and phonon modes of an unfolded Brillouin zone. In Supplementary Information, we use Houston's method [START_REF] Houston | Normal Vibrations of a rystal Lattice[END_REF] to show how the obtained coefficients of the low frequency end of the phonon spectrum are related to the phonon group velocities.

IV.3.2. High frequency phonon spectrum from specific heat measurements

The moment trace method has been widely used for the approximate calculation of the phonon spectrum [START_REF] Roll | Frequency Spectrum of r ystalline Solids[END_REF], [START_REF] Roll | Frequency Spectrum of r ystalline Solids. II. General Theory and Applications to Simple Cubic Lattices[END_REF]. The essence of this method is the expansion of

   g
in a series of Legendre polynomials in which the coefficients of successive terms are linear combinations of the even spectrum moments, which are defined as:

             1 0 2 1 2 2 2 3 1 3 1 dx x g x Tr rN rN L n n L j n n j n     k k k D k (IV.15)
Where   k D is the dynamical matrix. However, it has been shown that the expansion in Legendre polynomials gives a fairly acceptable indication of the gaps in the phonon spectrum, a satisfactory description of the phonon spectrum at high frequency, and a poor description of the low frequency end of the phonon spectrum [START_REF] Peaslee | Frequency Spectrum of r ystalline Solids. III. Body-entered ub ic Lattices[END_REF]. This can be understood if we realize that the low frequency end of the phonon spectrum is a slowly varying function of  and would be poorly reproduced by a linear combination of a few Legendre polynomials. Furthermore, the low frequency end of the phonon spectrum contributes very little to the moments, which is an additional reason this approximation might be expected to be a poor one at low frequencies. Therefore, the approximation presented below can be accurately employed for the calculation of the gaps in the spectrum and the phonon spectrum at high frequencies. We expand the phonon spectrum as:

              0 n L n n P a g    (IV.16)
Where the coefficients n a are given by: Thus, the approximate calculation of the phonon spectrum at high frequency can be made possible when the dimensionless spectrum moments n u 2 are determined.

           1 1 1 1 2 dx x P x g n a n L n  (IV.
Thirring made use of the moments of the phonon spectrum to find an expansion for the specific heat at high temperatures in inverse powers of the temperature [149]. He used the result:

            1 2 2 ! 2 1 2 1 1 n n n n x n x B x e x ,  2  x (IV.20)
Where n B 2 are the Bernoulli numbers, to expand the second term on the right hand in the expression of the total energy:

                                   rN j B j j j T k k k k T E 3 1 1 exp 2       (IV.21)
Then, upon interchanging the order of summations, Thirring found that: This curves fitting technique allowed deducing only three spectrum moments for each measured samples. The obtained moments are presented in Table IV.I. The fact that each investigated sample is characterized by a different spectrum moments set suggests that, besides the low frequency end of the phonon spectrum, the phonon spectrum at high frequencies can also be modified by surface structuration. Unfortunately, three moments were not sufficient to precisely describe the entire features of the phonon spectrum at high frequencies. In fact, it has been shown that at least fourteen spectrum moments are required to synthesize the main gaps in the high frequency phonon spectrum [START_REF] Peaslee | Frequency Spectrum of r ystalline Solids. III. Body-entered ub ic Lattices[END_REF]. However, we believe that fitting equation (IV.22) to specific heat curves in the full temperature range (up to a temperature close to the material melting point) might provide enough spectrum moments for accurate experimental description of the phonon spectrum at high frequencies. Such studies are under consideration.

                                           n n n n n B v v u T n n B rNk T T E T C 2 1

IV.4. Conclusion

In summary, we have measured the heat capacities of bare and surface-structured Si substrates originated from the same Si wafer over a 3-300 K temperature range. The results showed that the heat capacity curves of the surface-structured substrates significantly differ from that of the bare Si substrate in many aspects. We attributed this difference to an important modification in the phonon spectrum due to a phonon hybridization mechanism. In order to determine the phonon spectra of the measured samples at low frequencies we took advantage of the fact that at low temperatures the expression for the heat capacity contains an exponential factor that cuts out contributions from all except small values of frequency and temperature, and expanded the low frequency end of the phonon spectrum in terms of polynomials of even coefficients. The obtained results showed that surface structuration may strongly modify the bulk phonon spectrum and group velocity of long wavelength phonons, which determine the elastic properties and are the principal heat carriers in semiconductor materials. We have also modeled the measured heat capacity at high temperatures by employing Thirring's e pansion technique, which consists in a formal high temperature expansion for the heat capacity in terms of the moments of the phonon spectrum. We have found that the surface structuration may also affect the phonon spectrum moments, and consequently alter the phonon spectrum at high frequencies. Thus, the results presented in this chapter clearly demonstrate the possibility of modifying mechanical, thermal, and infrared optical properties of bulk materials by surface structuration, and may open the door for the development of higher-performance technological metamaterials, either completely new or through surface structuration of existing ones.

: Specific heat of free-surface ZnO nanowires Chapter V and thermal interface conductance

In this chapter, we report calorimetric measurements of specific heat of free-surface zinc oxide nanowires (NWs) and thermal conductance of the interface between vertically aligned ZnO NWs and silicon (Si) substrate. Heat-pulse calorimetric measurements were carried out on vertically aligned free-surface ZnO NWs on Si substrate. The entire temperature response of the heat-pulse calorimeter was fitted to a model that takes into account the effect of the conductance of the thermal bond of ZnO NWs to the Si substrate. Linear least squares method was used to determine the specific heat of the measured ZnO NWs and the thermal conductance of the interface between the Si substrate and ZnO NWs from 1.8 to 300 K. It is found that at low temperatures (below 4 K) the ZnO NW specific heat exhibits a clear contribution from an essentially two-dimensional crystal. However, above 25 K, the ZnO NW specific heat begins to show enhancement compared to bulk ZnO, and the enhancement factor increases as the temperature increases and the NW diameter decreases. The thermal conductance of the interface between Si substrate and ZnO NWs is found to be orders of magnitude lower than that between bulk ZnO and Si substrate, suggesting the formation of thin layer of low crystallinity between ZnO NWs and Si. Furthermore, the recorded data demonstrated transition from specular to diffusive elastic transmission, and then from diffusive elastic to diffusive inelastic transmission as the temperature increases.

V.1.Introduction

Let us recall that the successful processing and control of low-dimensional materials have paved the way for the development of low-dimensional quantum devises. Appropriate thermal management is indeed essential for the success of these devices. However, the design of optimal and robust thermal management strategy requires clear understanding of the laws that govern the thermal properties and thermal transport in low-dimensional materials. Therefore, intensive research has been dedicated to understanding the thermal properties and fundamentals of heat transport in low-dimensional materials. Although significant progress has been achieved in this field, there are still important issues that need to be addressed.

The strong difference between the specific heat of nanoparticles and that of their bulk counterparts has been clearly demonstrated by a series of measurements and theoretical approaches [START_REF] Novotny | Effect of Size and Surface on the Specific Heat of Small Lead Particles[END_REF]- [START_REF] Lee | Quantum Size Effect on the Lattice Specific Heat of Nanostructures[END_REF]. The results demonstrated that the specific heat curves of spherical nanoparticles deviate significantly from the Debye T 3 law, and are strongly enhanced over their bulk counterparts up to a few tens of degrees Kelvin. The deviation of the nanoparticle specific heat value from the bulk specific heat value, however, weakens to become negligible as the temperature increases [START_REF] Novotny | Effect of Size and Surface on the Specific Heat of Small Lead Particles[END_REF]- [START_REF] Neeleshwar | Size-dependent properties of dSe quantum dots[END_REF]. On the other hand, it was predicted that the specific heat values of cube-like nanoparticles would be lower than the bulk values at low temperatures [START_REF] Wang | Surface and Size Effects on the Specific Heat apacity of Nanoparticles[END_REF], [START_REF] Lee | Quantum Size Effect on the Lattice Specific Heat of Nanostructures[END_REF]. The observed size-and shape-dependent specific heat was basically attributed to size-and shape-dependent phonon density of states and surface density of states determining the fundamental thermal properties of the measured nanoparticles [START_REF] Lin | Surface/interface influence on specific heat capacity of solid, shell and core-shell nanoparticles[END_REF].

The observed anomalous specific heat of single-wall carbon nanotube also revealed quantum size effects on the nanotube phonon density of states [START_REF] Hone | Quantized phonon spectrum of single-wall carbon nanotubes[END_REF]. Thin multiwalled carbon nanotube was found to exhibit a linear specific heat over a few tens of degrees Kelvin, indicating a weak inter-wall coupling and a constant phonon density of states [START_REF] Yi | Linear specific heat of carbon nanotubes[END_REF]. Moreover, it was found that when multiwalled carbon nanotubes are grouped together in the form of a rope, they exhibit a specific heat larger than that of graphite and of isolated nanotube below 50K [START_REF] Mizel | Analysis of the low-temperature specific heat of multiwalled carbon nanotubes and carbon nanotube ropes[END_REF]. Calorimetric measurements on titanium dioxide nanotubes offered further information about the phonon density of states and specific heat in nanotubes, which are rolled-up versions of two-dimensional materials [START_REF] Dames | Low-dimensional phonon specific heat of titanium dioxide nanotubes[END_REF]. It was shown that the anatase nanotube specific heat always exceed that of bulk anatase. From 95 K down to 60 K, the enhancement is approximately 20-30%, and from 50 K down to 5 K, the nanotubes specific heat curves exhibit a T 2.6 dependence rather than T 3 dependence. However, below about 3 K, the nanotubes specific heat curves show an additional transition to nearly constant specific heat values, exceeding the bulk specific heat values by factors of 25 to 50 at about 1.5 K. The experimental observations were attributed to transition from three-dimensional density of states to low-dimensional density of states as the average phonon wavelength becomes comparable to the wall thickness. Multiwalled carbon-like nanotubes, such as multiwalled boron nitride nanotubes, showed the same behavior as the multiwalled carbon nanotubes [162].

In the case of nanowires, theoretical approaches predicted that phonon quantum confinement mechanisms have observable effects only below a few degrees Kelvin and only in nanowires of diameters not exceeding 6 nm [START_REF] Lee | Quantum Size Effect on the Lattice Specific Heat of Nanostructures[END_REF], [START_REF] Prasher | Appro imate Analytical Models for Phonon Specific Heat and Ballistic Thermal Conductance of Nanowires[END_REF]- [START_REF] Huang | A theoretical study of the specific heat and Debye temperature of low-dimensional materials[END_REF]. Nevertheless, the presence of free-surface gives rise to distinct size effects on the thermal properties of nanowires even in the absence of phonon quantum effects [START_REF] Lee | Quantum Size Effect on the Lattice Specific Heat of Nanostructures[END_REF], [START_REF] Pascual-Gutiérrez | Thermal conductivity and phonon transport properties of silicon using perturbation theory and the environment-dependent interatomic potential[END_REF]- [166]. Experimental investigation has been carried out to explore the nanowire specific heat [START_REF] Kurtz | Specific Heat of Superconducting n Nanowires[END_REF]- [START_REF] Pradhan | Specific heat and thermal conductivity measurements for anisotropic and random macroscopic composites of cobalt nanowires[END_REF]. However, no strong conclusions could be drawn from the reported experimental results because the measurements were performed at extremely low or high temperatures [START_REF] Kurtz | Specific Heat of Superconducting n Nanowires[END_REF], [168], or on nanowires embedded in a matrix, which makes their surfaces clamped and masks the lookedfor contribution from the surface [START_REF] Pradhan | Specific heat and thermal conductivity measurements for anisotropic and random macroscopic composites of cobalt nanowires[END_REF]. Therefore, in this work, we tackle this issue and present a thorough experimental study of specific heat of free-surface zinc oxide nanowires (ZnO NWs) from 1.8 to 300 K. We demonstrate that the low temperatures specific heat curve of a free-surface nanowire exhibits a bulk specific heat curve with an additional contribution from a two-dimensional crystal.

As for the thermal transport in low-dimensional materials, efficient experiments have been designed to measure the thermal conductivity in low-dimensional materials [126], [START_REF] Hochbaum | Enhanced thermoelectric performance of rough silicon nanowires[END_REF], [START_REF] Boukai | Silicon nanowires as efficient thermoelectric materials[END_REF]- [START_REF] Seol | Two-dimensional phonon transport in supported graphene[END_REF], and precise theories have been developed to construe the experimental results [START_REF]Ballistic-Diffusive Heat-onduction Equations[END_REF]- [START_REF] Yang | How does folding modulate thermal conductivity of graphene?[END_REF]. Ultrahigh thermal conductivity has been measured in graphene and several technologies have been established to tailor the thermal conductivity of graphene nanoribbon [START_REF] Xiong | Heat conduction in twodimensional disk models[END_REF]- [START_REF] Ouyang | Thermal transport in graphyne nanoribbons[END_REF]. Moreover, the experimental and theoretical advancements revealed interesting physical phenomena in graphene nanoribbon, such as the resonant splitting of phonon transport in periodic T-shaped graphene nanoribbon and the important role of the so-called edge modes in the heat transport [START_REF] Yang | Resonant splitting of phonon transport in periodic T-shaped graphene nanoribbons[END_REF], [START_REF] Jiang | onditions for the e istence of phonon localized edge-modes[END_REF].

The phonon processes resulting in ballistic and diffusive thermal transport modes in single-wall and multiwall carbon nanotubes also have been in the spotlight of research in the last decade [192], [START_REF] Zhang | Anomalous vibrational energy diffusion in carbon nanotubes[END_REF]. However, the thermal transport in semiconductor nanowires have attracted the greatest attention because their low thermal conductivity values compared to those of their bulk counterparts make them potential candidates for applications in solid state thermoelectric devices [START_REF] Hochbaum | Enhanced thermoelectric performance of rough silicon nanowires[END_REF], [START_REF] Boukai | Silicon nanowires as efficient thermoelectric materials[END_REF]. It is demonstrated that the high rate of phonon scattering from high surface to volume ratio and phonon confinement mechanisms make the thermal conductivity of nanowires orders of magnitude smaller than that of their bulk counterparts [126], [START_REF] Bui | Diameter-Dependent Thermal Transport in Individual ZnO Nanowires and its Correlation with Surface Coating and Defects[END_REF], which markedly rise their thermoelectric figure of merit. Theoretical approaches have been developed to understand the dependence of the nanowire thermal conductivity on many parameters such as shape, size, length, surface asperities, purity, and elastic field amount in order to further lower the nanowires thermal conductivity and achieve a desired thermoelectric figure of merit value for efficient conversion of waste heat into electrical power [START_REF] Nuo | Ultralow Thermal Conductivity of Isotope-Doped Silicon Nanowires[END_REF]- [START_REF] Donadio | Temperature Dependence of the Thermal onductivity of Thin Silicon Nanowires[END_REF]. It is also predicted that core/shell nanowires and vertical ultrathin nanowires array on ultrathin plate induce localized phonon modes, which reduce the effective thermal conductivity of the system beyond the thermal conductivity of the individual nanowires [START_REF] Davis | Nanophononic etamaterial Thermal onductivity Reduction by Local Resonance[END_REF], [START_REF] Zen | Engineering thermal conductance using a two-dimensional phononic crystal[END_REF]- [START_REF] Honarvar | Spectral energy analysis of locally resonant nanophononic metamaterials by molecular simulations[END_REF], [START_REF] Hen | Impacts of Atomistic oating on Thermal onductivity of Germanium Nanowires[END_REF]- [START_REF] Anufriev | Heat conduction engineering in pillar-based phononic crystals[END_REF] without introducing undesirable obstacles for the electronic transport. Moreover, vertically aligned nanowires grown on substrates have shown thermal conductivity values below the thermal conductivity values of the individual nanowires [START_REF] Pradhan | Specific heat and thermal conductivity measurements for anisotropic and random macroscopic composites of cobalt nanowires[END_REF], [START_REF] Zhang | Ultralow thermal conductivity of silicon nanowire arrays by molecular dynamics simulation[END_REF]- [START_REF] Xie | Thermal conductivity of a n na nowire/silica aerogel nanocomposite[END_REF], and consequently enhanced thermoelectric figure of merit [START_REF] Hang | Enhancement of thermoelectric figure-of-merit in laterally-coupled nanowire arrays[END_REF], [START_REF] Zhang | High thermoelectric figure-of-merits from large-area porous silicon nanowire arrays[END_REF]. The observed suppression in the thermal conductivity and enhancement in the thermoelectric figure of merit were attributed to enhanced phonon-boundary scattering rate or to strong lateral coupling between the nanowires. However, the thermal resistance between the nanowires and the underlying substrate, which may have the major contribution to the suppression of the thermal conductivity [START_REF] Anufriev | Aluminium nanopillars reduce thermal conductivity of silicon nanobeams[END_REF], has never been measured. In this work, we also address this issue and report calorimetric measurements of thermal conductance interfaces between Si and ZnO NWs 1.8 to 300 K. We demonstrate that the thermal conductance of the interface between Si and ZnO NWs is orders of magnitude lower than that of the interface between Si and bulk ZnO in a wide temperature range because of the formation of a thin layer of low crystallinity between the NWs and the Si substrate. We also demonstrate that transitions between several regimes of phonon transport across the interface between ZnO NWs and Si occur as the excited phonon wavelengths change.

V.2.Experiments

The synthesis of vertically aligned ZnO NWs on Si substrate has been accomplished in two steps. 15-20 nm thick ZnO seed layer has first been synthesized, then ZnO NWs have been grown on it. The ZnO seed layer consists of ZnO nanoparticlas. It acts as nucleation layer for the growth of ZnO NWs. It was prepared by dissolution of 2.197 g of zinc acetate (Zn(CH 3 COO 2 ) 2 ,2H 2 O) in 20 ml of ethanol by stirring constantly for 24 hours. Then, the obtained solution was spincoated onto a clean Si substrate. After spin-coating, the seed layer was annealed at 400° C on a hot plate for 10 minutes. The spin-coating was repeated twice, and the sample was annealed after each spin-coating step. Chemical bath deposition (CBD) technique was used to grow the ZnO NWs. An amount of 0.025 M zinc acetate was dissolved in 250 ml of deionized water. Then, the solution was stirred for a few minutes and 0.3 ml of ammonium hydroxide (28% concentration) was added to the solution under constant stirring. The obtained solution was then heated to 87°C in a three-neck flask. Then, the Si substrate with the ZnO seed layer was introduced into the solution for NWs growth. The concentration of zinc acetate and dipping time were varied to obtain ZnO NWs of different diameters and lengths. Three samples were selected to carry out the calorimetric measurements. These The calorimetric measurements were performed in the heat-pulse calorimeter of a Quantum Design Physical Property Measurement System. The design of this calorimeter can be found in Ref. [66]. However, the thermal response of the calorimeter was analyzed by taking into account the conductance of the thermal bond of the ZnO NWs to the Si substrate instead of using the standard relaxation method [66], [START_REF] Shepherd | Analysis of the lumped τ 2 effect in rela ation calorimetry[END_REF]. The drawings in Figure V.2 describe the measurement procedure. Before loading any samples, a small amount of Apiezon N grease was affixed to the sample platform, and the temperature response of the addenda (platform plus grease) was recorded in a measurement cycle (a heating period followed by a cooling period). Then, the heat balance equation:

        0 1 1 1 T t T dt t dT c t P w     (V.1)
Was fitted to the recorded thermal response using linear least squares method to obtain the specific heat of the addenda ( 1 c ) and the thermal conductance of the wires ( w  ) that connect the sample platform to a heat sink maintained at temperature 0 T . In equation (V.1),   t P is the input power in the sample platform and   t T 1 is the temperature of the platform at instant t. As a second step, a bare Si substrate was weighted, placed onto the Apiezon N grease, and then the thermal response of the calorimeter was recorded in a measurement cycle. In the presence of the Si substrate, the thermal response should be modeled with the following set of heat balance equations [66]:

              t T t T T t T dt t dT c t P G w 2 1 0 1 1 1        (V.2a)         t T t T dt t dT c G 1 2 2 2 0     (V.2b)
Where 2 c is the specific heat of Si, G  is the thermal conductance of the Apiezon N grease, and   t T 2 is the temperature of the Si substrate at instant t. The specific heat of the Si substrate and the thermal conductance of the Apiezon N grease were obtained by eliminating   t T 2 in the above heat balance equations, integrating over the measurement time period, and then fitting the resulting equation for   t T 1 to the measured thermal response of the calorimeter using linear least squares method [66]. The measured specific heat of the Si substrate is shown in Figure V.3. The measurement uncertainties are mainly determined by the scatter in the data points, and are estimated to not exceed 2%. The overall specific heat curves of the samples WP1, WP2, and WP3 measured with the same technique used for the Si substrate are also plotted in Figure V.3. The presence of ZnO NWs on the Si substrate significantly alters the thermal response of the calorimeter. Thus, the ZnO NWs have a measurable effect on the temperature rise and drop in a measurement cycle, and consequently can be measured using a calorimetric technique. )

Temperature (K)

After measuring the grease heat conductance and Si substrate specific heat, the mass of ZnO NWs was deduced from electronic scanning measurements of their concentration, average diameter, and average length in each sample, and the thermal responses of the calorimeter were measured when the samples are placed onto the Apiezon N grease. In this step, the measurement time was set to be longer than the time it takes for the Si substrate and ZnO NWs to reach thermal equilibrium. In that case, the calorimeter thermal response should satisfy the following heat balance equations:

              t T t T T t T dt t dT c t P G w 2 1 0 1 1 1        (V.3a)               t T t T t T t T dt t dT c I G 3 2 1 2 2 2 0        (V.3b)       t T t T dt dT c I 2 3 3 3 0     (V.

V.3. Results And Discussion

In this section, we present the experimental results and develop physical studies on the NWs specific heat and the thermal conductance of the interface between the Si substrate and ZnO NWs.

V.3.1. Specific heat of ZnO nanowires

The measured specific heat curves of the ZnO NWs are shown in Figure V.5 together with calculated specific heat of bulk ZnO [START_REF] Grimvall | Lattice instabilities in metallic elements[END_REF]. The uncertainties on the ZnO NWs data are dominated by the scatter in the data points in each measurement step and the error on the evaluation of the weight of the ZnO NWs in each measured sample. They are estimated as 10%. It is worth noting here that Figure V.5 shows constant-pressure specific heat curves because measurements of constant-volume specific heat (which is a quantity that can be described with easier and more quantifiable models) are hard to arrange experimentally. However, one can show that at room temperature and below, the difference between constantpressure and constant-volume specific heat is smaller than the experimental error [START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF]. Consequently, the modeling of the measured constant-pressure specific heat curves with models established for constant-volume specific heat in what follows is justified. The results in Figure V.5 indicate that above 25 K, the specific heat of ZnO NW begins to show remarkable enhancement compared to bulk ZnO, and the enhancement factor increases in the entire measured temperature range as the ZnO NW average diameter decreases. T curves are given in Table V.I. The good agreement between the measured and calculated T c versus 2 T curves demonstrates that besides a contribution from a three-dimensional crystal, the NW specific heat exhibits an appreciable contribution from an essentially twodimensional crystal.

                       Q D c L D c T pN T pN T c    (V.

V.3.2. Thermal conductance between Si substrate and ZnO nanowires

The curves of the thermal conductance of the interface between Si and ZnO NWs (Si/ZnO NWs) obtained from the calorimetric measurements are plotted in Figure V.7. The thermal conductance of the interface between Si and bulk ZnO (Si/ZnO) obtained from first principles calculations within the diffuse mismatch model [START_REF] Reddy | Diffuse mismatch model of thermal boundary conductance using e act phonon dispersion[END_REF] is also plotted in Figure V.7 for the sake of comparison. The results show that the thermal conductance of Si/ZnO NWs is independent of the ZnO NWs average size.

Furthermore, Figure V.7 shows a noticeable difference in the thermal conductance of Si/ZnO NWs and Si/ZnO, both in magnitude and in temperature dependence. The thermal conductance of Si/ZnO NWs is about five orders of magnitude lower than that of Si/ZnO. The low thermal conductance value of Si/ZnO NWs compared to that of Si/ZnO can be attributed to the formation of a thin layer of low crystallinity at the interface between ZnO NWs and Si leading to phonon scattering at a high rate [START_REF] Srivastava | The physics of phonons[END_REF]. This conclusion is supported by recent transmission electron microscopy [START_REF] Anufriev | Heat conduction engineering in pillar-based phononic crystals[END_REF], which demonstrated that the thermal conductivity in phononic metamaterials is reduced mainly by phonon scattering in an intermixing region formed at the interface between the nanostructures and the underlying substrate. It can also be noticed from Figure V.7 that the thermal conductance of Si/ZnO NWs drops rapidly between 1.8 and 3 K, follows the behavior of Si/ZnO between 3 and 70 K, and then grows almost linearly at higher temperatures. We believe that the rapid drop between 1.8 and 3 K is due to a transition from specular to diffusive phonon transmission as the temperature increases, or alternatively as the excited phonon wavelengths change. This is supported by measurements of reflection of heat pulses from free solid surfaces, which showed that the phonon transmission changes from specular to diffusive when phonons of frequencies greater than 100 GHz are excited at a few degrees Kelvin [START_REF] Eisenmenger | Phonon Scattering in Condensed Matter : Proceedings of the Fourth International Conference University of Stuttgart[END_REF]. The near linear growth of the thermal conductance of Si/ZnO NWs above 70 K can be attributed to a significant contribution of anharmonic phonon processes (which are not taken into account in the diffused mismatch model) to the transfer of energy across the interface between Si and ZnO NWs. The contribution of these anharmonic phonon processes to the interface thermal conductance and the associated linear growth of the measured interface thermal conductance curves have been previously observed in the cases of interfaces between materials with highly asymmetric phonon spectra [START_REF] Lyeo | Thermal conductance of interfaces between highly dissimilar materials[END_REF]. Therefore, we believe that the strong mismatch between the phonon spectrum of ZnO NWs and that of Si promotes the contribution of anharmonic phonon processes to the thermal transport across the interface between the Si and the ZnONWs. As the temperature increases, the phonon anharmonic processes enhance, leading to an excess conductance at the interface between Si and ZnONWs, which is revealed in the linear temperature dependence of the interface thermal conductance above 70 K.

V.4. Conclusion

Heat pulse calorimetric measurements were performed on free-surface vertically aligned zinc oxide nanowires (ZnONWs) on Si substrate from 1.8 to 300 K. The thermal response of the calorimeter was analyzed by taking into account the effect of the conductance of the thermal bond of ZnO NWs to the Si substrate. Linear least squares techniques were used to deduce the ZnO NW specific heat and the thermal conductance of the interface between Si and ZnO NWs. The results indicated that above 25 K, the ZnO NW specific heat begins to show enhancement compared to bulk ZnO, and the enhancement factor increases with the temperature and the NW diameter. However, at low temperatures (below 4 K) the ZnO NW specific heat exhibits a clear contribution from an essentially two-dimensional crystal. Theoretical modeling demonstrated that phonon modes associated with quadratic dispersions give rise to contribution from three-dimensional crystal, whereas phonon modes associated with linear dispersions give rise to contribution from two-dimensional crystal to the NW specific heat at low temperatures. The thermal conductance of the interface between Si and ZnO NWs is found to be five orders of magnitude lower than the theoretical value of the thermal conductance of the interface between Si and bulk ZnO. The low thermal conductance of the interface between Si and ZnO NWs was attributed to the possible formation of thin layer of low crystallinity between ZnO NWs and the Si substrate. A sharp drop in the measured thermal conductance between 1.8 and 3 K was interpreted as a transition from specular to diffusive phonon transmission. A near linear temperature dependence in the measured thermal conductance above 70 K was

Conclusion and perspectives

This dissertation aimed at providing fundamental understanding of the role of surfaces in the thermal, thermodynamic and optical properties of solids. An attempt is made to use various experimental techniques and analysis methods to describe how the presence of free-surfaces modifies the materials properties. The adequacy and range of validity of these experimental and analysis methods are evaluated, and suggestions are made concerning possible experimental and theoretical investigations of surface effects. A major part of the manuscript is devoted to describe how the presence of surface modifies the phonon spectrum and the related material properties.

First, a review the general theory of heat transfer and photon-phonon interaction was presented. Certain aspects related to the phonon anharmonicity were discussed. A special consideration was given to the effect of quantum phonon confinement on the light-matte interaction strength and to the boundary conductance. Then, a detailed description of the samples growth procedures as well as the experimental techniques used in the research work reported in this manuscript was presented.

The usual approach to analyze experimental phonon excitation is to assume cyclic boundary conditions. Such a treatment, while mathematically convenient, eliminates the possibility of studying the dynamical properties of atoms in the neighborhood of a free surface of a real crystal. The reason is that the crystal atomic configuration in the surface layers differ from the atomic configuration in the bulk of the crystal. The forces acting on atoms located at the surface layers are different from the forces acting on atoms in the bulk. An atom in the surface layers has fewer nearest neighbors than an atom in the volume of a crystal. Therefore, one would expect that the phonon dynamical properties and the resultant thermal and thermodynamic properties of the surface and the volume differ. Thus, when the crystal size becomes small enough so that the ratio of surface to volume is no longer negligible, the modification of the frequency distribution function of the crystal due to the presence of free surfaces alter the thermal and thermodynamic properties of the crystal and gives rise to distinct size effect. Furthermore, selection rules determining physical properties of infinite crystals can be relaxed for finite crystals or for atoms in the surface layers for which the symmetry properties no longer hold. Consequently, one would expect that the thermal and thermodynamic properties of finite (nanosized) crystals would show specific features that bulk crystals do not exhibit. The third, fourth, and fifth chapter have presented the fundamental understanding of the surface effect on phonon related properties.

In the third chapter (III) of this dissertation, we demonstrate that due to the presence of confined phonons on the surface, phonons and photons of different momenta can interact with each other within the same nanostructure. Raman spectra from large arrays of dielectric Si resonators attached to Si substrates showed an appreciable Raman enhancement accompanied by a downshifting and broadening relative to the Raman spectrum from a bare Si substrate. FDTD simulations demonstrated that the collective lattice resonance induces an efficient field confinement in the resonators, which results in an appreciable Raman enhancement. A spatial correlation model demonstrated that the spectrum downshift and broadening is the result of the relaxation of the phonon wavevector, which is due to phonon confinement in nanosized Si crystallites in the surface layers of the resonators. The effect of resonators size and shape on the confinement of photons and phonons was investigated. It was demonstrated that, as the resonators increase in height and their shape becomes cylindrical, the amplitude of their coherent oscillation increases and hence their ability to confine the incoming electric field increases. However, it was found that the resonators shape and size do not have a remarkable effect on the phonon confinement. The results presented in the third chapter show that large arrays of long cylindrical dielectric Si resonators drastically suppress the light scattering losses by field confinement and promote the interaction of the confined field with the confined phonons. The results highlighted in this chapter are expected to help to gain more insight into the confinement of phonons and photons and rationally design metasurfaces for high efficiency optical devices. For instance, the observed amplified photon-phonon non-conservative interaction mechanism has great potential for strongly enhancing the photovoltaic effect in Si.

In the fourth chapter (IV), we present experimental evidence on the change of the phonon spectrum and vibrational properties of a bulk material through phonon hybridization mechanisms through surface nanostructuring. We demonstrate that the phonon spectrum of a bulk material can be altered by hybridization mechanisms between confined phonon modes in nanostructures introduced on the surface of a bulk material and the underlying bulk phonon modes. We also show that the shape and size of the nanostructures made on the surface of the bulk substrate have strong effects on the phonon spectrum of the bulk material.

In the fifth chapter (V), we demonstrate that at low temperatures (below 4 K) the nanowire specific heat exhibits a clear contribution from an essentially twodimensional crystal (the nanowire surface). However, above 25 K, the nanowire specific heat begins to show enhancement compared to its bulk counterpart, and the enhancement factor increases as the temperature increases and the NW diameter decreases. We also demonstrate that the thermal conductance of the interface between a substrate and vertically aligned nanowires is orders of magnitude lower than that between bulk materials, suggesting that a thin layer of low crystallinity forms at the between nanowires and a substrate. Furthermore, we present clear evidence on transition from specular to diffusive elastic transmission, and then from diffusive elastic to diffusive inelastic transmission occur as temperature increases.

It is to be noted that in the fourth and fifth chapter, calorimetric measurements were performed in the heat-pulse calorimeter of a Quantum Design Physical Property Measurement System. However, the thermal response of the calorimeter was analyzed by taking into account the conductance of the thermal bond of the nanostructures to the substrate instead of using the standard relaxation method.

Perspectives: In the third chapter, we demonstrate that arrays of resonators, depending on their shape and size and the incident light wavelength, either forward scatter the incoming light toward the substrate on which they are attached or promote the confinement of the incident electric field in them. We demonstrate that such optical resonance in dielectrics is beneficial for energy harvesting through photovoltaic conversion. Let us consider now what would happen when resonators made of wide bandgap semiconductor material (such as silicon carbide) are attached to a substrate made of small bandgap semiconductor material (such as silicon). If the size and shape of the resonators were properly optimized, the resonators would forward scatter the wavelengths of the visible spectrum toward the substrate and confine the wavelength of the ultraviolet spectrum in them. Consequently, an important amount of the solar energy will contribute to the photovoltaic generation of electricity, and the photovoltaic efficiency of the socalled tandem solar cells can be remarkably enhanced. The design of highefficiency surface-structured tandem solar cell will be the subject of future research work.

In the fourth chapter, we demonstrate that phonon wave-packets confined in lowdimensional resonators can hybridize with running phonon wave-functions in the substrate on which the resonators are attached and change the phonon spectrum of the substrate. If the size and shape of these resonators were properly optimized to give a high density of singularities in the resultant phonon spectrum, the resultant system would present a high thermal resistance and becomes suitable for highefficiency thermoelectric applications. The design of metamaterials for highefficiency thermoelectric applications will also be the subject of future research work.

Finally, in the fifth chapter, we demonstrate that calorimetric measurements analyzed using a model that takes into account the effect of the thermal bond between the nanostructures and the substrate to the temperature response of the calorimeter can provide accurate description of the boundary thermal resistance of the nanostructures. This unique measurement technique can later be used to 123 examine experimentally the possibility of thermal rectification in nanostructure and the design of nanosized thermal diodes for many novel technological applications.

Résumé en français

Cette thèse menée en collaboration entre l'UTT (Université de Technologie de Troyes) et l'AUB (American University of Beyrout) avait pour cadre, le contrôle du transport thermique via les phonons et leur interaction avec des photons dans des nanostructures. Le manuscrit comprend cinq chapitres. Dans le premier, nous introduisons la physique des phonons et excitations élémentaires optiques de la matière. Le deuxième chapitre fournit une description des procédés de croissance, techniques de structuration et techniques de caractérisation utilisées. Dans le troisième chapitre, nous démontrons qu'à la fois, phonons et photons peuvent être confinés et interagir dans une même nanostructure. Dans le quatrième chapitre, nous montrons expérimentalement que le spectre de phonons d'un matériau peut être modifié par des mécanismes d'hybridation entre des modes de surface introduits par une nanostructuration et les modes normaux du matériau massif. Nous montrons que la forme et la taille des nanostructures sur la surface du matériau ont des effets sur le spectre de phonons du substrat. Dans le cinquième chapitre, nous montrons qu'à basse température (inférieure à 4 K), la chaleur spécifique de nanofils est équivalente à celle d'un cristal essentiellement bidimensionnel. Encore plus étonnant à l'interface entre les nanofils et le substrat, nous avons mis en évidence une transition entre une transmission élastique spéculaire et une transmission élastique diffuse. Lorsque la température augmente on observe alors une transition entre une diffusion élastique et une diffusion inélastique. L'ensemble de ces résultats laisse entrevoir des perspectives intéressantes pour le contrôle des propriétés thermiques de matériaux massifs par nanostructuration de surface.

Nous avons privilégié pour le résumé la traduction des chapitres correspondant aux résultats. Le I du résumé correspond au troisième chapitre, le II au quatrième chapitre et enfin le III au cinquième chapitre.

I. Interaction entre phonons confinés et photons présents dans des résonateurs périodiques en silicium

Dans ce travail, nous démontrons que les phonons et les photons de différents moments peuvent être confinés et interagir les uns avec les autres au sein de la même nanostructure. L'interaction entre des phonons confinés et des photons confinés dans des réseaux de résonateurs de silicium est observée au moyen de la spectroscopie Raman. Les spectres Raman provenant de réseaux de résonateurs en silicium présentent une exaltation Raman accompagnée d'un décalage vers le rouge et d'un élargissement. L'analyse de l'intensité Raman et de la forme des raies à l'aide de simulations temporelles à différences finies (FDTD) et d'un modèle de corrélation spatiale a démontré une interaction entre les photons confinés dans les résonateurs et les phonons confinés dans des régions fortement défectueuses. Nous montrons que l'e altation Raman est due à la résonance collective du réseau induisant le confinement du champ dans les résonateurs, tandis que le rétrécissement et l'élargissement du spectre sont des signatures du relâchement du vecteur d'onde phonon dû au confinement des phonons dans les régions défectueuses des résonateurs. Nous constatons que lorsque les résonateurs augmentent en hauteur et que leur forme devient cylindrique, l'amplitude de leur oscillation cohérente augmente et, par conséquent, leur capacité à confiner le champ électrique entrant augmente.

EXPERIENCES

Les résonateurs sous la forme de piliers et de cônes tronqués ont été fabriqués en Si par gravure ionique réactive (RIE) en utilisant un masque dur. Une technique spéciale appelée MUZ utilisant ZnO comme masque dur a été employée [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. Alors que le ZnO peut être facilement dissous chimiquement, il est difficile à attaquer par gravure sèche. Pour définir les modèles de masque, une lithographie interférentielle utilisant une source laser à 355 nm a été utilisée. Un réseau carré avec des motifs circulaires d'une période de 660 nm a été créé par une double exposition de l'échantillon, ce qui a permis de contrôler le diamètre des caractéristiques en modifiant le temps d'exposition [START_REF] Gwiazda | Simple and Versatile High Aspect Ratio Nanostructuring via inc ide a sking[END_REF]. Ensuite, la gravure au Si a été effectuée dans un réacteur Plassys MU400 en utilisant un mélange de gaz SF 6 /O 2 . Une sélectivité d'environ 100 a été démontrée avec Si utilisant ZnO comme masque dur. Trois séries d'échantillons ont été obtenues. Le premier ensemble est constitué de deux échantillons (S S,1 et S S,2 ) dans lesquels les résonateurs ont presque la même hauteur. Cependant, les résonateurs du premier échantillon sont en forme de cône, tandis que les résonateurs du second échantillon sont en forme de pilier. Le second ensemble comprend trois échantillons (S M,1 , S M,2 , et S M,3 ). La hauteur des résonateurs de cet ensemble est supérieure à celle des résonateurs du premier ensemble. Les échantillons du deuxième ensemble, comme dans le premier ensemble, ne diffèrent que par le fait que les formes de leurs résonateurs sont différentes. Le troisième ensemble comprend trois échantillons (échantillons S L,1 , S L,2 , et S L,3 ). Les résonateurs dans cet ensemble sont les plus élevés, et encore une fois la seule différence entre les échantillons est la forme de leurs résonateurs. Dans la figure 1, nous présentons des micrographies électroniques à balayage (MEB) de quatre réseaux différents en tant qu'illustration de la diversité des résonateurs de Si étudiés dans ce travail.

FIG. 1 Micrographies électroniques à balayage (MEB) de réseaux périodiques de résonateurs de Si de différentes tailles et formes.

Dans le tableau I, nous présentons une description structurale détaillée de tous les échantillons étudiés.

Les mesures Raman ont été effectuées dans la configuration de rétrodiffusion en utilisant une ligne laser de 532 nm provenant d'un laser Ar-ion. Tableau I Description structurale détaillée et les paramètres obtenus des distributions de taille des cristallites dans la couche de surface des résonateurs de Si dans les échantillons étudiés. Les indices S, M et L se réfèrent à "court", "moyen" et "long", respectivement.

RESULTATS ET DISCUSSION

Les échantillons étudiés dans ce travail sont constitués de résonateurs de Si disposés périodiquement dans un réseau bidimensionnel sur un substrat de Si, et la structuration de surface de Si a été faite à basse température pour empêcher toute diffusion d'impuretés. Par conséquent, nous n'attendons pas de champs élastiques induits par le stress ou l'impureté qui contribuent aux spectres Raman mesurés. Afin d'exclure l'effet du chauffage laser et de la résonance Fano induite par laser, nous avons diminué successivement la puissance du laser d'excitation jusqu'à ce que la position et l'élargissement du pic Raman deviennent indépendants de la puissance du laser. Il convient de noter ici que nous avons trouvé des effets de la résonance Fano et du chauffage laser pour une puissance laser aussi faible que 0,2 mW/μm 2 , ce qui est cinq fois plus faible que la puissance seuil signalée pour les effets du chauffage laser et de la résonance Fano. Nous abordons la raison derrière cette observation à la fin de cette section. Sur la figure 2, nous illustrons les spectres Raman des échantillons étudiés (voir Tableau I.) avec le spectre Raman d'un substrat de Si nu. Tous les spectres de la figure 2 sont enregistrés en utilisant une puissance laser de 0,1 mW/μm 2 et un long temps d'intégration pour éviter tout effet de la résonance Fano et du chauffage laser. Le côté gauche de la figure 2 présente les spectres Raman "tels que mesurés", tandis que le côté droit de la figure 2 présente les spectres Raman des échantillons mesurés normalisés sur le spectre Raman du substrat de Si nu. Comme on peut le remarquer, la plupart des échantillons structurés présentent un pic Raman élargi et rétrogradé par rapport au pic Raman du substrat Si nu. Puisque nous avons écarté les effets du chauffage laser et de la résonance Fano, nous l'attribuons avec confiance au confinement des phonons. En d'autres termes, nous attribuons le ramollissement et l'élargissement observés de la résonance Raman à des cristallites nanométriques noyés dans du Si hautement défectueux dans les couches superficielles des résonateurs induites par le processus de structuration. Cette hypothèse est étayée par des mesures Raman et thermiques précédemment rapportées démontrant la présence de phonons confinés induits par des défauts dans de grands réseaux de résonateurs de Si [START_REF] Cloutier | Enhancement of Radiative Recombination in Silicon via Phonon Localization and Selection-Rule Breaking[END_REF], [START_REF] Guico | Phonon localization in periodic unia ially nanostructured silicon[END_REF], [START_REF] Iskandar | odi fication of the phonon spectrum of bulk Si through surface nanostructuring[END_REF].

Outre l'élargissement et le décalage du pic Raman, la figure 2 montre que les résonateurs induisent une remarquable amélioration Raman. Si les spectres Raman de la figure 2 étaient corrigés pour la densité surfacique des résonateurs, l'amélioration de Raman serait d'ordres de grandeur. Comme mentionné ci-dessus, le confinement des phonons brise la conservation de l'impulsion totale et permet à tous les modes phonons dont les vecteurs d'onde sont égaux ou inférieurs à D 1

(où D est la taille des cristallites dans lesquels les phonons sont confinés) de contribuer à la réponse de l'échantillon sous excitation optique. Cela impliquerait en effet une amélioration Raman de l'ordre de 3  D si l'on considère une dispersion de phonons dans une zone de Brillouin tridimensionnelle. Cependant, d'un autre côté, l'efficacité de la diffusion Raman est proportionnelle au volume d'interaction, qui est à peu près égal à 3 D . Par conséquent, nous croyons que l'amélioration Raman observée n'est pas due à l'effet de confinement des phonons. On sait également que des cristallites nanométriques isolées, ou des cristallites nanométriques noyées dans un milieu amorphe avec un indice de réfraction n très différent, peuvent présenter une amplification Raman due aux résonances électromagnétiques induites par des résonances de de champ local et les phonons de surface [START_REF] Hayashi | Raman study of gas-evaporated germanium microcrystals[END_REF]- [START_REF] Hayashi | ptical Study of Electromagnetic Surface ode s in i crocrystals[END_REF], [START_REF] Hayashik | Amorphous-like Raman spectra of semiconductor microcrystals[END_REF]. Cependant, comme le Si cristallin et le Si amorphe ont des indices de réfraction similaires, l'e altation Raman observée dans nos expériences ne peut pas être également attribuées à des résonances électromagnétiques induisant un champ local et aux modes de surface contribuant aux spectres Raman mesurés. Néanmoins, il a été démontré théoriquement et expérimentalement que, comme les résonances plasmoniques localisées dans les nanorésonateurs plasmoniques, les résonances localisées de Mie dans les résonateurs diélectriques aux points d'un réseau bidimensionnel induisent des oscillations hautement cohérentes dans les résonateurs diélectriques, ce qui conduit à un confinement fort dans les résonateurs diélectriques [START_REF] Evlyukhin | ptica l response features of Si-nanoparticle arrays[END_REF], [START_REF] Tsoi | E perimental demonstration of the optical lattice resonance in arrays of Si nanoresonators[END_REF]. Nous considérons qu'il s'agit de la seule explication raisonnable de l'amélioration Raman observée dans nos expériences. Par conséquent, nous pouvons conclure des mesures de Raman illustrées sur la figure 2 que de grands réseaux de résonateurs de Si pourraient assurer une interaction entre des phonons confinés et des photons confinés. Dans ce qui suit, nous examinerons l'effet de la forme et de la taille des résonateurs sur le confinement des phonons et des photons.

A. Effet de la forme du résonateur sur les résonateurs "courts"

Sur la figure 3 Plusieurs modèles peuvent être utilisés pour quantifier la taille des cristallites confinant les phonons dans les résonateurs de type pilier à partir du décalage vers les basses fréquences et de l'élargissement du spectre Raman. Les plus populaires sont le modèle de force microscopique [START_REF] Cheng | alculations on the size effects of Raman intensities of silicon quantum dots[END_REF], le modèle de polarisation de liaison [START_REF] Xie | omparison of models for Raman spectra of Si nanocrystals[END_REF]- [START_REF] Thonhauser | Predicted Raman spectra of Si[111] nanowires[END_REF] et les modèles de corrélation spatiale [START_REF] Fauchet | The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors[END_REF], [START_REF] Parayanthal | Raman Scattering in Alloy Semiconductors: "Spatial Correlation" Model[END_REF]. Les modèles de corrélation spatiale, cependant, ont l'avantage d'être moins affectés par certaines hypothèses et approximations fortes telles que l'approximation du milieu continu. Par conséquent, nous avons adopté un modèle de corrélation spatiale pour déduire du décalage et de l'élargissement du spectre Raman mesuré la distribution de taille des cristallites dans les couches superficielles des résonateurs.

B. Effet de la forme du résonateur sur les résonateurs de hauteur "moyenne"

Les spectres Raman et les simulations FDTD de la distribution de champ pour les échantillons du second ensemble (échantillons S M,1 , S M,2 , et S M,3 ) sont représentés sur la figure 4. Nous rappelons ici que la hauteur des résonateurs de cet ensemble est supérieure à celle des résonateurs du premier ensemble (voir tableau I).

Les simulations FDTD de la figure 4 montrent qu'une quantité notable du champ électrique incident est confinée dans les résonateurs en forme de cône de l'échantillon S M,1 . Par conséquent, le spectre Raman correspondant présente une amélioration significative du signal Raman avec décalage et élargissement.

L'e altation Raman est due à un champ électrique confiné dans les résonateurs en forme de cône, alors que le décalage et l'élargissement du spectre Raman sont dus à une interaction lumineuse avec des phonons confinés dans des cristallites nanométriques dans les résonateurs en forme de cône. Ainsi, contrairement aux résonateurs en forme de cône du premier ensemble, les résonateurs en forme de cône du second ensemble peuvent favoriser l'interaction entre un champ électrique confiné et des phonons confinés dans des cristallites nanométriques. Les simulations FDTD de la figure 4 Raman Shift (cm -1

) Sur la figure 7, nous montrons les spectres Raman des échantillons S M,3 et S L,3 (les échantillons avec des résonateurs en forme de piliers, qui montrent le confinement de champ le plus fort) mesurés avec une puissance laser de 1 mW/μm 2 les effets du chauffage laser et de la résonance Fano) et une puissance laser de 0,1 mW/μm 2 . Les spectres sont normalisés à titre de comparaison. Les spectres mesurés avec une puissance laser de 1 mW/μm 2 sont fortement déformés. Un décalage et un élargissement asymétrique par rapport aux spectres mesurés avec une puissance laser de 0,1 mW/μm 2 peuvent être clairement observés. Ceci met davantage en évidence la force du champ électrique qui peut être confinée dans des réseaux de résonateurs diélectriques.

CONCLUSION

Les

EXPERIENCES

Un réseau carré avec des motifs circulaires de la période 700 nm a été créé par une double exposition des échantillons, le diamètre des caractéristiques variant de 280 nm à 460 nm en changeant le temps d'exposition. Des structures d'une hauteur variant entre 400 nm et 1 μm ont été obtenues en augmentant le temps de gravure.

Les nanostructures étudiées dans ce travail sont mentionnées sur la figure 8. Différentes techniques pour mesurer la chaleur spécifique sont mises en oeuvres selon les tailles considérées et la précision requise [START_REF] Stewart | e asurement of low-temperature specific heat[END_REF]. Les mesures de chaleur spécifique des échantillons examinés dans ce travail ont été réalisées en utilisant le système PPMS (Physical Property Measurement System) de Quantum Design inc.

qui utilise une technique de mesure de temps de relaxation de grande précision.

Après chaque cycle de mesure, qui est une période chauffante suivie d'une période de refroidissement, le système rends compte de la réponse globale en température avec un modèle incluant tant la relaxation thermique de la plate-forme type au bain de température, que la relaxation entre la plate-forme type et l'échantillon lui-même [66]. De plus, même si le contact thermique entre l'échantillon et la plate-forme est faible, les valeurs de chaleurs spécifiques restent correctes.

Le système suppose d'abord que la plate-forme et l'échantillon sont en bon contact thermique l'un avec l'autre et qu'ils sont à la même température pendant la période de mesure. Dans ce cas simple, la température T de la plate-forme, comme fonction de temps t, obéit à l'équation : Le temps de mesure est prolongé de façon qu'il soit bien supérieur au temps nécessaire à l'échantillon pour atteindre l'équilibre thermique. La reproductibilité des mesures est vérifiée en mesurant chaque échantillon plusieurs fois. Toutes les mesures pour un même échantillon étaient identiques aux incertitudes standards de mesures près ; associées au fit (méthode des moindres carrés linéaires).

    t P T T dt dT C b w total      (
            t T t T T t T t P dt dT C p s g b p w p platform        (2) Et       t T t T dt dT C p s g s sam ple     (3) Ici, platform

RESULTATS ET DISCUSSION

Nous avons mesuré la chaleur spécifique à pression constante du substrat de silicium nu (Si) et huit substrats structurés en Si avec des structures de Si en forme de cône tronqué sur un intervalle de température de 3-300 K. Tous les échantillons mesurés ont été obtenus de la même référence de Si et les surfaces de structuration ont été faites aux températures basses et avec le soin approprié pour empêcher la diffusion d'impuretés ou la formation de défauts dans la structure de Si. Les courbes mesurées sont présentées dans la figure 9.

FIG. 9 Mesures de la chaleur spécifique des échantillons étudiés L'incertitude de mesure est indiquée par la taille des symboles. On peut remarquer que les données mesurées pour les échantillons structurés de surface diffèrent de ceux du substrat de Si nu. Cette différence dans la chaleur spécifique provient de différences dans les propriétés de vibration harmonique des échantillons ou celles du gaz électronique dans ces derniers. Dans la discussion suivante, par commodité, nous nommons simplement chaleur spécifique (à volume constant), la mesure directe de la chaleur spécifiques à volume constant fois la masse mesurée de l'échantillon. Le volume constant est difficile à obtenir expérimentalement, nous mesurons la chaleur spécifique à pression constante. Cependant, on peut aisément montrer qu'à température ambiante et en-dessous, le ratio de la chaleur spécifique à pression constante et celle à volume constant est très proche de l'unité.

À quelques degrés Kelvin, où le potentiel chimique de cristal coïncide avec l'énergie de Fermi, le volume constant de cristal de la chaleur spécifique est donné par l'équation ( ) :

3 2 2 2 5 4 3 2                    D B F B v t v T rNk T n k T E C     (4)
Où t E est l'énergie totale de l'échantillon, B k la constante de Boltzmann, n la concentration des porteurs libres, F  l'énergie de Fermi, r le nombre d'atomes par unité de cellule, N le nombre de cellules unitaires dans l'échantillon, et D  la température de Debye. Le premier et le second terme du côté droit de l'équation 4 représentent la contribution du gaz d'électrons et la contribution de la dynamique harmonique du réseau à la chaleur spécifique totale, respectivement. La contribution de la dynamique du réseau domine complètement la chaleur spécifique à des températures élevées. Cependant, bien en deçà de la température ambiante, la contribution de la dynamique du réseau décroît comme le cube de la température et à des températures très basses elle peut devenir comparable à la contribution électronique, qui ne diminue que linéairement avec la température. Afin de distinguer ces deux contributions, nous écrivons Eq.4 comme Les petites valeurs obtenues de  démontrent clairement l'absence de contribution observable du gaz d'électrons à la chaleur spécifique dans tous les échantillons mesurés. Par conséquent, nous pouvons raisonnablement considérer que la dynamique harmonique de réseaux des échantillons est le seul déterminant de leurs chaleurs spécifiques.

                               j k B j B j B v v T k k T k k k T E C , 2 2 2 sinh 2     (5)
Où l'addition écrase tous les états de phonon

       k et polarisations          j . Ceci
                    L d g T k h T k rNk C B B B v      0 2 2 2 csc 2 3   (6)
Où L  est la fréquence qui détermine la limite supérieure de la bande de phonon. Dans la suite, nous nous servons des mesures des chaleurs spécifiques et de l'Eq.6 pour évaluer l'effet des modes phonon limités sur quelques fonctions importantes de    g .

A. Effet des mécanismes d'hybridation des phonons sur l'extrémité basse fréquence du spectre phonon

A basse température, l'expression de la chaleur spécifique, Eq.6 prend la forme :

                          L d g T k n n T k rNk T C n B B L B v      0 1 2 exp 3   (7)
Étant donné que le facteur exponentiel dans l'équation 7 va réduire les contributions de toutes les valeurs, sauf petites, de  etT , nous devons considérer seulement l'extrémité basse fréquence du spectre de phonon. De plus, pour de petites valeurs de , le spectre phonon d'un cristal tridimensionnel se développe sous la forme:

  ... 4 4 2 2       a a g (8)
Lorsque cette expansion est substituée dans l'équation 7, et que la limite supérieure de l'intégrale est éliminée à l'infini, on constate que : Le bon accord entre les spectres de phonons dérivés et calculés à basses fréquences démontre clairement l'adéquation de l'approche expérimentale utilisée pour déterminer l'extrémité basse fréquence du spectre phonon. Les spectres de phonons à basse fréquence des échantillons structurés en surface tels que dérivés en utilisant l'approche décrite ci-dessus sont présentés et comparés au spectre de phonons de substrat de Si nu sur la figure 13.

                         ... 21
                     ... 21
On peut remarquer que des structures tronconiques de faible hauteur et de grand sommet (S4 et S5) ne modifient pas le spectre des phonons aux basses fréquences. Cela peut être attendu car ces structures peuvent être considérées comme une rugosité de surface fortement corrélée. Cependant, des structures en forme de cône disposées périodiquement et présentant des sommets aigus (S1 et S3) augmentent la fréquence des phonons, et des structures en forme de piliers (S2) et des structures tronconiques (S6, S7 et S8) affaiblissent le spectre des phonons à basses fréquences. L'amplification du spectre des phonons à basses fréquences est certainement due à la planéité dans les branches des phonons acoustiques. Ceci peut être attribué à des mécanismes d'hybridation entre les résonances locales et les modes phononique du bulk. La réduction du spectre des phonons aux basses fréquences, d'autre part, peut être due à l'augmentation des fréquences des modes phonon. Ceci peut être attribué à un mécanisme d'hybridation entre les modes de phonons acoustiques de la zone de Brillouin repliée et les modes de phonon du bulk. Ainsi, des structures coniques de sommets aigus semblent aplatir les branches de phonons acoustiques et réduire les vitesses de groupe de phonons acoustiques dues au mécanisme d'hybridation entre les résonances locales et les modes de phonon du bulk, alors que les structures de type pillier et les structures de cône tronqué de hauteur dépassant un micromètre semble avoir pour effet d'augmenter les pentes des branches de phonons acoustiques et par conséquent d'augmenter les vitesses du groupe de phonons acoustiques dues à un mécanisme d'hybridation entre les modes de phonon d'une zone de Brillouin repliée et les modes de phonon d'une zone de Brillouin non pliée. Dans l'Information Complémentaire, nous utilisons la méthode de Houston [START_REF] Houston | Normal Vibrations of a rystal Lattice[END_REF] pour montrer comment les coefficients obtenus de l'extrémité basse fréquence du spectre phonon sont liés aux vitesses du groupe phonon.

B. Spectre de phonons à haute fréquence à partir de mesures de chaleur spécifiques

La méthode des moments est largement utilisée pour le calcul approximatif du spectre des phonons [START_REF] Roll | Frequency Spectrum of r ystalline Solids[END_REF], [START_REF] Roll | Frequency Spectrum of r ystalline Solids. II. General Theory and Applications to Simple Cubic Lattices[END_REF]. L'essence de cette méthode est l'expansion de    g en une série de polynômes de Legendre dans laquelle les coefficients des termes successifs sont des combinaisons linéaires des moments pairs du spectre, définis comme :

             1 0 2 1 2 2 2 3 1 3 1 dx x g x Tr rN rN L n n L j n n j n     k k k D k (15) Où   k D
est la matrice dynamique. Cependant, il a été montré que l'expansion des polynômes de Legendre donne une indication assez acceptable des gaps dans le spectre des phonons, une description satisfaisante du spectre des phonons à haute fréquence et une mauvaise description de l'extrémité basse fréquence du spectre des phonons [START_REF] Peaslee | Frequency Spectrum of r ystalline Solids. III. Body-entered ub ic Lattices[END_REF]. Cela peut être compris si l'on se rend compte que l'extrémité basse fréquence du spectre phonon est une fonction lentement variable qui sera mal reproduit par une combinaison linéaire de quelques polynômes de Legendre. De plus, l'extrémité basse fréquence du spectre des phonons contribue très peu aux moments, ce qui est une raison supplémentaire pour laquelle cette approximation pourrait être mauvaise aux basses fréquences. Par conséquent, l'approximation présentée ci-dessous peut être employée avec précision pour le calcul des écarts dans le spectre et le spectre de phonons aux hautes fréquences. Nous développons le spectre des phonons :

              0 n L n n P a g    (16)
Où les coefficients n a sont donnés par : Ainsi, le calcul approximatif du spectre de phonons à haute fréquence peut être rendu possible lorsque les moments spectraux sans dimension n u 2 sont déterminés.

           1 1 1 1 2 dx x P x g n a n L n  (
Thirring a utilisé les moments du spectre du phonon pour développer une expression de la chaleur spécifique à des températures élevées en puissance inverse de la température [149]. Il a utilisé le résultat :

            1 2 2 ! 2 1 2 1 1 n n n n x n x B x e x  2  x (20)
Où n B 2 sont les nombres de Bernoulli, pour développer le second terme dans la partie droite de l'expression de l'énergie totale :

                                   rN j B j j j T k k k k T E 3 1 1 exp 2       (21)
Puis, en changeant l'ordre des sommations, Thirring trouva que :

                                           n n n n n B v v u T n n B rNk T T E T C 2 1 2 2 ! 2 2 1 1 1 3 (22)
Où le paramètre Les moments du spectre de phonons peuvent donc être obtenus en ajustant l'équation 22 aux courbes de chaleur spécifiques mesurées à des températures élevées. Puis, à partir de la connaissance des moments spectraux, le spectre de phonons à haute fréquence peut être approché en utilisant les équations ( 16) et [START_REF] Zhang | Dielectric spheres with maximum forward scattering and zero backscattering: a search for their material composition[END_REF]. Les meilleurs ajustements de l'équation 22 aux courbes de chaleur spécifiques mesurées sont représentés sur la figure 14 ette technique d'ajustement des courbes a permis de déduire seulement trois moments spectraux pour chaque échantillon mesuré. Les moments obtenus sont présentés dans le tableau II. Le fait que chaque échantillon étudié se caractérise par des moments spectraux différents suggère que, outre l'extrémité basse fréquence du spectre phonon, le spectre phonon aux hautes fréquences peut également être modifié par structuration de surface. Malheureusement, trois moments n'étaient pas suffisants pour décrire précisément toutes les caractéristiques du spectre des phonons aux hautes fréquences. En fait, il a été démontré qu'au moins quatorze moments du spectre sont nécessaires pour synthétiser les principaux gaps du spectre des phonons haute fréquence [START_REF] Peaslee | Frequency Spectrum of r ystalline Solids. III. Body-entered ub ic Lattices[END_REF]. Cependant, l'ajustement de l'équation 22 à des courbes de chaleur spécifique dans la plage de température complète (jusqu'à une température proche du point de fusion du matériau) pourrait sans doute fournir suffisamment de moments spectraux pour une description expérimentale précise du spectre des phonons aux hautes fréquences.

CONCLUSION

En résumé, nous avons mesuré les chaleurs spécifiques des substrats de Si nus et structurés en surface issus du même substrat de silicium sur un intervalle de température de 3-300 K. Les résultats ont montré que les courbes de la chaleur spécifique des substrats structurés en surface diffèrent sensiblement de celles du substrat en Si nu dans de nombreux aspects. Nous avons attribué cette différence à une modification importante dans le spectre phonon due à un mécanisme d'hybridation du phonon. Afin de déterminer les spectres de phonons des échantillons mesurés aux basses fréquences, nous avons profité du fait qu'à basse température l'expression de la chaleur spécifique contient un facteur exponentiel qui exclut les contributions autres que les petites valeurs de fréquence et de température, et nous avons développé l'extrémité basse fréquence du spectre phonon en termes de polynômes à coefficients pairs. Les résultats obtenus montrent que la structuration de la surface peut fortement modifier le spectre de phonons du bulk et la vitesse de groupe des phonons à longue longueur d'onde qui déterminent les propriétés élastiques et qui sont les principaux supports de chaleur dans les matériaux semi-conducteurs. Nous avons également modélisé la chaleur spécifique mesurée à des températures élevées en utilisant la technique de dilatation de Thirring, qui consiste en une expansion formelle à haute température de la chaleur spécifique en termes des moments du spectre phonon. Nous avons trouvé que la structuration de la surface peut également affecter les moments du spectre des phonons et, par conséquent, modifier le spectre des phonons aux hautes fréquences. Ainsi, les résultats présentés dans ce document démontrent clairement la possibilité de modifier les propriétés optiques mécaniques, thermiques et infrarouges des matériaux bulk par la structuration de surface, ouvrant la porte au développement de métamatériaux technologiques de haute performance via la structuration de surface. Les mesures calorimétriques ont été effectuées comme précédemment avec le dispositif PPMS. Le détail de ce calorimètre peut être trouvé dans la Ref. [66]. Cependant, la réponse thermique du calorimètre a été analysée en prenant en compte la conductance de la liaison thermique des ZnO NWs au substrat Si au lieu d'utiliser la méthode de relaxation standard [66], [START_REF] Shepherd | Analysis of the lumped τ 2 effect in rela ation calorimetry[END_REF].

III. Chaleur spécifique des nanofils

Les dessins de la FIG.17 décrivent la procédure de mesure. Avant le chargement des échantillons, une petite quantité de colle thermique (Apiezon N) a été fixée sur la plate-forme d'échantillonnage et la réponse en température de l'addenda (plateforme plus colle thermique) a été enregistrée dans un cycle de mesure (période de chauffage suivie d'une période de refroidissement). Ensuite, l'équation du bilan thermique : l'instant t. Dans un second temps, un substrat de Si nu a été pesé, placé sur la colle thermique Apiezon N, puis la réponse thermique du calorimètre a été enregistrée dans un cycle de mesure. En présence du substrat de Si, la réponse thermique doit être modélisée avec l'ensemble des équations de bilan thermique suivant [66]:

        0 1 1 1 T t T dt t dT c t P w     (
              t T t T T t T dt t dT c t P G w 2 1 0 1 1 1        (24a)         t T t T dt t dT c G 1 2 2 2 0     (24b) Où 2
c est la chaleur spécifique de Si, G  est la conductance thermique de la colle thermique Apiezon N, et   t T 2 est la température du substrat de Si à l'instant t. La chaleur spécifique du substrat de Si et la conductance thermique de la colle thermique Apiezon N ont été obtenues en éliminant   t T 2 dans les équations de bilan thermique ci-dessus, en intégrant sur la période de temps de mesure, puis en ajustant l'équation résultante pour   t T 1 à la réponse thermique mesurée du 157 calorimètre par la méthode des moindres carrés linéaires [66]. La chaleur spécifique mesurée du substrat de Si est représentée sur la figure 18. Les incertitudes de mesure sont principalement déterminées par la dispersion dans les points de données et ne dépassent pas 2%. Les courbes de chaleur spécifiques globales des échantillons WP1, WP2 et WP3 mesurées avec la même technique utilisée pour le substrat Si sont également reportées sur la figure 18. La présence de ZnO NW sur le substrat de Si altère significativement la réponse thermique du calorimètre. Ainsi, les ZnO NW ont un effet mesurable sur l'élévation de température et la chute dans un cycle de mesure, et peuvent par conséquent être mesurés en utilisant une technique colorimétrique. Après avoir mesuré la conductance thermique de la colle thermique et la chaleur spécifique du substrat en Si, la masse de ZnO NWs a été déduite des mesures électroniques de leur concentration, diamètre moyen et longueur moyenne dans chaque échantillon, et les réponses thermiques du calorimètre ont été mesurées sur la colle thermique Apiezon N. Dans cette étape, le temps de mesure est également fixée pour être bien supérieur au temps nécessaire pour que le substrat Si et ZnO NW atteignent l'équilibre thermique. Dans ce cas, la réponse thermique du calorimètre doit satisfaire aux équations de bilan thermique suivantes : )

Temperature (K)               t T t T T t T dt t dT c t P G w 2 1 0 1 1 1        (25a)               t T t T t T t T dt t dT c I G 3 2 1 2 2 2 0        (25b)       t T t T dt dT c I 2 3 3 3 0     (25c)

RESULTATS ET DISCUSSION

Dans cette section, nous présentons les résultats expérimentaux et développons des études physiques sur la chaleur spécifique des NWs et la conductance thermique de l'interface entre le substrat de Si et les ZnO NW.

A. Chaleur spécifique des nanofils de ZnO

Les courbes de chaleur spécifiques mesurées des ZnO NW sont représentées sur la figure 20 ) Temperature (K) quasi linéaire de la conductance thermique de Si / ZnO NWs au-dessus de 70 K peut être attribuée à une contribution significative des processus de phonons anharmoniques (qui n'est pas prise en compte dans le diffuse mismatch model) au transfert d'énergie entre Si et ZnO NWs. La contribution des processus de phonons anharmoniques à la conductance thermique de l'interface et la croissance linéaire associée des courbes de conductivité thermique de l'interface mesurées ont été observées précédemment dans le cas d'interfaces entre matériaux avec des spectres de phonons très dissemblables [START_REF] Carron | Resonances of two-dimensional particle gratings in surface-enhanced Raman scattering[END_REF]. Par conséquent, nous suggérons que le fort désaccord entre le spectre de phonons de ZnO NWs et celui du Si favorise la contribution des processus de phonons anharmoniques au transport thermique à travers l'interface entre Si et ZnO NWs. Au fur et à mesure que la température augmente, les processus anharmoniques des phonons s'accroissent, conduisant à une sur-conductance à l'interface entre Si et ZnO NW, qui se révèle dans la dépendance linéaire de la température de la conductance thermique de l'interface au-dessus de 70 K.

                       Q D c L D c T pN T pN T c    (26)

CONCLUSION

Des mesures calorimétriques à impulsion de chaleur ont été effectuées sur des nanofils d'oxyde de zinc (ZnO NWs) alignés verticalement et de surface libre sur une plage de température allant de 1.8 à 300 K. La réponse thermique du calorimètre a été analysée en tenant compte de l'effet de la conductance de la liaison thermique entre ZnO NWs et substrat de Si. Un méthode des moindres carrés linéaire a été utilisés pour déduire la chaleur spécifique des ZnO NWs et la conductance thermique de l'interface entre Si et ZnO NW. Les résultats ont indiqué qu'au-dessus de 25 K, la chaleur spécifique de ZnO NW devient supérieure au ZnO massif, et le facteur d'amélioration augmente à mesure que la température augmente et que le diamètre NW diminue. Cependant, à basse température (inférieure à 4 K), la chaleur spécifique des ZnO NWs présente la contribution claire d'un cristal essentiellement bidimensionnel. La modélisation théorique de la chaleur spécifique montre que les modes phononiques de dispersions quadratiques donnent lieu à une contribution de cristal tridimensionnel, tandis que les modes phononiques de dispersions linéaires donnent lieu à une contribution de cristal bidimensionnel aux basses températures. La conductance thermique de l'interface entre Si et ZnO NWs est inférieure de cinq ordres de grandeur à la valeur théorique de la conductance thermique de l'interface entre Si et ZnO massifs. La faible conductance thermique de l'interface entre Si et ZnO NW a été attribuée à la formation possible d'une couche mince de faible cristallinité entre les ZnO NW et le substrat Si. Une forte baisse de la conductance thermique mesurée entre 1.8 et 3 K a été interprétée comme une transition de la transmission des phonons spéculaire à la diffusion des phonons. Une dépendance quasi linéaire de la température dans la conductance thermique mesurée au-dessus de 70 K a été interprétée comme des processus de phonons anharmoniques contribuant au transfert de l'énergie thermique à travers l'interface entre Si et ZnO NW.

CONCLUSION GENERALE

Cette dissertation visât à renforcer la compréhension du rôle des surfaces dans les propriétés thermiques, thermodynamiques et optiques des solides. On a tenté d'utiliser diverses techniques expérimentales et méthodes d'analyse pour décrire comment la présence de surfaces libres modifie les propriétés des matériaux. L'adéquation et la portée de validité de ces méthodes expérimentales et d'analyse ont été évaluées, et des suggestions ont été faites concernant d'éventuelles recherches expérimentales et théoriques sur les effets de surface. Une partie importante du manuscrit a été consacrée à décrire comment la présence de surface modifie le spectre des phonons et les propriétés matérielles associées. Tout d'abord, une revue de la théorie générale du transfert de chaleur et de l'interaction photon-phonon a été présentée. Certains aspects liés à l'anharmonicité des phonons ont été discutés. Une attention particulière a été accordée à l'effet du confinement quantique des phonons sur la force d'interaction lumière-matte et sur la conductance limite. Ensuite, une description détaillée des procédures de croissance des échantillons ainsi que les techniques expérimentales utilisées dans les travaux de recherche rapportés dans ce manuscrit ont été présentées. L'approche habituelle pour analyser l'excitation expérimentale des phonons consiste à supposer des conditions aux limites cycliques. Un tel traitement, bien que mathématiquement pratique, élimine la possibilité d'étudier les propriétés dynamiques des atomes au voisinage d'une surface libre d'un cristal réel. La raison en est que la configuration atomique cristalline dans les couches de surface diffère de la configuration atomique dans la masse du cristal. Les forces agissant sur les atomes situés sur les couches de surface sont différentes des forces agissant sur les atomes dans la masse. Un atome dans les couches de surface a moins de voisins plus proches qu'un atome dans le volume d'un cristal. Par conséquent, on peut s'attendre à ce que les propriétés dynamiques des phonons et les propriétés thermiques et thermodynamiques résultantes de la surface et du volume diffèrent. Ainsi, lorsque la taille du cristal devient suffisamment petite pour que le rapport surface/volume ne soit plus négligeable, la modification de la fonction de distribution de fréquence du cristal due à la présence de surfaces libres modifie les propriétés thermiques et thermodynamiques du cristal qui deviennent très sensibles à la taille du cristal. De plus, les règles de sélection déterminant les propriétés physiques des cristaux infinis peuvent être relâchées pour les cristaux finis ou pour les atomes dans les couches de surface pour lesquelles les propriétés de symétrie ne sont plus maintenues. Par conséquent, on pourrait s'attendre à ce que les propriétés thermiques et thermodynamiques des cristaux finis (nanométriques) présentent des caractéristiques spécifiques que les cristaux en masse ne présentent pas. Les troisième, quatrième et cinquième chapitres ont présenté la compréhension fondamentale de l'effet de surface sur les propriétés liées aux phonons.

Dans le troisième chapitre (III) de cette thèse, nous démontrons qu'en raison de la présence de phonons confinés à la surface, les phonons et les photons de différents moments peuvent interagir les uns avec les autres au sein de la même nanostructure. Les spectres Raman provenant de grands réseaux de résonateurs diélectriques Si attachés à des substrats de Si présentaient une amélioration appréciable de Raman accompagnée d'un décalage et d'un élargissement par rapport au spectre Raman à partir d'un substrat de Si nu. Les simulations FDTD ont démontré que la résonance du réseau collectif induit un confinement efficace du champ dans les résonateurs, ce qui entraîne une amélioration appréciable de l'intensité Raman. Un modèle de corrélation spatiale a démontré que le rétrécissement et l'élargissement du spectre sont le résultat de la relaxation du vecteur d'onde phonon, qui est due au confinement des phonons dans les cristallites de Si nanométriques dans les couches superficielles des résonateurs. L'effet de la taille et de la forme des résonateurs sur le confinement des photons et des phonons a été étudié. Il a été démontré que, lorsque les résonateurs augmentent en hauteur et que leur forme devient cylindrique, l'amplitude de leur oscillation cohérente augmente et, par conséquent, leur capacité à confiner le champ électrique entrant augmente. Cependant, il a été constaté que la forme et la taille des résonateurs n'ont pas un effet remarquable sur le confinement des phonons. Les résultats présentés dans le troisième chapitre montrent que des réseaux de résonateurs diélectriques longs cylindriques suppriment de manière drastique les pertes de diffusion de la lumière par confinement de champ et favorisent l'interaction du champ confiné avec les phonons confinés. Les résultats mis en évidence dans ce chapitre devraient aider à mieux comprendre le confinement des phonons et des photons et à concevoir des métasurfaces de conception rationnelle pour les dispositifs optiques à haut rendement. Par exemple, le mécanisme d'interaction non conservatif photonphonon amplifié observé a un grand potentiel pour renforcer fortement l'effet photovoltaïque dans le silicium.

Dans le quatrième chapitre (IV), nous présentons des preuves expérimentales sur le changement du spectre des phonons et les propriétés vibrationnelles d'un matériau massif à travers des mécanismes d'hybridation de phonons à travers la nanostructuration de surface. Nous démontrons que le spectre de phonons d'un matériau massif peut être modifié par des mécanismes d'hybridation entre des modes de phonons confinés dans des nanostructures introduites sur la surface du matériau et les modes de phonons de masse sous-jacents. Nous montrons également que la forme et la taille des nanostructures faites sur la surface du substrat ont un fort effet sur le spectre de phonons du matériau massif.

Dans le cinquième chapitre (V), nous démontrons qu'à basse température (inférieure à 4 K) la chaleur spécifique de nanofils (ZnO) présente une contribution claire d'un cristal essentiellement bidimensionnel liée à la surface du nanofil. Cependant, au-dessus de 25 K, la chaleur spécifique du nanofil commence à augmenter par rapport à la chaleur spécifique du massif. Cette augmentation en fonction de la température est d'autant plus importante que le diamètre du nanofil diminue. Nous démontrons également que la conductance thermique de l'interface entre un substrat et des nanofils alignés verticalement est inférieure d'un ordre de grandeur à celle des matériaux massifs, suggérant qu'une couche mince de faible cristallinité se forme entre les nanofils et un substrat. De plus, à mesure que la température augmente, on observe clairement deux transitions. Une première transition entre une transmission élastique spéculaire vers une transmission élastique diffusive , puis de la transmission élastique diffusive à la transmission inélastique diffusive qui se produit lorsque la température augmente.

Il convient de noter que dans les quatrième et cinquième chapitres, des mesures calorimétriques ont été effectuées dans le calorimètre à impulsion thermique d'un système de mesure de propriétés physiques Quantum Design. Cependant, la réponse thermique du calorimètre a été analysée en prenant en compte la conductance de la liaison thermique des nanostructures au substrat au lieu d'utiliser la méthode de relaxation standard. performed over the wavevectors within the Brillouin zone only. Upon using Eqs.A1-A4, it can be found that if the first order Raman spectrum from an infinite crystal is given by:

      2 2 0 inf 2 1 2 1 . 1                q I A5
The first order Raman spectrum from a finite crystal is given by:

      q d L C I 3 2 0 . , ,   q q q   A6
Where  is the zone center optical phonon damping,   0 q  is the zone center optical phonon frequency, and   q ,  L is given by:

      2 2 2 1 2 1 . 1 ,                q q L A7
It can be clearly noticed from Eqs.A5 and A6 that the Raman scattering from an infinite material is a process in which the total momentum is conserved (it involves the contribution of only 0  q phonons to keep the total momentum conserved).

However, the Raman scattering process from a finite crystal does not require momentum conservation and involves contribution from optical phonons away from the Brillouin zone center. In order to account for a possible distribution of crystal sizes in the resonators, we express the first order Raman spectrum as:

              0 3 2 0 . , , dD q d L C D F I q q q  

A8

It has been demonstrated that a lognormal function best describes the size distribution of Si nanocrystals embedded in a given matrix [START_REF] Adu | onfined Phonons in Si Nanowires[END_REF]- [START_REF] Zhang | Size-dependent evolution of phonon confinement in colloidal Si nanoparticles[END_REF]. Therefore, we write the size distribution   . The most physically accepted forms are that of Faraci et al. [START_REF] Faraci | odi fied Raman confinement model for Si nanocrystals[END_REF]- [START_REF] Faraci | Raman and photoluminescence spectroscopy of Si nanocrystals Evidence of a form factor[END_REF] and that of Roodenko et al. [START_REF] Roodenko | odifie d phonon confinement model for Raman spectroscopy of nanostructured materials[END_REF]. In our calculation, we adopt the form of the phonon weighting function established by Roodenko et al. because it has the advantage of easier numerical implementation. In order to express   q  with taking into account the anisotropy of the phonon dispersion in silicon, we use the method of Paillard et al. [START_REF] Paillard | Improved one-phonon confinement model for an accurate size determination of silicon nanocrystals[END_REF], which is based on Brout sum rule to accurately average over different phonon branches in high symmetry crystallographic directions [START_REF] Brout | Sum Rule for Lattice Vibrations in Ionic rystals[END_REF].

APPENDIX 2: THE USE OF HOUSTON'S METHOD FOR DERIVING A RELATIONSHIP BETWEEN THE COEFFICENTS OF THE LOW FREQUENCY END OF THE PHONON SPECTRUM AND THE PHONON GROUP VELOCITIES

The method of Houston for calculating the phonon spectra and, by extension, thermodynamic properties of cubic crystals has attracted a lot of interest because it gives best results at the lowest frequencies and therefore is valuable for deriving an accurate relationship between the coefficients of the low frequency end of the phonon spectrum and the phonon group velocities in cubic crystals. The phonon spectrum of the jth phonon branch is given by: where m  is a normalization constant and mn  is the usual Kronecker symbol. The first few m  and Kubic harmonics as well as the method for generating these functions are reported in Ref. [START_REF] Betts | Houston's e thod and Its Application to the Calculation of Characteristic Temperatures of Cubic[END_REF]. Now, if we substitute Eq.A13 into Eq.A12 and make use of Eq.A14, we obtain:

   j g =                  k d v r
      

APPENDIX 3: MODEL FOR SPECIFIC HEAT OF CYLINDRICAL NANOWIRE

The equation of motion of a solid within the continuum medium approximation (which is valid at low temperatures) can be written as:

    U U t U             2 2 2 A23
where  is the material mass density, U is the displacement vector, and  and  are the Lamé constants. As boundary conditions, we consider azimuthal symmetry, and that the radial component of the displacement vector as well as the derivative of the z-component with respect to the radial component of the displacement vector blow up to zero at the boundary of the nanowire (when the r-coordinate is equal to the NW radius R). Considering these boundary conditions, and solving the equation of motion in cylindrical coordinates using the separation of variables technique, we obtain a displacement vector of the form:

      t i L z n n R r z J t z r U m m                 
  0 2 2 2 2 2 2 2 2 2                                                               L n R z L n R z m m          A25
When R tends to infinity, Eq.A25 reduces to the dispersions of an infinite crystal. The solutions of Eq.A25 are:

                      2 2 2 2 L n R z E m L     A26a                       2 2 2 2 L n R z I C m T     A26ab 2 2 2 4 2 1                       L n R z S EI m x B     A26c 2 2 2 4 2 2                       L n R z S EI m y B     A26d
Where L, T, and B 1 [START_REF] Dennard | Design of ion-implanted S FET's with very small physical dimensions[END_REF] refer to longitudinal, torsional, and doubly degenerate bending modes. In Eq.A26a-A26d, C is the NW torsional rigidity, E is the NW Young's modulus, I x(y) is the area moment of inertia with respect to an axis perpendicular to the NW axis (which is assumed to be along the z-axis), L is the NW length, and S is the NW cross section area. Consider now Eq.A26a, which describes the dispersion relation of the longitudinal modes. According to Eq.A26a, the number of longitudinal modes with frequencies less than  is equal to the number of lattice points ( m z and n) which obey:

2 2 2 2                           L n R z E m A27
That is, the number of points lying within an octant of the ellipse given by:

              E L y R x 2 2 2 2 2 2 2 A28
The NWs investigated in this work are large enough to consider that this number is equal to the actual area of the first octant of the ellipse. Hence, the number of longitudinal modes of frequencies less than  in the NW is given by:

  E RL N L 4 2   

A29

In the same sense, we can find that the number of torsional modes with frequencies less than  in the NW is given by:

  C I RL N T 4 2    

A30

And the number of bending modes with frequencies less than  is equal to twice (because of the isotropic nature of the x-y plane and the degeneracy of the bending modes) the volume of the first octant of the ellipsoid specified by:

A37

It can be clearly seen from Eq.A37 that the density of states in the NW is the sum of densities of surface-like states (first term in Eq.A37) and volume-like states (second term in Eq.A37).

At this point, it is possible to calculate the NW specific heat in terms of its length and radius from:

A39

The first and second term in Eq.A39 describe contributions to the NW specific heat from modes of linear dispersions and modes of quadratic dispersions, respectively. In the limit of low temperatures, the upper limits on the integrals in Eq.A39 can be removed to infinity, and the expression of the NW specific heat reduces to:

Figure I. 1 20 Figure I. 2 20 Figure I. 3 23 Figure I. 4

 1202203234 Figure I.1 These figures represent the diffusive and the ballistic transport regimes of heat carriers, respectively. The black segments represent the trajectories of the heat carriers modified either by their interactions with other carriers or with the interfaces.[3] ............................................................................................................ 20 Figure I.2 Thermal conductivity of a suspended monolayer of graphene as a function of its size.[2] .............................................................................................. 20 Figure I.3 Energy density over a plane surface separating a semi-infinite medium of glass at K 300 from vacuum at K 0 at different distances from the surface.[4] 23 Figure I.4 Schematic of a two-dimensional lattice. Vectors a  represent a conventional Bravais lattice, and the vectors b  a primitive cell.[3] ........................ 25 Figure I.5 First Brillouin zone of a face-centered cubic lattice. Each letter represents a point of high symmetry of the crystal, while the lines represent the directions [5]. ........................................................................................................... 26 Figure I.6 Schematic of a monatomic linear chain that consists of atoms of mass

Figure I. 7 29 Figure I. 8

 7298 a) Dispersion relation for a monatomic linear chain. (b) Group velocity for all phonons of a monatomic linear chain. The dotted horizontal line represents the speed of sound in the material .[3] ..................................................................... Schematic of a diatomic linear chain that consists of atoms of masses 1 m and 2

Figure I. 9 36 Figure I. 13 38 Figure I. 14 40 Figure I. 15

 9361338144015 Figure I.13 Mie resonances of a spherical particle. (A) Scattering efficiency (dimensionless ratio of scattering cross section to geometrical cross section of the particle) versus dielectric permittivity  (lossless particle, q=0.5) for plasmonic   0  

  .................................................................................. 41 Figure I.

43 Figure II. 1 53 Figure II. 10 64 Figure

 431531064 Figure II.10 XRD of ZnO nanowires samples. ....................................................... 54 Figure II.11 Schematic representing the three different forms of scattering. ......... 56 Figure II.12 Diagram of Raman Microscope System used for Raman spectroscopy and delivery.[55] ...................................................................................................... 58 Figure II.13 a) Heat flow diagram for the simple model. tot C is the heat capacity of

  Figure II.[START_REF] Schuller | General properties of dielectric optical antennas[END_REF] A simplified picture of the Dynacool. In green is the bucket with the liquid helium which flows to the pot in the bottom of the annulus. Around the sample chamber flows helium gas. In the bottom of the sample chamber the puck is placed in the isothermal region, under which a block heater is attached for temperature control. To achieve high vacuum, the sample chamber is exposed to a two stage cryopump.[61] ......................................................................................... 65 Figure II.17 The puck 1792 (left) and the platform (right). 1: The sapphire platform 2: Au-Pd wires 3: Gold contacts 4: Thermometer 5: Heater.[60] ............. 66 Figure II.18 The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas. [61] .................................................................................................................................. 67 Figure II.19 The results of the calibration (Pass 1 and Pass 2) of puck 1505. The resistance of the platform thermometer (orange circles) and heater (blue triangles) measured during the calibration. The inset shows the thermal conductivity (black circles) of the eight Au-Pd wires during Pass 2 of the calibration.[60] ................... 67 Figure II.20 The dialogue box. 1) Temperature range: Temperature startpoint, endpoint and spacing. 2) Temperature rise: Height of temperature pulse. 3) Repetitions at each temperature: At each temperature point the specific heat is measured three times in order to check stability and reproducibility. The results are not averaged. 4) Relaxation measurement: The measurement time is always set to

  Figure II.[START_REF] Schuller | General properties of dielectric optical antennas[END_REF] A simplified picture of the Dynacool. In green is the bucket with the liquid helium which flows to the pot in the bottom of the annulus. Around the sample chamber flows helium gas. In the bottom of the sample chamber the puck is placed in the isothermal region, under which a block heater is attached for temperature control. To achieve high vacuum, the sample chamber is exposed to a two stage cryopump.[61] ......................................................................................... 65 Figure II.17 The puck 1792 (left) and the platform (right). 1: The sapphire platform 2: Au-Pd wires 3: Gold contacts 4: Thermometer 5: Heater.[60] ............. 66 Figure II.18 The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas. [61] .................................................................................................................................. 67 Figure II.19 The results of the calibration (Pass 1 and Pass 2) of puck 1505. The resistance of the platform thermometer (orange circles) and heater (blue triangles) measured during the calibration. The inset shows the thermal conductivity (black circles) of the eight Au-Pd wires during Pass 2 of the calibration.[60] ................... 67 Figure II.20 The dialogue box. 1) Temperature range: Temperature startpoint, endpoint and spacing. 2) Temperature rise: Height of temperature pulse. 3) Repetitions at each temperature: At each temperature point the specific heat is measured three times in order to check stability and reproducibility. The results are not averaged. 4) Relaxation measurement: The measurement time is always set to

  .......................................................................................... 69 Figure II.

76 Figure III. 2 83 Figure III. 6 Figure III. 7 87 Figure IV. 1 91 Figure IV. 2 93 Figure 94 Figure IV. 4 99 Figure IV. 6 100 Figure IV. 7 103 Figure V. 1 109 Figure V. 2 110 Figure V. 3 111 Figure V. 4 113 Figure V. 5 115 Figure V. 7

 7628367871912939449961007103110921103111411351157 Figure III.6 Raman shift and spectrum broadening versus the incident laser power………………………………………………………………………………87 Figure III.7 Rama spectra of samples S M,3 and S L,3 measured with a laser power of 1mW/µm 2 and 0.1 mW/µm 2 . The spectra are normalized for the sake of comparison. .............................................................................................................. 87 Figure IV.1 Phonon hybridization mechanism and sketch of typical investigated Si nanostructured samples. ........................................................................................... 91 Figure IV.2 Heat capacity measurements of the samples investigated. .................. 93 Figure IV.3 T C v versus 2T . Symbols: Measurements. Solid lines: Extrapolation



  over that of the transverse optical phonon   TO  . The frequency range bounded by TO  and LO  is known as the Reststrahlen band. Within the Reststrahlen band, the real part of the material response to an electromagnetic excitation

Figure I. 1

 1 Figure I.1 These figures represent the diffusive and the ballistic transport regimes of heat carriers, respectively. The black segments represent the trajectories of the heat carriers modified either by their interactions with other carriers or with the interfaces [7].

Figure I. 2

 2 Figure I.2 Thermal conductivity of a suspended monolayer of graphene as a function of its size [6].

Figure I. 3

 3 Figure I.3Energy density over a plane surface separating a semi-infinite medium of glass at 300 K from vacuum at 0 K at different distances from the surface[START_REF] Joulain | Surface Electromagnetic Waves Thermally Excited: Radiative Heat Transfer, oherence Properties and asim ir Forces Revisited in the Near Field[END_REF].

  Vectors a  are used to define a lattice in Figure I.4. The system is invariant for any translation that is defined by a combination of these vectors. The lattice nodes correspond to geometric points generated by the translation of the Bravais lattice vectors (denoted by crosses in Figure I.4). The lattice unit cell is defined by the smallest vectors a  . The volume of the lattice unit cell reproduces the studied structure when repeated in the three directions of space. An atomic arrangement is associated to each lattice node, and is repeated in each translation.

Figure I. 4

 4 Figure I.4 Schematic of a two-dimensional lattice. Vectors a  represent a conventional Bravais

Figure I. 4 .

 4 Figure I.4. The set of vectors b  also defines a lattice leaving the system invariant by a translation carried by these vectors. The primitive cell defined by vectors b contains only a single node. The crystal periodicity allows to study several physical properties in the reciprocal space, i.e. in the space associated with the wave vector.

  zone is therefore included in the following interval  

Figure I. 5

 5 Figure I.5 shows to the first Brillouin zone of a face-centered cubic lattice. The letters denote the directions of the crystal's high symmetry, with  being the origin. They are often used to establish the electrons or phonons dispersion relations.

Figure I. 5

 5 Figure I.5 First Brillouin zone of a face-centered cubic lattice. Each letter represents a point of high symmetry of the crystal, while the lines represent the directions [9].

Figure I. 6

 6 Figure I.6 Schematic of a monatomic linear chain that consists of atoms of mass m, separated by a distance " a " and interacting with a harmonic potential and force constant C .with a force constant C . We note that   t x n is the position of the atom n at time t .

13 ).

 13 Substituting equation (I.13) into equation (I.12), we deduce the dispersion relation of phonons, i.e., the value that the angular frequency  can have as a function of the wave vector k : that the chain consists of N atoms and use cyclic periodic boundary conditions we find the following quantized values for the wavevectors n being an integer.As it has been stated in the previous section, the aim is to study the dispersion relation in the first Brillouin zone, precisely in the interval   Thus, the number of the vibration modes is equal to the number of degrees of freedom in the system. Figure I.7.a. shows a plot of the dispersion relation of the monatomic linear chain, relation is symmetric with respect to a

Figure I. 7

 7 Figure I.7 a) Dispersion relation for a monoatomic linear chain. (b) Group velocity for all phonons of a monoatomic linear chain. The dotted horizontal line represents the speed of sound in the material .[7] Figure I.8 shows a schematic of a diatomic linear chain where two atoms of mass 1 m and 2

.

  Our interest will be centered on the interval   a 2 , 0  , noting that the other part being symmetrical to the first.

Figure I. 8

 8 Figure I.8 Schematic of a diatomic linear chain that consists of atoms of masses 1 m and 2m , separated by a distance " a " and interacting with a harmonic potential of force constant C .

Figure I. 9

 9 Figure I.9 shows the obtained dispersion relations and the associated group velocities with 1 1 . 10

Figure I. 9

 9 Figure I.9 (a) The dispersion relations for a diatomic linear chain. (b) The group velocity of all the phonons of a diatomic linear chain. The dotted horizontal line represents the speed of sound in the material [7]. In order to understand the natures of these branches, it is essential to analyze the amplitudes of vibration   k u 1 and   k u 2 of each atom for all modes of the chain.

Figure I. 9

 9 Figure I.9.a. shows the amplitudes of vibration of the two atoms of the unit cell, for the acoustic branch. The amplitudes conserving the same sign signifies that the atoms vibrate in phase. The center of gravity of the unit cell undergoes a major shift enabling an efficient transport of energy through the system. We note that when we get closer to the edge of the first Brillouin, the light atom's contribution becomes less and less dominant, and eventually drops to zero near the edge of the Brillouin zone. This means that the energy associated with this mode cannot flow in the crystal.

Figure I. 10

 10 Figure I.10 (a) amplitude of vibration of the normal modes for the acoustic branch. (b) Amplitude of vibration of the normal modes for the optical branch [7].

  l , corresponds to the sum of l R  , with the vector that corresponds to the atom position in the unit cell

  represents the variation of the potential energy that is detected when the b atom of the unit cell l is displaced in the direction  , whereas the b atom of the unit cell l moves in the  direction.By inserting the equation (I.21) into equation (I.22), we get:

  By inserting equation (I.24) in equation (I.23), and by making the necessary change of index:

Figure I. 11

 11 Figure I.11 shows the dispersion relations for the Silicon crystal. Noting that the points of the reciprocal space are represented in Figure I.5. Having two atoms in its elementary unit cell due to its diamond structure, 6 branches are present: three acoustic branches and three optical. Unlike the one-dimensional case in the previous section, the phonons have a polarization with respect to the wave-vector

Figure I. 11

 11 Figure I.11 Phonons dispersion relations of a bulk crystal of silicon[START_REF] Wei | Phonon dispersions of silicon and germanium from first-principles calculations[END_REF].

Figure I. 12

 12 Figure I.12 (a) Inelastic process corresponding to the destruction of two phonons   k  ,  and

Figure I. 13

 13 Figure I.[START_REF] Wei | Phonon dispersions of silicon and germanium from first-principles calculations[END_REF].C shows the field structure of the four major resonant modes in highindex dielectric particles: magnetic dipole, electric dipole, magnetic quadrupole, and electric quadrupole. According to the Mie theory, the maximum achievable scattering efficiency for a specific multipolar excitation of a subwavelength particle depends only on the resonance frequency and not the type of material, which suggests that various plasmonic effects observed for the scattering of light by metallic nanoparticles can be realized with high-index dielectric nanoparticles[START_REF] Schuller | General properties of dielectric optical antennas[END_REF].

Figure I. 13

 13 Figure I.13 Mie resonances of a spherical particle. (A) Scattering efficiency (dimensionless ratio of scattering cross section to geometrical cross section of the particle) versus dielectric permittivity  (lossless particle, q=0.5) for plasmonic   0  

Figure I. 14

 14 Figure I.14 Demonstration of optical magnetic response and examples of resonant silicon nanostructures. (A to C) Dark-field optical microscope images (top left), scanning electron microscope (SEM) images (top right), and dark-field scattering spectra (bottom) of spherical Si nanoparticles with approximate diameters of 100 nm (A), 140 nm (B), and 180 nm (C) [33]. Abbreviations for resonances are as in Figure I.13. (D) Dark-field microscope image of laser-printed Si nanoparticle arrays. Si is crystallized inside the marked areas by additional laser irradiation, producing a change in color [30]; scale bar, 10 mm. (E) Lithographically fabricated Si nanodisk arrays with varied spectral positions of the electric and magnetic resonances [25]. (F) Monodispersed silicon nanoparticles resonant at near-IR frequencies prepared by chemical synthesis [34][15].

  It can be shown by using the Mie theory for a spherical dielectric nanoparticle (Figure I.15.A to C), that when the magnetic and electric dipole coefficients, 1

Figure I. 15

 15 Figure I.15 Directional scattering of light by a single dielectric nanoparticle. (A) Scattering efficiency of a dielectric sphere calculated by Mie theory versus size parameter q for refractive index n=4. Abbreviations for resonances are as in Fig. 1. (B) Electric 1 a and magnetic 1 b dipole scattering coefficients of the sphere (log scale). (C) Forward (FS) and backward (BS) scattering efficiencies of the sphere (log scale). The dashed line in (A) to (C) marks the wavelength at which the first Kerker condition is realized. (D) Experimental forward and backward scattering from a subwavelength spherical dielectric particle with diameter of 18 mm, recorded using a 3D angular measurement setup in the microwave spectral range at mm 18  

Figure I. 16

 16 Figure I.16 Dielectric nanoantennas. (A) Magnetic and electric near-field enhancement in a nanogap between two silicon nanocylinders (height, 15 nm; diameter, 140 nm; gap, 30 nm). Top: Schematics of the measured system; p and m indicate electric and magnetic dipole moments, respectively. Bottom: Scanning near-field optical microscopy images for the two orthogonal excitation polarizations [44]. Arrows indicate polarization direction for electric (E) and magnetic (H) fields. Color varies from minimal (blue) to maximal (red) values. Dashed circles mark the positions of silicon nanocylinders. Inset shows a SEM image of the studied dimer nanoantenna. (B) Magnetic Fano resonance in all-dielectric oligomers [45]. Top: SEM images of fabricated heptamer for different diameters (d) of the central Si disk. Bottom: Theoretical and experimental demonstrations of the magnetic Fano resonances. Blue, green, red, and cyan curves cor-respond to different diameters of the central Si nanodisk: 400 nm, 380 nm, 350 nm, and 320 nm, respectively. Arrows indicate the spectral positions of the Fano resonance. (C) Nonradiating anapole mode in silicon nanodisks [46]: Experimental dark-field scattering spectra of an amorphous Si nanodisk (diameter, 310 nm; height, 50 nm) on a quartz substrate. A strong dip in the nanodisk scattering at the wavelength of anapole excitation is clearly observed. Inset shows experimental near-field scanning optical microscopy images of the anapole mode at three different wavelengths (640, 650, and 600 nm) near the scattering minimum. White dashed circles indicate the nanodisks' positions. (D) Nonlinear spectroscopy of an array of Si nanodisks[START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by ag netic Response[END_REF]. Shown is the negative logarithm of the normalized transmission spectrum of a sample marked by a gray area indicating a resonance at 1.24 nm. The third-harmonic generation (THG) spectrum of the sample normalized over the spectrum of the substrate (purple dots) is strongly enhanced within the spectral band of the resonance. The inset shows a schematic of the THG at a single silicon nanodisk[START_REF] Kuznetsov | ptically resonant dielectric nanostructures[END_REF].

Figure II. 1 ,

 1 ZnO layer is first deposited on the silicon substrate followed by the deposition of a photoresist layer by spin-coating on top of it (Figure II.1a). Specific patterns are created in the photoresist lithographically by simple or multiple exposures of a 355 nm laser interference pattern followed by the development step. A square lattice with circular patterns of period 700 nm was created by a double exposure of the sample, whereby the diameter of the features could be controlled by changing the exposure time. (Figure II.1b). ZnO is then selectively wet-etched by means of a photoresist mask (Figure II.1c). After transferring the pattern to the ZnO layer, the photoresist can then be removed by a simple step which involves chemical dissolution (Figure II.1d). Afterwards, the resulting ZnO pattern is transferred to the substrate by reactive-ion etching (Figure II.1e). Finally, the residual mask is etched off in an acidic solution consisting of ferric chloride (10 -2 Mole of FeCl 3 .H 2 O), Figure II.1 (f) [57].

Figure II. 1

 1 Figure II.1 Schematic representing the various stages involved in the ZnO mask fabrication process and its transfer to the Si substrate [57].

Figure II. 2

 2 Figure II.2 SEM images of Si nanostructures obtained by laser interferential lithography and etched at different substrate temperatures.

Figure

  Figure II.2 show SEM images of arrays of Si resonators having different shapes and sizes. Varying the laser interference conditions, we show that a variety of shapes can be obtained from nanocones to truncated cones as well as pillar-like structures. Working at a higher laser power when the interferential patterning is performed, pillar like structures are determined where the bottom and top of the resonators have nearly identical diameters when etching is performed for 2 minutes only, as seen in Figure II.4. Cone-like structures are obtained if etching time is increased to 5 min as shown in Figure II.4b. It should also be mentioned that in all cases the arrays of silicon resonators display a high degree of periodicity in spacing and uniform shapes within the same sample. This is testimony to the high reliability and controllability of one laser based interferential lithography and RIE processes.

Figure II. 3

 3 Figure II.3 Scanning Electron Micrograph SEM of periodic silicon resonators arrays displaying multiple shapes that vary from nanocones-like to truncated cones and to pillar-like structures.

Figure II. 4

 4 Figure II.4 SEM images of Si resonate as obtained when laser interferential lithography is performed at 200 W, for 2 different etching durations.

Figure II. 5

 5 Figure II.5 The set-up used for ZnO nanowires growth.

  Figure II.7 (a) and (b), that the average nanowires diameter for each process increased when increasing the time of the dip-coating process.

Figure II. 6 Figure II. 7

 67 Figure II.6 (a) and (b) respectively. SEM images of ZnO nanowires prepared at a dip-coating process duration that exceed 60 min, while fixing the concentration of the Zinc Acetate and 0.025 M the temperature of the mi ed solution at C.

Figure II. 8

 8 Figure II.8 (a), (b), and (c) respectively. SEM images of ZnO nanowires prepared with varying Zinc Acetate concentration: (a)0.023 M, (b) 0.025 M, (c) 0.03 M and a varying dip-coating process duration, while fixing the temperature of the mi ed solution at .

Figure II. 9

 9 Figure II.9 XRD of undoped p-type (100) Si substrates.

Figure II. 10

 10 Figure II.10 XRD of ZnO nanowires (002) samples.

m

  of a molecule is measured as a shift from the incident beam frequency   0  . If the energy is transferred from the incident photon to the molecule (which is the predominant case, because at room temperature most of the molecules occupy the ground vibrational state), then the molecule is excited to higher energy state and the photon is scattered with a redreferred to as "Stokes shift"(Figure II.11).

  referred to as "anti-Stokes shift". These phenomena are shown in Figure II.11.It should be noted however that, the majority of the incident photons are scattered off by the electron cloud without any loss of energy, and therefore there is no change in frequency, since the electrons are comparatively light. The scattering is elastic in this case and this is known as Rayleigh Scattering(Figure II.11).

Figure II. 11

 11 Figure II.11 Schematic representing the three different forms of scattering.

 n N

 n is the number of molecules in the excited vibrational energy level n.  m N is the number of molecules in the ground vibrational energy level m.  m n g is the degeneracy of the levels in n and m.  m n E E  is the energy difference between the vibrational energy levels.

  Raman spectroscopy system consists of a spectrograph with the following main components: an excitation source, a light detector and a light collection and delivery system. The excitation source consists of a Cobolt samba 25 mW Ar-ion laser emitting at 532 nm, while an intensified Charged Coupled Device (ICCD) camera is used for detection, and an optical microscope for light collection. The Raman spectroscopy system used in shown in Figure II.12.

Figure II. 12

 12 Figure II.12 Diagram of Raman Microscope System used for Raman spectroscopy and delivery [59].

C

  is the total heat capacity of the sample and sample platform.

 0 T

 0 is the thermal conductance of the wires (W.K -1 ). is the temperature of the thermal bath (puck frame).  pl T is the temperature of the platform.  t P is the power applied by the heater.Therefore, that   t P is equal to 0 P during the heating portion and equal to zero during the cooling process.

Figure II. 13

 13 Figure II.13 a) Heat flow diagram for the simple model. tot C is the heat capacity of the whole

Figure II. 14

 14 Figure II.14 An example of a temperature response curve with base temperature 0 T . The heater

  10b) Where:  pl C is the heat capacity of the sample platform.

  Figure II.15 Setup for the specific heat measurement. The sample is placed with grease on a platform. Under the platform are the heater and thermometer connected. The wires from the platform to the sample holder provide a link to the thermal bath. The thermal conductance of the wires is w  , the temperature of the sample is   t T , and the temperature of the thermal bath is b T

Figure

  Figure II.17, shows the puck and platform. The sample is mounted on a thin sapphire (Al 2 O 3 ) platform of 3x3 mm 2 which is embedded in the puckframe , while a RuO 2 heater (R=0.1-10 K and a Lakeshore Cernox thermometer (R=0.1-10 K are mounted on the backside of the platform.. The platform is connected to the puckframe with eight Au-Pd wires (R=10 K. Four wires are for the 4-point probe method of the heater and four wires for the 4-point probe method of the thermometer. In each 4-point probe method, two wires are for determining the voltage and two for the current. These eight wires act as a thermal link to the bath. The current supplied to the thermometer is ∼1 nA and to the heater is ∼10 nA. The contacts between the heater and thermometer to the wires are provided by a gold layer evaporated on the sapphire platform.

Figure II. 17

 17 Figure II.17 The puck 1792 (left) and the platform (right). 1: The sapphire platform 2: Au-Pd wires 3: Gold contacts 4: Thermometer 5: Heater [64].

  Figure II.19.

Figure II.

  Figure II.[START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF] The baffle assembly with the baffles and the charcoal holder at the end. The charcoal holder is placed just above the puck and will adsorb He gas[START_REF] Design | Physical Property easurement System, Torque a gnetometer ption Use r ' s anu al[END_REF].

Figure II. 19

 19 Figure II.19 The results of the calibration (Pass 1 and Pass 2) of puck 1505. The resistance of the platform thermometer (orange circles) and heater (blue triangles) measured during the calibration. The inset shows the thermal conductivity (black circles) of the eight Au-Pd wires during Pass 2 of the calibration [64].

Figure II. 20

 20 Figure II.20 The dialogue box. 1) Temperature range: Temperature startpoint, endpoint and spacing. 2) Temperature rise: Height of temperature pulse. 3) Repetitions at each temperature: At each temperature point the specific heat is measured three times in order to check stability and reproducibility. The results are not averaged. 4) Relaxation measurement: The measurement time is always set to 2 . During this time data acquisition takes place. 5) Determination of initial time constant: In general determined by applying a trial heat pulse, or, if known, it can be fixed. 6) Measurement stability: A heat pulse will only be applied when the temperature is within 1% of the requested tem-perature rise. 7) Extra wait time: When measuring at high temperatures it is necessary to wait longer before the measurement starts[START_REF] Design | Physical Property easurement System, Torque a gnetometer ption Use r ' s anu al[END_REF].

  Figure II.21. The specific heat of the grease is obtained by subtracting the 'empty puck' curves from the 'addenda' curves. In order to obtain an average result for the specific heat of Apiezon N Grease, four different combinations were made (Addenda # 1-Puck # 1, Addenda # 2-Puck # 1, etc.) as shown in Figure II.22. The solid line in Figure II.22 represents the average and the uncertainty in the mean is indicated by the colored area around the average curve. Other works also managed to determine the specific heat of Apiezon N Grease, and these results are summarized in Figure II.23.The first major difference between the measured curve here and the literature is the absence of an anomaly at around ∼ 280 K, also the specific heat of the grease measured show a plateau around 3 mJ/gK 2 , whereas the literature values all attain a value around 6 mJ/gK 2[START_REF] Wielemaker | The evolution of the ferromagnetic ordering of o dope d URhGe[END_REF]. This could be due to a change in the composition of the N Grease.

Figure II. 21

 21 Figure II.21The heat capacity of the puck with 1 mg of grease (blue circles) and of the puck with 0.3 mg grease (yellow squares). The heat capacity of the empty puck is measured twice (red diamonds and green triangles)[START_REF] Wielemaker | The evolution of the ferromagnetic ordering of o dope d URhGe[END_REF].

Figure II. 22

 22 Figure II.22The average specific heat of Apiezon N Grease (solid blue line) calculated from two addenda measurements and two empty measurements (four different combinations). The uncertainty in the mean is indicated by the colored area around the average curve. The inset shows the low temperature behavior[START_REF] Wielemaker | The evolution of the ferromagnetic ordering of o dope d URhGe[END_REF].

Figure II. 23

 23 Figure II.23 The specific heat of Apiezon N Grease measured by several groups. The anomaly around 290 K reduces through the years. All the curves saturate (if the anomaly around 280 K is discarded) around C/T= 7 mJ/gK 2 . Picture taken from Moriya et al. [67].

  Figure III.1, we present scanning electron micrographs (SEM) of four different arrays as an illustration of the diversity of the Si resonators investigated in this work.

Figure III. 1

 1 Figure III.1 Scanning electron micrographs (SEM) of periodic arrays of Si resonators of different sizes and shapes.

Figure III. 2

 2 Figure III.2 Raman spectra from the investigated structured Si substrates and a bare Si substrate. Left-hand side: As measured Raman spectra. Right-hand side: Normalized Raman spectra. The position of the Raman peak of the bare Si substrate at 520.8 cm -1 is highlighted to illustrate the downshift of the Raman peaks corresponding to the structured Si substrates.

  the broadening and downshift of the Raman peak, Figure III.2 shows that the resonators induce a remarkable Raman enhancement. If the Raman spectra in Figure III.2 were corrected for the areal density of the resonators, the Raman enhancement would be of orders of magnitude. As mentioned above, the phonon confinement breaks the conservation of the total momentum and allows all the phonon modes whose wavevectors equal to or smaller than D 1 (where D is the size of the crystallites in which the phonons are confined) to contribute to the sample's response to an optical e citation. This would indeed imply a Raman enhancement of the order of 3 

  Figure III.2 that large arrays of Si resonators could ensure interaction between confined phonons and confined photons. In what follows, we shall examine the effect of resonators shape and size on the phonon and photon confinement. III.3.1. Effect of the resonator shape for the "short" height resonators The Raman spectra of the samples of the first set (samples S S,1 and S S,2 ) are shown together with finite-difference time-domain (FDTD) simulations of the electric field distribution on the surfaces of the measured samples in Figure III.3.

Figure III. 3 III. 3 . 2 .

 332 Figure III.3 Finite-difference time-domain (FDTD) simulations of the electric field distribution in the samples of the first set, and the Raman spectra corresponding to the samples of the first set. The dashed lines in the FDTD simulations indicate the interface between the resonators and the Si substrate. The symbols indicate the measured Raman spectra. The solid lines indicate the calculated Raman spectra using the spatial correlation model. The agreement between the calculated and measured Raman spectra is demonstrated in Figure III.3. We show in Table III.I. the parameters of the crystallites size distribution obtained by fitting the spatial correlation model to the measured Raman spectra.

Figure III. 4

 4 Figure III.4 Finite-difference time-domain (FDTD) simulations of the electric field distribution in the samples of the second set, and the Raman spectra corresponding to the samples of the second set. The dashed lines in the FDTD simulations indicate the interface between the resonators and the Si substrate. The symbols indicate the measured Raman spectra. The solid lines indicate the calculated Raman spectra using the spatial correlation model.

III. 3 . 3 .

 33 .I). The FDTD simulations in FigureIII.4 demonstrate that a noticeable amount of the incident electric field is confined in the cone-like resonators in sample S M,1 . Therefore, the corresponding Raman spectrum exhibits a significant Raman enhancement accompanied by a downshift and broadening. The Raman enhancement is due to electric field confined in the conelike resonators, whereas the downshift and broadening of the Raman spectrum is due to light interaction with phonons confined in nanosized crystallites in the conelike resonators. Thus, unlike the cone-like resonators of the first set, the cone-like resonators of the second set can promote interaction between confined electric field and phonons confined in nanosized crystallites. The FDTD simulations in Figure III.4 also demonstrate that the field confinement gets stronger as the resonators become more pillar-like. The results of the FDTD simulations are supported by the Raman measurements, which demonstrate that the Raman enhancement gets stronger as the diameter of the resonators at the top approaches the diameter of the resonators at the bottom. The spatial correlation model was fitted to the Raman spectra of the samples of the second set with adjusting the parameters determining the lognormal size distribution function. A very satisfactory agreement between the calculated and measured Raman spectra was obtained. The calculated Raman spectra are plotted in Figure III.4 using solid lines, and the crystallites size in each of the measured samples obtained from fitting the spatial correlation model to the measured Raman spectra is indicated in Table III.I. We found a size distribution only for the sample having pillar-like resonators (sample S M,3 ). The other samples of the second set showed monodisperse crystallites. Effect of resonator shape for "large" height resonators The samples in the third set have the highest resonators. The Raman spectra and FDTD simulations of the field distribution for the samples of this set (samples S L,1 , S L,2 , and S L,3 ) are shown in Figure III.5.

Figure III. 5

 5 Figure III.5 Finite-difference time-domain (FDTD) simulations of the electric field distribution in the samples of the third set, and the Raman spectra corresponding to the samples of the third set. The dashed lines in the FDTD simulations indicate the interface between the resonators and the Si substrate. The symbols indicate the measured Raman spectra. The solid lines indicate the calculated Raman spectra using the spatial correlation model.

  By comparing the FDTD simulations of the electric field distributions and the Raman enhancements shown in Figure III.3, Figure III.4, and Figure III.5, we notice that the field confinement gets stronger as the resonators height is increased. On the other hand, by comparing the FDTD simulations and Raman enhancements shown in each of Figure III.3, Figure III.4, and Figure III.5, we notice that the field confinement gets stronger as the resonators shape becomes cylindrical. Moreover, we can notice from Figure III.5 that, although the volume of the resonators of sample S L,2

Figure III. 3 ,

 3 Figure III.4, and Figure III.5 with the corresponding crystallites size distributions in Table

Figure III. 6

 6 Figure III.6 Raman shift and spectrum broadening versus the incident laser power.

1 )Figure III. 7

 17 Figure III.7 Rama spectra of samples S M,3 and S L,3 measured with a laser power of 1mW/µm 2 and 0.1 mW/µm 2 . The spectra are normalized for the sake of comparison.

  The nanostructures investigated in this work are sketched in Figure IV.1.

Figure IV. 1

 1 Figure IV.1 Phonon hybridization mechanism and sketch of typical investigated Si nanostructured samples.

  Where total C is the total heat capacity of the sample and sample platform, w  the thermal conductivity of the supporting wires, b T the temperature of the thermal bath, and   t P the power applied by the heater. The heater power   t P is equal to 0 P during the heating portion of the measurement and equal to zero during the cooling portion. The solution of this equation is given by exponential functions with a characteristic time constant  equal to total total C  , where total  is the thermal conductivity of the sample and the sample platform.



  the thermal conductivity of the grease between the sample platform and the sample.

  Figure IV.2.

Figure IV. 2

 2 Figure IV.2 Heat capacity measurements of the samples investigated.

Figure

  Figure IV.3 T C v versus 2 T . Symbols: Measurements. Solid lines: Extrapolation of T C v down to

  hand, the Debye approximation for the phonon spectrum of a solid can be expressed as:It must be emphasized, however, that D  is an artificial limiting frequency which leads to correct normalization of    g and bears no simple relation to the true maximum frequency of the crystal. Since the specific heat at low frequencies is mainly a function of the small  behavior of    g and the behavior of    g in the Debye approximation and for exact discrete lattice models is the same for small  , the Debye approximation can be accurately used for the analysis of the specific heat at low temperature and the estimation of the low frequency end of the phonon spectrum. Upon using the Debye characteristic temperature from equation (IV.10) and (IV.6) that:

  coefficients of the low frequency end of the phonon spectrum ( 2 a and 4 a ) can be obtained by fitting the

Figure IV. 4

 4 Figure IV.4 Temperature-dependent Debye temperatures of the measured samples. Symbols: Values obtained by using the heat capacity measurements in equation (IV.11). Solid lines: Best fits of the temperature-dependent Debye temperature curves with equation (IV.13) and equation (IV.14).

Figure IV. 5 (

 5 after proper normalization).

Figure IV. 5

 5 Figure IV.5 Low frequency end of the phonon spectrum of Si. Symbols: Values obtained from the analysis of heat capacity measurements at low temperatures. Solid lines: Values obtained from first principles calculations.

Figure IV. 6

 6 Figure IV.6 Low frequency ends of the phonon spectra of the samples investigated.

17

 17

g

  is an even function of  , only the even coefficients n a 2 are non- vanishing. The even moments of the phonon spectrum may be given in terms of the power of x, we find upon combining equation (IV.17) and equation (IV.18) that the coefficient n a 2 in equation (IV.16) can be expressed as:

.

  relation to the Debye temperature. By using equation (IV.22), Thirring clearly demonstrated that one can find a high temperature expansion for the specific heat which converges, This limitation is imposed by the finite radius of convergence of the series in equation (IV.20). The moments of the phonon spectrum can hence be obtained by fitting equation (IV.22) to the measured specific heat curves at high temperatures. Then, from the knowledge of the spectrum moments, the high frequency phonon spectrum can be approximated by using equation (IV.16) and equation (IV.19). The best fits of equation (IV.22) to the measured specific heat curves are shown in Figure IV.7 in solid lines.

Figure IV. 7

 7 Figure IV.7 Heat capacity. Symbols: measurements of the investigated samples. Solid lines: Best fit of the heat capacity measurements at high temperatures using Equ.(IV.22).

  samples will be referred to in what follows as WP1, WP2, and WP3. The X-ray diffraction patterns from the samples investigated are illustrated in chapter II (Figure II.10).

Figure V. 1

 1 Figure V.1 Top-view electron scanning micrographs of the samples investigated.

Figure V. 2

 2 Figure V.2 Models for sample holder, Si substrate on sample holder, and ZnO nanowires on Si substrate on sample holder with the effect of the conductance of the thermal bond of ZnO NWs to the Si substrate.

Figure V. 3

 3 Figure V.3 Specific heat of Si and the samples investigated. No separation between the contribution of the nanowires and the contribution of the substrate to the thermal response was made.



  are the ZnO NWs specific heat and the thermal conductance of the interface between Si and ZnO NWs, respectively, and   t T 3 is the temperature of ZnO NWs at instant t. The values of 3 c and I  in the measured temperature range were obtained by eliminating   t T 2 and   t T 3 in equation (V.3) integrating over the measurement time period, and then fitting the resulting equation for   t T 1 to the measured thermal response of the calorimeter using linear least squares method. The measured   t T 1 and variation of   WP3 are initially (before the arrival of the heating pulse) at 1.80, 42.30, and 79.40 K are shown in Figure V.4 by symbols. The linear least squares fitting of the measured   t T 1 and 1 T  data are shown in the same figure by solid lines. The good agreement between measured and calculated of the deduced results.

Figure V. 4

 4 Figure V.4 Temperature variation of the sample platform in a measurement cycle when the samples investigated are loaded in the calorimeter. The holder temperature before the arrival of the heat pulse is indicated. The symbols indicate the measured data. The solid lines indicate the data fitting using the heat balance equations.

Figure V. 5 2 T

 52 Figure V.5 Filled symbols: Calorimetric measurements of specific heat of free-surface ZnO nanowires. Open symbols: Specific heat of bulk ZnO obtained from first principles calculations.

Figure V. 6

 6 Figure V.6 C/T vs T 2 plots corresponding to bulk Si and the investigated ZnO nanowires. Fiiled symbols: Data obtained from calorimetric measurements. Open symbols: Data obtained from first principles calculations. Solid line: Data fitting using the model outlined in the appendix 3.The results in Figure V.6 also show that the deviation of the NW specific heat from the three-dimensional behavior increases with the NW diameter. In order to understand the physical phenomena responsible for this deviation from the threedimensional behavior, we developed a theoretical model based on the solution of the solid equation of motion in cylindrical coordinate within the continuum medium approximation with appropriate boundary conditions. The developed model is detailed in Appendix 3. The model demonstrates that the NW specific heat at low temperatures is of the form:



  is the Debye temperature corresponding to the phonon modes associated with linear dispersions giving rise to a two-dimensional crystal contribution, and Q D,  is the Debye temperature corresponding to phonon modes associated with quadratic dispersions giving rise to a three-dimensional crystal contribution. In fact, the low-temperature specific heat of any three-dimensional solid is characterized by only one parameter, namely the Debye temperature D  . All experimentally measured low-temperature specific heat curves can be brought into coincidence with each other by assigning the proper value of D  to each measured solid. Analogously, a Debye temperature determining the contribution from three-dimensional crystal and a Debye temperature determining the contribution from two-dimensional crystal characterize the low-temperature specific heat of any NW. The theoretical model in Appendix 3 showing that they depend on the diameter and length of the NW. The solid lines in Figure V.6 represent the best fit of the measured T c curves using equation (V.4). The best fit was obtained with considering  equal to zero (no contribution from electron gas) and adjusting L D, that gave the best agreement between experimental and theoretical T c versus 2

Figure V. 7

 7 Figure V.7 Filled symbols: calorimetric measurements of thermal conductance of the interface between Si and ZnO nanowires. Open symbols: Thermal conductance of the interface between Si and bulk ZnO obtained from first principles calculations within the diffuse mismatch model.

FIG. 2

 2 FIG.2 Les spectres Raman des substrats de Si structurés étudiés et un substrat de Si nu. Partie gauche: mesurée par les spectres Raman. Côté droit : spectres Raman normalisés. La position du pic Raman du substrat de Si nu à 520,8 cm -1 est mise en évidence pour illustrer le décalage basse fréquence des pics Raman correspondant aux substrats de Si structurés.

FIG. 3

 3 FIG.3 Simulations de domaine temporel à différence finie (FDTD) de la distribution du champ électrique dans les échantillons du premier ensemble, et les spectres Raman correspondant aux échantillons du premier ensemble. Les lignes pointillées dans les simulations FDTD indiquent l'interface entre les résonateurs et le substrat Si. Les symboles indiquent les spectres Raman mesurés. Les lignes continues indiquent les spectres Raman calculés en utilisant le modèle de corrélation spatiale.

FIG. 5

 5 FIG.5 Simulations électromagnétiques aux différences finies (FDTD) de la distribution du champ électrique dans les échantillons du troisième ensemble, et spectres Raman correspondant aux échantillons du troisième ensemble. Les lignes pointillées dans les simulations FDTD indiquent l'interface entre les résonateurs et le substrat Si. Les symboles indiquent les spectres Raman mesurés. Les lignes continues indiquent les spectres Raman calculés en utilisant le modèle de corrélation spatiale.

FIG. 6 FIG. 7

 67 FIG.6 Décalage Raman et élargissement du spectre par rapport à la puissance laser incidente.

FIG. 8

 8 FIG.8 Mécanisme d'hybridation de Phonon et croquis d'échantillons nanostructurés de Si typiques étudiés.

C

  est la chaleur spécifique de la plate-forme, sample C la chaleur spécifique de l'échantillon et g  la conductivité thermique de la colle thermique entre la plate- forme et l'échantillon.

  FIG.10 : Variation de T C v en fonction de 2 T . Symboles : Mesures. Lignes pleines : Extrapolation

  aboutit à une chaleur spécifique pouvant être exprimée comme la moyenne sur le spectre phonon    g comme :



  part, l'approximation de Debye pour le spectre phonon d'un solide peut être exprimée comme : Il faut toutefois souligner que  est une fréquence limite artificielle qui conduit à une normalisation correcte de    g et qui n'a pas de relation simple avec la fréquence maximale vraie du cristal. Puisque la chaleur spécifique aux basses fréquences est principalement fonction du petit comportement de  et que le comportement de    g dans l'approximation de Debye et pour des modèles exacts de réseaux discrets est identique aux petit  , l'approximation de Debye peut être utilisée avec précision pour l'analyse de la chaleur spécifique à basse température et l'estimation de l'extrémité basse fréquence du spectre de phonon. Lors de l'utilisation de la température caractéristique de Debye à partir de l'équation 10 et de l'équation 6 que : sans dimension. Dans la limite où la borne supérieure de l'intégrale de l'équation 11 peut être rejetée à l'infini, l'expression de la chaleur spécifique se réduit à : contribution de la dynamique harmonique du réseau à la chaleur spécifique à basse température dans l'équation 4. De plus, on voit dans l'équation 11 que la chaleur spécifique de tout solide dans l'approximation de Debye est caractérisée par un seul paramètre T D  . Par conséquent, afin d'obtenir une description précise de la chaleur spécifique solide, il est nécessaire d'utiliser une valeur différente D  à chaque température [140]-[142]. La détermination de la dépendance à la température de D  peut être effectuée en égalant les valeurs mesurées de la chaleur spécifique à l'expression de Debye pour la chaleur spécifique, Eq.11, et considérer l'équation résultante comme une équation pour D  . La dépendance en température de D  des échantillons structurés en surface et celle du substrat de Si nu sont reportées ensemble sur la figure 10 avec des symboles. Maintenant, en comparant l'équation 9 et l'équation 12, nous remarquons que : est la température de Debye à des températures extrêmement basses. D'autre part, l'équation 12 donne la chaleur spécifique rigoureuse à basse température d'un cristal. Toutefois, compte tenu de l'équation 9, si tel est le cas, D  doit être une fonction de la température dont la valeur à toute température est déterminée en égalant l'équation 12 et l'équation 6. Lorsque cela est fait, on constate que, à basse température :

et 4 a

 4 peut être obtenu en extrapolant les courbes de   T D  en T =0, et les coefficients de l'extrémité basse fréquence du spectre de phonons ( 2 en utilisant l'équation 13 et l'équation 14 sont représentés sur la figure 11 par des lignes continues, et les valeurs obtenues de sont reportées dans le tableau II. Selon l'équation 8, les coefficients 2 a et 4 a peuvent être utilisés pour tracer l'extrémité basse fréquence des spectres de phonons des échantillons mesurés.

FIG. 12

 12 FIG.[START_REF] Srivastava | The physics of phonons[END_REF] Extrémité basse fréquence du spectre de phonons de Si. Symboles : Valeurs obtenues à partir de l'analyse des mesures de la chaleur spécifique à basse température. Lignes pleines : Valeurs obtenues à partir des calculs des premiers principes.

FIG. 13

 13 FIG.[START_REF] Wei | Phonon dispersions of silicon and germanium from first-principles calculations[END_REF] Extrémités basse fréquence des spectres de phonons des échantillons étudiés.

  17) Puisque    g est une fonction paire de , seuls les coefficients pairs n a 2 sont nonnuls. Les moments pairs du spectre de phonons peuvent être donnés en termes des moments adimensionnels n u 2 comme : la puissance paire de x, on constate, en combinant l'équation 17 et l'équation 18, que le coefficient n a 2 de l'équation 16 peut être exprimée comme :

.

  de relation simple avec la température de Debye. En utilisant l'équation 22, Thirring a clairement démontré qu'on peut trouver un développement à haute température pour la chaleur spécifique qui converge, théoriquement, pour Cette limitation est imposée par le rayon de convergence fini de la série dans l'équation 20.

FIG. 14

 14 FIG.[START_REF] Peierls | ur kinetischen Theorie der Wärmeleitung in Kristallen[END_REF] Chaleur spécifique. Symboles : mesures des échantillons étudiés. Lignes pleines : meilleur ajustement des mesures de chaleur spécifique à haute température en utilisant Eq.22.

FIG. 16

 16 FIG.[START_REF] Schuller | General properties of dielectric optical antennas[END_REF] Images au microscope électronique à balayage des échantillons étudiés.

FIG. 17

 17 FIG.[START_REF] Evlyukhin | ptica l response features of Si-nanoparticle arrays[END_REF] Modèles pour porte-échantillon, substrat Si sur porte-échantillon et nanofils ZnO sur substrat Si sur porte-échantillon avec effet de la conductance de la liaison thermique des ZnO NW sur le substrat Si.

  23) a été ajusté à la réponse thermique enregistrée en utilisant la méthode des moindres carrés linéaires pour obtenir la chaleur spécifique de porte-échantillon ( 1 c ) et la conductance thermique des fils ( w  ) qui relient la plate-forme d'échantillon à un radiateur maintenu à la température 0 T . Dans l'Eq.23,   t P est la puissance d'entrée dans la plate-forme d'échantillonnage et   t T 1 est la température de la plate-forme à

FIG. 18

 18 FIG.[START_REF] García-Etxarri | Strong magnetic response of submicron Silicon particles in the infrared[END_REF] Chaleur spécifique de Si et les échantillons étudiés. Aucune séparation entre la contribution des nanofils et la contribution du substrat à la réponse thermique n'a été faite.

FIG. 19

 19 FIG.[START_REF] Zhang | Dielectric spheres with maximum forward scattering and zero backscattering: a search for their material composition[END_REF] Variation de température de la plate-forme d'échantillon dans un cycle de mesures lorsque les échantillons étudiés sont chargés dans le calorimètre. La température du support avant l'arrivée de l'impulsion de chauffage est indiquée. Les symboles indiquent les données mesurées. Les lignes continues indiquent l'ajustement des données en utilisant les équations de bilan thermique.



  FIG.21 représentent le meilleur ajustement de la mesure T c courbes en utilisant Eq.26. Le meilleur ajustement a été obtenu en considérant  égal à zéro (pas de contribution du gaz d'électrons) et en ajustant L D,  et Q D,  . Les valeurs de

n

  peut également remarquer d'après la figure 2 que la conductance thermique de l'interface Si / ZnO NW chute rapidement entre 1.8 et 3 K, suit le comportement de Si / ZnO entre 3 et 70 K, puis croît presque linéairement à des températures plus élevées. Nous pensons que la chute rapide entre 1.8 et 3 K est due à une transition entre transmission phononique spéculaire et transmission diffusive lorsque la température augmente, ou bien lorsque les longueurs d'onde des phonons excités changent. Ceci est étayé par des mesures de réflexion de pulses de chaleur sur des solides de surfaces libres, qui montrent que la transmission de phonons passe d'une nature spéculaire à un comportement diffusif lorsque les phonons de fréquences supérieures à 100 GHz sont portés à quelques degrés Kelvin[START_REF] Eisenmenger | Phonon Scattering in Condensed Matter : Proceedings of the Fourth International Conference University of Stuttgart[END_REF]. La croissance

  the most probable size and  is the skewness of the size distribution.

  Several attempts have been made to establish a universal form for the phonon weighting function   D r, 

x

  the volume of the first Brillouin zone, and the integral is carried out through the first Brillouin zone. If we change to spherical polar coordinates  are simple zeros of   x f, Eq.A10 is transformed into: distribution function per unit solid angle for the jth branch of the total spectrum.The idea behind Houston's method is as follows. The function in  and  because the normal mode frequencies are invariant against any real orthogonal transformation of axes which takes the crystal to itself. in terms of Kubic harmonics which have the symmetry of the lattice[START_REF] Der Lage | A e thod for btaining Electronic Eigenfunctions and Eigenvalues in Solids with An Application to Sodium[END_REF],

,.C

  Houston reasonably considered that there exist directions in reciprocal space along which the cubic crystal dynamical matrix factors into equations of low degree in2  which can be solved exactly for  as a function of the wavevector k . These directions are high symmetry directions such as (100),[START_REF] Iskandar | odi fication of the phonon spectrum of bulk Si through surface nanostructuring[END_REF], and (111) directions. Along these directions the equation Since these equations can be inverted exactly to obtain k for simple enough models, it follows that we can obtain special directions. If we then retain as many terms in Eq. A13 as the number of directions   the (100),[START_REF] Iskandar | odi fication of the phonon spectrum of bulk Si through surface nanostructuring[END_REF], and (111) directions, it follows from Eq.A13, Eq.A14, and Eq.A15 that: superscripts A, B, and C stand for (100),[START_REF] Iskandar | odi fication of the phonon spectrum of bulk Si through surface nanostructuring[END_REF], and (111) respectively. Indeed, Houston's method can be used for the appro imate evaluation of any integral of the form: dealing here with the low frequency end of the phonon spectrum we need to consider only the acoustic branches of the phonon spectrum. From the general form of a three-dimensional dynamical matrix on can readily find that the leading term in the expansion of j C and j D have the symmetry of the lattice. InvertingEq.A18 we find the frequency distribution function per unit solid angle for the jth branch of the spectrum: Eq.A17, Eq.A19, and Eq.A12 we find that: notice from Eq.A18 that the coefficients phonon group velocities for a given direction    , of propagation. Thus, by applying Houston's method to the evaluation of the integral in Eq.A20 we find that the coefficients of the low frequency end of the phonon spectrum can be expressed in terms of to the phonon group velocities as: indicate crystallographic directions.

  number of modes in the NW is thus:

196

 196

k

  the number of atoms per unit cell, c N is the total number of unit cells, are the cut-off frequencies for linear (quadratic) dispersions. It is a common practice to relate the cut-off frequency to Debye temperature according to is the Boltzmann constant. This allows us to establish two Debye temperatures for NWs: a Debye temperature for the modes of linear dispersions given by: temperature for the modes of quadratic dispersions given by:

2 g

 2 are the first and second term in Eq.A37. Upon introducing .A34 and Eq.A35 in Eq.A36, we can write the expression of the NW specific heat in the form:

  deviation of the NWs specific heat from the Debye 3 T law.

  .......................................................................................................... 77 Table IV.I The characteristics of the samples investigated and the parameters of their phonon spectra obtained from the analysis of the specific heat measurements. .................................................................................................................................. 95 Table V.I The structural characteristics and Debye temperatures of the ZnO nanowires in the measured samples. ...................................................................... 109

  of the differences S between the modelled T at the measured time i t

					P 			e	t			C tot	T	0		t 	0 t	(II.8)
		T	t	P 	e	t		e t	t		tot	t 	t	0
	Where	 tot C 		is a relaxation constant. The measured   t T	curve is fitted to
	equation (II.8). The three unknown variables  	w	0 , T , C tot		are varied in such a way
	that the sum

and the measured temperature i T are minimal:

  Figure III.2, we illustrate the Raman spectra of the samples investigated (see Table III.I) together with the Raman spectrum of a bare Si substrate. All the spectra in Figure III.2 are recorded by using a laser power of 0.1 mW/µm 2 and long integration time to avoid any effect of Fano resonance and laser heating. The lefthand side of Figure III.2 presents the "as measured" Raman spectra, whereas the right-hand side of Figure III.2 presents the Raman spectra of the measured samples normalized to the Raman spectrum of the bare Si substrate. As can be noticed, most of the structured samples present a broadened and downshifted Raman peak relative to the bare Si substrate Raman peak. Since we have ruled out the effects of laser heating and Fano resonance, we confidently attribute this to phonon confinement.

Table IV .

 IV I.

	Sample Structure bottom	Structure top diameter	Structure height	Structure volume			0	a	2	a	4	Spectrum moments
		diameter (nm)	(nm)	(nm)	(nm 3 )		(K)	X 10 -42	X 10 -70
	Si	-	-	-	-	0.007	625	5.05	-4501	u2=0.850
													u4=0.720
													u6=0.300
	S1	134	15	450	2 378 696	0.0070	600	6.1894	-3901	u2=0.695
													u4=0.450
													u6=0.145
	S2	453	413	592	87 236 076	0.0070	750	3.1690	-230	u2=0.840
													u4=0.780
													u6=0.355
	S3	334	43	800	26 759 458	0.0070	590	6.5094	-475	u2=0.760
													u4=0.540
													u6=0.190
	S4	380	235	875	66 185 340	0.0005	650	4.8681	-345	u2=0.690
													u4=0.460
													u6=0.150
	S5	265	60	1000	23 489 950	0.0070	625	5.4760	-4101	u2=0.930
													u4=0.870
													u6=0.400
	S6	450	100	1150	77 525 343	0.0048	870	2.0302	-128	u2=0.670
													u4=0.530
													u6=0.205
	S7	291	159	1240	50 717 463	0.0070	855	2.1390	-120	u2=0.580
													u4=0.380
													u6=0.123
	S8	650	190	1280	195 063	0.0048	933	1.6461	-85	u2=0.610
					582								u4=0.445
													u6=0.160

Effet de la forme du résonateur sur les résonateurs de grande taille

  montrent également que le confinement du champ devient plus fort à mesure que les résonateurs deviennent plus semblables à des piliers. Les résultats des simulations FDTD sont appuyés par les mesures Raman, qui démontrent que l'amélioration Raman devient plus forte lorsque le diamètre des résonateurs au sommet s'approche du diamètre des résonateurs au fond. Le modèle de corrélation spatiale a été ajusté aux spectres Raman des échantillons du second ensemble en ajustant les paramètres déterminant la fonction de distribution de taille log-normale. Un accord très satisfaisant entre les spectres Raman calculés et mesurés a été obtenu. Les spectres Raman calculés sont représentés sur la figure4en utilisant des lignes pleines, et la taille des cristallites dans chacun des échantillons mesurés obtenus en ajustant le modèle de corrélation spatiale aux spectres Raman mesurés est indiquée dans le tableau I. Les autres échantillons du second ensemble ont montré des cristallites monos disperses.

	Les échantillons du troisième ensemble ont les résonateurs les plus intenses. Les
	spectres Raman et les simulations FDTD de la distribution de champ pour les
	échantillons de cet ensemble (échantillons S L,1 , S L,2 , et S L,3 ) sont représentés sur la
	figure 5.
	Les spectres Raman et les simulations FDTD démontrent que les résonateurs des
	échantillons dans le troisième ensemble sont remarquablement plus efficaces dans
	le confinement du champ électrique que les résonateurs des échantillons du premier
	et second ensemble. Cependant, semblablement au premier et au deuxième
	ensemble, le confinement de champ devient plus fort alors que les résonateurs
	deviennent plus semblables à des piliers.

C.

D. Effet de la taille et de la forme du résonateur sur l'interaction entre les photons confinés et les phonons confinés

  Les spectres Raman et les simulations FDTD réalisées sur les échantillons des trois ensembles d'échantillons décrits dans le tableau I ont révélé des informations importantes sur l'effet de la forme et de la taille des résonateurs sur le confinement des photons et des phonons. En comparant les simulations FDTD des distributions de champ électrique et les améliorations Raman montrées sur les FIG.3, FIG.4 et FIG.5, nous remarquons que le confinement de champ devient plus fort lorsque la hauteur des résonateurs est augmentée. D'autre part, en comparant les simulations FDTD et les améliorations Raman montrées dans chacune des FIG.3, FIG.4 et FIG.5, nous remarquons que le confinement de champ devient plus fort lorsque la forme des résonateurs devient cylindrique. De plus, on peut remarquer d'après la figure5que, bien que le volume des résonateurs de l'échantillon S L,2 soit supérieur à celui des résonateurs de l'échantillon S L,3 , le confinement du champ électrique dans l'échantillon S L,2 est plus faible que dans l'échantillon S L,3 . Ces résultats suggèrent ce qui suit. Alors que le volume des résonateurs ne présente pas d'effet appréciable sur le confinement du champ électrique dans les résonateurs, la hauteur et la forme des résonateurs sont des paramètres clés déterminant la force du confinement du champ dans les résonateurs diélectriques. Le confinement du champ électrique devient plus fort à mesure que les résonateurs augmentent en hauteur et que leur forme devient cylindrique. Il apparait ainsi que les résonateurs cylindriques les plus élevés peuvent atteindre la plus grande amplitude d'oscillation, ce qui est le principal déterminant du confinement sur le terrain. A mesure que la hauteur des résonateurs diminue et que leur forme s'écarte de la forme cylindrique idéale, les résonateurs perdent de leur caractère résonant, et par conséquent perdent leur efficacité à confiner le champ électrique incident.Le confinement du phonon dans les cristallites de taille moyenne comparable a été observé dans tous les échantillons mesurés. Ceci suggère que la taille et la forme des résonateurs n'ont pas d'effet appréciable sur la longueur de corrélation spatiale dans les résonateurs. Cependant, en comparant les simulations FDTD de la FIG.3, FIG.4et FIG.5avec les distributions de taille des cristallites correspondantes dans le Tableau I., nous remarquons que l'asymétrie de la distribution de taille augmente à mesure que le champ électrique confiné se répand dans les résonateurs. Ceci peut être compris si l'on se rend compte que les dommages dans les couches de surface du résonateur induits par le processus de structuration ne sont pas uniformes sur la hauteur du résonateur. Ainsi, lorsque le champ électrique se propage dans les résonateurs, il interagit avec des phonons confinés dans des cristallites de tailles différentes.En conséquence, de longs résonateurs cylindriques disposés sur des points de réseau bidimensionnels sur une grande surface présentent un grand potentiel pour confiner une quantité importante du champ électrique incident et favoriser l'interaction entre le champ électrique confiné et les phonons confinés dans des cristallites nanométriques dans les couches superficielles des résonateurs. Une telle interaction pourrait améliorer de manière significative les processus photovoltaïques dans Si car elle amplifie fortement le nombre de photons interagissant avec Si et ne nécessite pas de conservation de l'impulsion totale.

E.

Confinement de champ électrique et chauffage dans les résonateurs

  Il est maintenant bien connu que les puissances laser élevées peuvent accumuler une quantité importante d'énergie dans les cristallites nanométriques (en raison de leurs faibles propriétés thermiques), et donc déformer leur spectre Raman en renforçant les effets du chauffage laser et de la résonance Fano. Nous avons trouvé que la puissance de seuil pour les effets du chauffage laser et de la résonance Fano dans les cristallites de Si nanométriques est d'environ 1 mW/μm 2[START_REF] Faraci | Quantum confinement and thermal effects on the Raman spectra of Si nanocrystals[END_REF],[START_REF] Gupta | Laser-Induced Fano Resonance Scattering in Silicon Nanowires[END_REF]. Comme mentionné précédemment, afin d'exclure l'effet du chauffage laser et de la résonance Fano induite par laser, nous avons diminué successivement la puissance du laser d'excitation jusqu'à ce que le décalage Raman et la forme de la ligne deviennent indépendants de la puissance laser.Sur la figure6, nous montrons le décalage Raman mesuré et l'élargissement du spectre par rapport à la puissance laser incidente. Comme on peut le constater, le décalage Raman et l'élargissement du spectre deviennent indépendants de la puissance du laser uniquement pour des puissances laser inférieures à environ 0,2 mW/μm 2 . Cela implique que les effets de la résonance Fano et du chauffage laser sur le décalage Raman mesuré et la forme de la ligne ne disparaissent complètement que pour une puissance laser inférieure à 0,2 mW/μm 2 , ce qui est cinq fois plus faible que la puissance seuil signalée de Résonance fano dans des cristallites de Si nanométriques. Sur la base de la discussion ci-dessus, nous attribuons l'existence des effets de la résonance Fano et du chauffage laser à de telles puissances laser au fait que le confinement du champ électrique dans les résonateurs augmente significativement la température des cristallites incorporées dans les résonateurs.

du spectre de phonon de Si massif par nanostructure de surface

  spectres Raman provenant de grands réseaux de résonateurs diélectriques Si gravés sur des substrats de Si présentent une amélioration appréciable, accompagnés d'un décalage spectral et d'un élargissement par rapport au spectre Raman d'un substrat de Si nu. Les simulations FDTD ont démontré que la résonance du réseau collectif induit un confinement efficace du champ dans les résonateurs, ce qui entraîne une amélioration appréciable du signal Raman. Un modèle de corrélation spatiale a démontré que le rétrécissement et l'élargissement du spectre sont le résultat de la relaxation du vecteur d'onde phonon, qui est due au confinement des phonons dans les cristallites de Si nanométriques dans les couches superficielles des résonateurs. L'effet de la taille et de la forme des résonateurs sur le confinement des photons et des phonons a été étudié. Il a été démontré que, lorsque les résonateurs augmentent en hauteur et que leur forme devient cylindrique, l'amplitude de leur oscillation cohérente augmente et, par conséquent, leur capacité à confiner le champ électrique entrant augmente. Cependant, il a été constaté que la forme et la taille des résonateurs n'ont pas un effet remarquable sur le confinement des phonons. Les résultats présentés dans ce travail montrent que les grands réseaux de résonateurs diélectriques longs cylindriques suppriment radicalement les pertes de diffusion de la lumière par confinement de champ et favorisent l'interaction du champ confiné avec des phonons confinés. Les résultats mis en évidence dans ce document devraient permettre de mieux comprendre le confinement des phonons et des photons et de concevoir rationnellement des métasurfaces pour les dispositifs optiques à haut rendement. Par exemple, le mécanisme d'interaction non conservatif photon-phonon amplifié observé à un grand potentiel pour renforcer fortement l'effet photovoltaïque dans le silicium.Dans ce travail, nous présentons la preuve expérimentale du changement du spectre de phonon et des propriétés vibrationnel d'un matériau massif par des mécanismes d'hybridation de phonons. Le spectre de photon dans un matériau fini est fortement affecté par la présence de surfaces libres, qui est le complément d'une contribution d'un cristal essentiellement bidimensionnel. Le spectre de phonon d'un matériau massif peut par conséquent être modifié par un mécanisme d'hybridation entre les modes des phonons enfermés dans des nanostructures présentes à la surface d'un matériau massif et les modes phononiques du matériau sous-jacent. Nous avons mesuré les chaleurs spécifiques de substrats nus et structurés de surface de silicium provenant d'un même substrat de silicium de référence. Nous avons alors déduit les principales caractéristiques des spectres de phonons des échantillons par une analyse rigoureuse des courbes de chaleurs spécifiques mesurées. Les résultats ont clairement montré que la forme et la taille des nanostructures en surface du substrat massif ont un fort effet sur le spectre de phonon du matériau massif.

	II.	Modification

  P pendant la partie chauffante de la mesure, et égale à zéro pendant le refroidissement. La solution de cette équation est donnée par des fonctions exponentielles avec un temps caractéristique constant  égale à

			1)
	Où total C est la chaleur specifique totale de la plate-forme et l'échantillon, w  la
	conductance thermique des fils, b T la température du bain thermique et   t P	la
	puissance appliquée par l'appareil de chauffage. La puissance d'appareil de
	chauffage   t P est égale à 0
	total C	total  , où total  est la conductivité thermique de l'échantillon et la plate-forme.
	Le système utilise généralement Eq.1 pour mesurer la plupart des échantillons aussi
	bien que le porte d'échantillon (le détenteur et la colle thermique). Cependant,
	quand le contact thermique entre la plate-forme et l'échantillon est faible, le
	système simule automatiquement l'effet de chaleur s'écoulant entre la plate-forme
	type et l'échantillon en utilisant un modèle à deux constantes de temps pour
	représenter la réponse en température. Dans ce cas, la température de la plate-forme
	p T et la température de l'échantillon s T obéit aux équations :

ZnO de surface libre et conductance thermique de l'interface Si-nanofils

  Nous présentons ici des mesures calorimétriques de la chaleur spécifique de nanofils d'oxyde de zinc de surface libre (ZnO NWs) et des mesures de la conductance thermique de l'interface entre ces ZnO NWs alignés verticalement et le substrat de silicium (Si). Des mesures calorimétriques à impulsion thermique ont été effectuées sur des ZnO NWs de surface libre alignées verticalement sur un substrat en Si. L'ensemble de la réponse en température du calorimètre à impulsion thermique est rendue compte par un modèle qui prend en compte l'effet de la conductance de la liaison thermique des ZnO NWs sur le substrat Si. La méthode des moindres carrés linéaires a été utilisée pour déterminer la chaleur spécifique des ZnO NWs mesurés et la conductance thermique de l'interface entre le substrat Si et ZnO NWs de 1,8 à 300 K. On trouve qu'à basse température (inférieure à 4 K) la chaleur spécifique des ZnO NW présente une contribution évidente d'un cristal essentiellement bidimensionnel. Cependant, au-dessus de 25 K, la chaleur spécifique des ZnO NWs commence à montrer une amélioration par rapport au ZnO en massif, et le facteur d'amélioration augmente à mesure que la température augmente et que le diamètre des NWs diminue. La conductance thermique de l'interface entre le substrat de Si et ZnO NWs est inférieure d'un ordre de grandeur à celle entre le substrat de ZnO et de Si, suggérant la formation d'une couche mince de faible cristallinité entre ZnO NWs et Si. De plus, les données enregistrées ont démontré une transition de la transmission élastique spéculaire à une transmission élastique diffusive, puis d'une transmission élastique à une diffusion inélastique diffusive lorsque la température augmente.La synthèse de ZnO NW alignés verticalement sur substrat de Si a été réalisée en deux étapes. Une couche de germes de ZnO de 15 à 20 nm d'épaisseur a d'abord été synthétisée, avant croissance des ZnO NWs proprement dits. La couche de germination de ZnO est constituée de nanoparticules de ZnO. Il agit d'une couche de nucléation pour la croissance de ZnO NWs. Elle a été préparée par dissolution de 2.197 g d'acétate de zinc (Zn(CH 3 COO 2 ) 2 ,2H 2 O) dans 20 ml d'éthanol en agitant constamment pendant 24 heures. Ensuite, la solution obtenue a été appliquée par centrifugation sur un substrat de Si propre. Après le revêtement par centrifugation, la couche de germe a été recuite à 400 ° C sur une plaque chauffante pendant 10 minutes. Le revêtement par centrifugation a été répété deux fois et l'échantillon a été recuit après chaque étape de revêtement par centrifugation. La technique de dépôt chimique en bain (CBD) a été utilisée pour faire pousser les ZnO NW. Une quantité de 0,025 M d'acétate de zinc a été dissoute dans 250 ml d'eau désionisée. Ensuite, la solution a été agitée pendant quelques minutes et 0,3 ml d'hydroxyde d'ammonium (concentration de 28%) a été ajouté à la solution sous agitation constante. La solution obtenue a été chauffée à 87 °C dans un ballon à trois cols. Ensuite, le substrat de Si avec la couche de germes de ZnO a été introduit dans la solution pour la croissance des ZnO. La concentration d'acétate de zinc et le temps d'immersion ont été variés pour obtenir des ZnO NW de différents diamètres et longueurs.

	Trois échantillons ont été sélectionnés pour effectuer les mesures colorimétriques.
	Ces échantillons seront désignés dans ce qui suit comme WP1, WP2 et WP3. Les
	diagrammes de diffraction des rayons X provenant des échantillons étudiés sont
	données sur la figure 15.
	1. EXPERIENCES
	FIG.15 Diagrammes de diffraction des rayons X des échantillons étudiés.
	La figure 15 montre que les ZnO NW ont une croissance avec une orientation
	hautement préférentielle le long de la direction cristallographique [002]. Les
	micrographies à balayage d'électrons en vue de dessus des échantillons sélectionnés
	sont représentées sur la figure 16.

  WP2 et WP3 sont initialement (avant l'arrivée de l'impulsion de chauffage) à 1.80, 42.30 et 79.40 K sont représentés sur la figure 19 par des symboles. L'ajustement linéaire des moindres carrés de la mesure  

	Où 3 c et I  sont la chaleur spécifique de ZnO NWs et la conductance thermique
	de l'interface entre Si et ZnO NWs, respectivement, et   t T 3 est la température de
	ZnO NWs à l'instant t. Les valeurs de 3 c et I  dans la plage de température mesurée ont été obtenus en éliminant   t t T 3 dans l'Eq.25, en intégrant sur la T 2 et   t T 1 à la réponse thermique mesurée du calorimètre par la méthode des moindres carrés période de temps de mesure, puis en ajustant l'équation résultante pour   linéaires. La valeur de   t t T 1 t T 1 lorsque les échantillons WP1, t T 1 et les données sont représentées sur la même figure par des lignes continues. Le bon T 1 mesuré et la variation de        accord entre valeur mesurée et calculée   t T 1 et 1 T  démontrent la fiabilité des résultats déduits.

  avec la chaleur spécifique calculée de masse ZnO[START_REF] Grimvall | Lattice instabilities in metallic elements[END_REF]. Les incertitudes sur les données de ZnO NW sont dominées par la dispersion observée dans les points de données dans chaque étape de mesure, et l'erreur sur l'évaluation de la masse des ZnO NW dans chaque échantillon mesuré. L'erreur relative est estimée à 10%. Il convient de noter ici que la figure 20 montre des courbes de chaleur spécifiques à pression constante car les mesures de chaleur spécifique à volume constant (qui peut être décrite avec des modèles plus faciles et plus quantifiables) sont difficiles à obtenir expérimentalement. Cependant, on peut montrer qu'à température ambiante et en dessous, la différence entre la chaleur spécifique à pression constante et à volume constant est inférieure à l'erreur expérimentale[START_REF] Tari | The Specific Heat of Matter at Low Temperatures[END_REF]. Par conséquent, la modélisation des courbes de chaleur spécifiques à pression constante mesurées avec des modèles établis pour la chaleur spécifique à volume constant dans ce qui suit est justifiée. Les résultats de la figure20indiquent qu'audessus de 25 K, la chaleur spécifique de ZnO NW commence à montrer une amélioration remarquable par rapport au ZnO massif, et le facteur d'amélioration augmente dans toute la gamme de température mesurée lorsque le diamètre moyen des ZnO NWs diminue. Symboles remplis : Mesures calorimétriques de la chaleur spécifique de nanofils de ZnO de surface libre. Symboles ouverts : Chaleur spécifique de masse ZnO obtenue à partir des calculs

	NWs mesurées ont été traitées de la même manière pour déterminer si la courbe de
	chaleur spécifique de NWs suit un comportement thermique spécifique
	tridimensionnel à basses températures. Les T c des courbes correspondant à ZnO
	NWs et ZnO en masse sont portées sur la Fig.21. Ils montrent que le T c par
	rapport à 2 T des courbes correspondant aux ZnO NW ne sont pas des droites, ce
	qui indique que la chaleur spécifique du ZnO aux basses températures s'écarte du
	comportement tridimensionnel.
		200
			bulk
			WP1
			WP2
	-1 )	150	WP3
	.g
	-1
	Specific heat (J.K	50 100
		0
		0	50	100	150	200	250	300
				Temperature (K)
	Les résultats de la figure 21 montrent également que l'écart de la chaleur spécifique
	du NW par rapport au comportement tridimensionnel augmente à mesure que le
	des premiers principes. diamètre NW augmente. Afin de comprendre les phénomènes physiques
	responsables de cette déviation observée du comportement tridimensionnel, nous A des températures plus basses, l'expression de la chaleur spécifique d'un matériau bulk est de la forme 3 T T c  avons développé un modèle théorique basé sur la résolution de l'équation du    , où les premiers et seconds termes sur le côté mouvement en coordonnées cylindriques dans l'approximation du milieu continu
	droit représentent respectivement la contribution du gaz d'électrons et la avec des conditions aux limites appropriées. Le modèle démontre que la chaleur
	contribution de la dynamique harmonique des phonons à la chaleur spécifique spécifique du NO aux basses températures est de la forme :
	totale à basse température. La pratique courante pour séparer ces deux contributions est de diviser l'expression de c par T, de tracer T c par rapport à 2 T , et puis de 3 , 2 , 980 . 25 2 3 212 . 2 7 3

FIG.20

trouver  en extrapolant linéairement T c jusqu'à l'origine de l'axe horizontal et notant où il intercepte l'axe vertical. Les courbes de chaleur spécifiques de ZnO FIG.21 : Variation de C/T par rapport à T 2 les tracés correspondant au Si massif et aux nanofils de ZnO étudiés. Symboles remplis : Données obtenues à partir de mesures calorimétriques. Symboles ouverts : Données obtenues à partir des calculs des premiers principes. Ligne continue : Ajustement des données en utilisant le modèle décrit dans l'annexe 3.

Conductivité thermique entre le substrat de Si et les nanofils de ZnO Les

  courbes de la conductance thermique de l'interface entre Si et ZnO NWs (Si / ZnO NWs) obtenues à partir des mesures calorimétriques sont reportées sur la figure22. La conductance thermique de l'interface entre Si et ZnO massif (Si / ZnO) obtenue à partir des calculs de premiers principes dans le « diffuse mismatch model »[START_REF] Reddy | Diffuse mismatch model of thermal boundary conductance using e act phonon dispersion[END_REF] est également reportée sur la figure 22 à des fins de comparaison. Les résultats montrent que la conductance thermique de Si / ZnO NWs est indépendante de la taille moyenne des ZnO NWs. En outre, la figure 22 montre une différence notable dans la conductance thermique de Si / ZnO NWs et Si / ZnO, à la fois en termes de magnitude et de dépendance à la température. La conductance thermique de Si / ZnONWs est inférieure d'environ cinq ordres de grandeur à celle de Si / ZnO. La faible valeur de conductance thermique de Si / ZnO NW comparée à celle de Si / ZnO peut être attribuée à la formation d'une couche mince de faible cristallinité à l'interface entre ZnO NWs et Si conduisant à une diffusion élevée des phonons[START_REF] Srivastava | The physics of phonons[END_REF]. Cette conclusion est soutenue par une étude récente en microscopie électronique en transmission[START_REF] Anufriev | Heat conduction engineering in pillar-based phononic crystals[END_REF], montrant que la conductivité thermique dans les métamatériaux phononiques est réduite principalement par la diffusion des phonons dans la région intermédiaire à l'interface entre les nanostructures et le substrat sous-jacent. Symboles remplis : mesures calorimétriques de la conductance thermique de l'interface entre nanofils Si et ZnO. Symboles ouverts : Conductance thermique de l'interface entre Si et masse ZnO obtenue à partir des calculs de premiers principes dans le modèle de mésappariement diffus.

	T . Les courbes sont données dans
	le tableau III. Le bon accord entre les valeurs mesurées et calculées T c par rapport
	à 2 T démontrent qu'en plus d'une contribution d'un cristal tridimensionnel, la
	chaleur spécifique de NW présente une contribution appréciable d'un cristal
	essentiellement bidimensionnel.

B.

FIG.

22

  are the zeros of the first kind Bessel's functions m J ,  is the azimuthal cylindrical coordinate, and n and n' are quantization integers. Upon substituting Eq.A24 in the equation of motion, we obtain the basic equation for determining the frequencies of the normal vibrational modes:

	,	,	.	.	cos	'	.	sin	exp
									A24
	Where m

z

FIG.4 Les simulations FDTD de la distribution du champ électrique dans les échantillons du second ensemble et les spectres Raman correspondant aux échantillons du second ensemble. Les lignes pointillées dans les simulations FDTD indiquent l'interface entre les résonateurs et le substrat Si. Les symboles indiquent les spectres Raman mesurés. Les lignes continues indiquent les spectres Raman calculés en utilisant le modèle de corrélation spatiale.

FIG.11Températures Debye dépendantes de la température des échantillons mesurés. Symboles : Valeurs obtenues en utilisant les mesures de la chaleur spécifique dans l'équation.11. Lignes pleines : correspond le mieux aux courbes de température de Debye dépendant de la température avec Eq.13 et Eq.14.Afin de tester la précision de l'approche décrite ci-dessus dans la détermination de l'extrémité basse fréquence du spectre de phonons à partir d'une mesure de chaleur spécifique, nous l'avons utilisée pour déterminer l'extrémité basse fréquence du spectre phonon du substrat de Si nu et comparer aux résultats calculés (first principle calculation) de la densité d'états des phonons du Si[START_REF] Ka Zan | First principles calculation of the thermal conductance of GaN/Si and GaN/Si interfaces as functions of the interface conditions[END_REF],[START_REF] Ka Zan | Interpolation Between the Acoustic ism atch ode l and the Diffuse Mismatch Model for the Interface Thermal Conductance: Application to InN/GaN Superlattice[END_REF]. Les courbes obtenues sont tracées ensemble sur la figure12(après normalisation appropriée).

Acknowledgements I would like to express my deepest gratitude to my doctoral research advisors, Pr. Gilles LERONDEL, Pr. Michel KAZAN and Pr. Malek TABBAL for their continuous guidance, supports and encouragements during the work of my research. Their research standards, intensity and dedication to science gave me freedom and opportunities to work on challenging research projects. They have given me proper exposure to academic and industrial research work in my field. It has been an invaluable experience working under their edification. I would like to thank my committee members, Pr. Sebastian VOLZ, Pr. Nadjib SEMMAR, Pr. Yannick De WILDE, Pr. Vladimir LYSENKO, Dr. Aurelien BRUYANT and Pr. Laurent DIVAY. Thank you for your time and constructive criticisms for this thesis.

I also thank Ms. Anisha GOKARNA and Ms. Agnieszka GWIAZDA of LNIO Laboratory of Nanotechnology, Instrumentation and Optics for synthesizing the sample for this project.

Sincere thanks to our lab engineers Mr. Sergei KOSTCHEEV and Mr. Regis DETURCHE for their technical support and training sessions. I would also like to thank all my colleagues and members of LNIO for providing a friendly working atmosphere.

I would like to thank the American university of Beirut and the National Council for Scientific Research (CNRS-Lebanon) for their financial support and assistance throughout.

I would like to Champagne-Ardennes Region and the European Regional Development Fund through the SYNAPS project and the Nano'mat platform for the clean room facilities and use of specific equipment.

this paper are expected to help to gain more insight into the confinement of phonons and photons and rationally design metasurfaces for high efficiency optical devices. For instance, the observed amplified photon-phonon non-conservative interaction mechanism has great potential for strongly enhancing the photovoltaic effect in Si. interpreted as the result of an anharmonic phonon processes that contribute to the transfer of thermal energy across the interface between Si and ZnO NWs. rystals," Phys. Rev., vol. 104, no. 1, pp. 37-42, Oct. 1956.

APPENDIX APPENDIX 1: SPATIAL CORRELATION MODEL USED IN THE CALCULATION OF THE PHONON CONFINEMENT IN THE SILICON RESONATORS

The wavefunction of a Raman active phonon mode of wavevector 0 q in an infinite crystal can be written in the form:

is the amplitude of the wavefunction having the crystal periodicity.

According to the spatial correlation model, it becomes:

is the phonon weighting function describing the strength of phonon confinement in the finite crystal (here we assume a finite spherical crystal). The function   r , ' 0 q  in Eq.A2 can be expanded in a Fourier series and written in the form:

Where the Fourier coefficients   q q , 0 C are given by:

The importance of Eq.A4 stems from the fact that it demonstrates that while the Raman spectrum from an infinite crystal is determined by a phonon wavefunction, a phonon wave packet formed by a superposition of eigenfunctions with wavevectors q centered at q 0 determines the Raman spectrum from a finite crystal.

In principle, the introduction of the phonon weighting function breaks the translational symmetry and imposes that the integral in Eq.A4 extends from   to   . However, if the size of the finite crystal under consideration is at least one order of magnitude greater than the lattice constant, the bulk phonon dispersion can be adopted within the spatial correlation length model, and the integration can be