Keywords: combination: optical development, 2, 1, 4 SMS and DAC combination: mechanical development, 2, 2, 2, 3 Perspectives: SPR linewidth and discontinuities

2 Optics of a single metal nano-object under high pressure 2.1 High-pressure microscopy setup . . . . . . . . .

Introduction

The physical properties of nanoparticles have been largely investigated over the last decades as a result of important developments in their synthesis, detection and modeling. It has been shown that nanosized materials, as compared to their bulk counterpart, exhibit new optical, electrical, magnetic, mechanical and thermal properties, due in particular to their high surface to volume ratio. From an optical point of view, noble metal nanoparticles show an extinction spectrum dominated by the surface plasmon resonance (SPR), responsible for the bright colors of nanoparticle colloids that are apparent in the stained-glass windows of medieval cathedrals. SPR is associated to the collective oscillation of electrons excited by the incident electromagnetic field, enhancing the electric field in and around the particle. It is influenced by the nanoparticle size, shape, composition and environment, whose control during fabrication has been improved during the last decades, increasingly allowing experimental and theoretical investigations on single nano-objects and their use in various applications such as sensing, energy harvesting and photothermal therapy. This thesis, organized in three chapters, aims to extend the knowledge and understanding of environmental effects on the optical and acoustic properties of single metal nanoparticles. The first chapter theoretically compares the physical properties of noble metals in the bulk state and at the nanoscale. The cases of spherical and elongated nanoparticles are discussed in details, allowing to connect SPR properties (spectral position, linewidth and area) with the nano-object features (size, shape, composition, environment). A final section deals with the spatial modulation spectroscopy (SMS) technique, consistently employed to optically detect single nanoparticles in far-field and quantitatively measure its extinction spectrum. The second chapter investigates experimentally and theoretically how the SPR of a single metal nanoparticle is affected by extreme conditions, i.e. by strongly modifying the pressure of the external environment. After the description of the diamond anvil cell (DAC) apparatus suitable for yielding a controlled high-pressure environment, its non-trivial combination with the SMS microscope is explained.

Introduction

The experimental results on the SPR properties evolution with increasing pressure are presented in details. In particular, the SPR spectral position redshift with pressure is successfully modeled both numerically, using the finite-element method (FEM), and analytically, under simplifying assumptions to provide further physical insights. The SPR spectral area dependence on pressure is explained through a simple model and the spectral linewidth change justified by considering the pressure dependence of its multiple contributions. The last chapter describes how the acoustic behavior of a single nanoparticle is affected by its morphology and interaction with the environment. After a theoretical introduction of the vibrational properties of single nano-objects, the principle of acoustic FEM simulations is explained. It follows the description of the investigated samples, composed of isolated gold nanodisks on a sapphire substrate, among which the most circular have been selected and investigated by SMS. The combination of the SMS microscope with a pump&probe (P&P) setup is introduced, allowing investigations of the ultrafast response of a gold nano-object. By focusing on the acoustic component of the P&P signal from single nanodisks, specific vibrational modes are characterized, with particular attention to the dependence of their damping rate on nanodisk morphology. FEM modeling results are compared to the data in order to unveil the origin of the observed phenomena, associated to a reduction of acoustic energy tranfer to the environment through the disk/substrate interface.

Chapter 1

Optical and electronic properties of metal nano-objects

Optical properties of metals are directly related to their electronic properties at both bulk and nanometric level. In this chapter, it is firstly presented the connection between electronic and optical properties for bulk metals with a focus on noble metals, widely used for plasmonics applications. Afterwards, the optical response of metallic nano-objects with simple geometries is described, with particular regard for their surface plasmon resonance (SPR). The last part deals with the optical detection of single nanoparticles via the spatial modulation spectroscopy (SMS) technique.

Electronic properties of bulk metals

Gold, silver and copper are noble metals. They are constituted by monovalent atoms with a resembling electronic configuration, comprehending the full electronic level (n -1)d and the level ns occupied by a single electron reported in Tab. (1.1). In bulk materials, these atoms are organized in a faced cubic centered METAL CONFIGURATION a ( Å) n e (×10 22 cm -3 )

Gold (Au)

[Xe]4f 14 5d 10 6s 1 4.07 5.90

Silver (Ag)

[Kr]4d 10 5s 1 4.08 5.86

Copper (Cu)

[Ar]3d 10 4s 1 3.61 8.47

Table 1.1: Electronic configuration, lattice periodicity a and electron density n e of noble metals [START_REF] Wyckoff | Crystal Structures[END_REF][START_REF] Ashcroft | Solid State Physics[END_REF].

Optical and electronic properties of metal nano-objects structure with lattice periodicity a, as illustrated in Fig. (1.1). Isolated atoms have quantized electronic levels, each one representing an orbital. In a solid, interactions between atoms induce a broadening of the energy levels, leading to a continuum of electronic states, called electronic bands.

The band structure of crystalline materials is theoretically described by the Bloch theorem (that allows to solve the Schrödinger equation of a free electron in a potential with periodicity defined through a). An ensemble of energy bands E p ( k) quantized by p emerges. Core electrons are weakly influenced by the presence of other atoms, while the electrons in the highest energy levels ( (n-1)d, ns and np ) are strongly affected. At T = 0 K, the energy bands are filled until the Fermi energy E F , corresponding to the energy of the electronic state with highest energy. Two energy bands are particularly important to understand the materials physical properties: the valence band, the highest energy band completely filled with electrons, and the conduction band, the lowest energy band partially filled. A sketch of the band structure of a metal is presented in Fig. (1.2). Each atom contributes to the conduction band with an electron (ns 1 ), and thus, the conduction electrons density is defined as

n e = N A ρ MM , ( 1.1)
where N A is the Avogadro number, ρ is the density and MM the molar mass.

The electrons of the conduction band may be considered as free electrons with effective mass m * , according to the dispersion relationship

E( k) = k 2 2m * . (1.
2)

The density of states g(E) of conduction electrons for a tridimensional system of volume V is therefore given by Thus, the Fermi energy is defined by the free electron density n e via

g(E) = V 2π 2 2m *
E F = 2 2m * (3πn e ) 2 3
(1.4) from which two constants can be deduced: the Fermi temperature T F = E F /k B (with k B the Boltzmann constant) and the Fermi velocity v F = 2E F m * . The main quantities of the free electrons model are summarized in Tab. (1.2). It follows that the volumetric heat capacity of free electrons is defined (under the hypothesis T e T F ) as electron mass m e = 9.11 × 10 -31 kg), Fermi energy (E F ), Fermi temperature (T F ) and Fermi velocity (v F ) for gold and silver [START_REF] Ashcroft | Solid State Physics[END_REF].

c e (T e ) = 1 V +∞ 0 E g(E) ∂f (E) ∂T e dE = π 2 n e k B 2T F T e = a e T e , (1.5)
METAL m * E F (eV) T F (×10 4 K) v F (×10
where f is the Fermi-Dirac distribution, describing the occupation probability of an energy state with energy E:

f (E) = 1 e (E-E F )/(k B Te) + 1 (1.6) (where the chemical potential has been posed equal to E F , accordingly with the condition T e T F ). In the gold and silver case, the proportionality constant results a e = π 2 ne 2T F k B ≈ 65 J m -3 K -2 .

Optical properties of bulk metals

In metals, optical properties are governed by the response of electrons to external electromagnetic fields.

Electromagnetism in a metal

The interaction between an electromagnetic field E(ω) of angular frequency ω and a material is described by the dielectric function (ω) = 1 (ω)+i 2 (ω) (being 1 (ω) = Re{ } and 2 (ω) = Im{ }) or using the complex refractive index ñ = n+iκ (with ñ2 = , n = Re{ñ} and κ = Im{ñ}). They link the electronic induction D to the electric field

D = 0 E = 0 (1 + χ) E (1.7)
where 0 is the vacuum permittivity and χ the electric susceptibility. Fig. (1.3) shows the dependence of 1,2 as a function of photon energy for gold and silver.

Optical properties of bulk metals 7

Supposing isotropy ( scalar) and a monochromatic electric field E( r, t) = E 0 e -iωt + c.c., the Helmholtz equation applies

∇ 2 E 0 + ω 2 c 2 (ω) E 0 = 0 . (1.8)
Considering the propagation along the axis z, the electric field becomes E( r, t) = E 0 ( r)e iωt( ñ c z-t) . The field intensity (I ∼ | E| 2 ) has the form

I(z) = I 0 e -αz , (1.9)
where α(ω) = 2 ω κ(ω)/c is the absorption coefficient. In metals, the dielectric function can be separated in two terms = D + ib , each one associated to different electron-photon mechanisms. Photons may be absorbed by conduction electrons leading to intraband transitions (internal to the conduction band), or by valence electrons yielding interband transitions (between valence and conduction band).

Classically, intraband transitions are described through the Drude model, based on the quantities D and χ D , while interband transitions are governed by ib and χ ib , related to by

= ib + D = 1 + χ ib + χ D , ( 1.10)
having defined ib = 1+χ ib and D = χ D .

Intraband transitions: the Drude model

The Drude model [START_REF] Drude | Zur Elektronentheorie der Metalle[END_REF] allows to calculate the dielectric function associated to intraband transitions D . It considers the quasi-free electrons of metals as free independent electrons interacting with their surrounding by scattering processes, described through the viscous force F = -m e γ v, with γ the scattering rate. The dynamics of an electron undergoing an electric field E(t) = E 0 e -iωt + c.c., is thus described by m e d 2 r dt 2 = -m e γ d r dt e E , (1.11) where r is the position vector. The Drude component of dielectric function D is defined through the polarization vector P NP = -n e e r = 0 χ D E, where n e is the conduction electrons density. It follows

(ω) = ib (ω) + D (ω) = 1 + χ ib (ω) + χ D (ω) = ib (ω) - ω 2 p ω(ω + iγ) , ( 1.12)
where ω p = nee 2 me 0 is the plasma frequency, resulting ω p ≈ 9 eV in gold and silver. The real D 1 and imaginary part D 2 of the Drude contribution result as

D 1 = - ω 2 p ω 2 +γ 2 ≈ - ω 2 p ω 2 D 2 = ω 2 p γ ω(ω 2 +γ 2 ) ≈ ω 2 p γ ω 3
for γ ω .

(1.13)

Optical and electronic properties of metal nano-objects

In noble metals, this approximation is valid in the optical frequency domain (ω/γ ∼ 10).

In the case ω = 0, γ is the scattering rate between electrons and lattice (electronphonon scattering) or other electrons (electron-electron scattering), and thus is restricted to two-bodies interaction processes. When ω = 0, γ takes into account the absorption of a photon with frequency ω by a conduction electron. As the momentum of a photon is negligible, this intraband absorption requires the presence of a third body (electron or phonon) in order to satisfy the energy and momentum conservation (see Fig.

Interband transitions

In noble metals, interband transitions occur between core electrons of d-bands and the empty states in the conduction band (Fig. (1.2) ). The energy threshold ω ib for this process is ≈ 2 eV in gold and ≈ 4 eV in silver [START_REF] Rangel | Band structure of gold from many-body perturbation theory[END_REF][START_REF] Sattler | Handbook of Nanophysics: Nanoelectronics and Nanophotonics[END_REF]. The energy threshold ω ib correspond to the step increase in 2 (Fig. (1.4b) ).

1.3 Optical properties of metal nano-objects 9

Optical properties of metal nano-objects

Metallic nanoparticles are small systems consisting of few tens to thousands or millions of atoms, whose optical response is dominated by the presence of an extinction resonance induced by the dielectric confinement, called surface plasmon resonance (SPR). Its properties (spectral position, linewidth and area) depend on the nanoparticle size, shape and environment, and are briefly discussed in this section.

Energetic considerations

When an electromagnetic wave E 0 (t) = E 0 e -iωt shines a nanoparticle, as illustrated in Fig. (1.5), the nanoparticle electrons oscillates along the light polarization at the frequency ω. This electronic periodic movement induces an electric field inside the nanoparticule, E int (t), and a scattered one, E sca (t), propagating through the external matrix (from the nanoparticles towards every direction). Absorbed electromagnetic energy can also be transformed in thermal energy via Joule effect. In an homogeneous environment, the power absorbed by the nanoparticle, W abs , W abs = -S P n dS , (1.14) is determined by the flux through the nanoparticule surface S (with n the unitary vector normal to the surface and directed towards the exterior) of the Poynting vector P , defined as a function of the total external electric field E m (t) = E 0 (t)+ E sca (t) and total external magnetic field H m (t) = H 0 (t)+ H sca (t)

P = 1 2 Re{ E m × H * m } . (1.15)

Optical and electronic properties of metal nano-objects

Considering a transparent environment, the development of the electric and magnetic fields yields [START_REF] Bohren | Extinction by a spherical particle in an absorbing medium[END_REF][START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF][START_REF] Lebedev | Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films[END_REF] W abs + W sca = W ext , (1.16) where

⎧ ⎨ ⎩ W sca = 1 2 Re S [ E sca × H * sca ] • n dS W ext = -1 2 Re S [ E 0 × H * sca + E sca × H * 0 ] • n dS .
(1.17)

The incident power loss (extinction), W ext , in the propagation direction is caused by the sum of the absorbed power W abs and scattered power W sca by the nanoparticle.

Convenient definitions are the extinction, absorption and scattering cross-sections of the nanoparticle. They are defined as the ratio of the powers over the average intensity I 0 (I 0 = n m c 0 E 0 2 /2, with n m the refractive index of the medium and c the speed of light) of the incident beam on the nano-object [START_REF] Lebedev | Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films[END_REF].

⎧ ⎪ ⎨ ⎪ ⎩ σ ext = Wext I 0 σ abs = W abs I 0 σ sca = Wsca I 0 , ( 1.18)
and therefore σ ext = σ abs + σ sca (1.19) Analytic computation of the cross-sections are possible in simple cases (spheroids in homogeneous matrix), while for more complicated geometries a numerical approach is required.

Optical response of a nanosphere in the quasi-static approximation

Consider the configuration illustrated in Fig. (1.6), where a sphere of diameter D and dielectric function (ω) = 1 + 2 is embedded in a transparent medium of dielectric function m and shined by the incident plane wave E i = E 0 e i( k • r-ωt) with k = n m ω/c. As shown in Fig. (1.7), if the sphere diameter is of the order of the incident wavelength (D ∼ λ), electric field is not uniform within the particle, creating an inhomogeneous charge distribution, inducing multipolar electron oscillations. On the other hand, if the particle is very small as compared to the optical wavelength (D λ), the field is uniform inside the particle and the charge distribution approaches that of an oscillating dipole (dipolar approximation). The latter case corresponds to the electrostatic problem of the polarization of a sphere in a uniform field, which can be solved by Poisson equation with proper boundary conditions. The incident field induces a polarization p s within the sphere and a polarization p 0 in the surrounding medium; applying field continuity conditions at the interface, the internal field E int is shown to be proportional to the incident field through a dielectric confinement factor [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]

E int = 3 m + 2 m E 0 , ( 1.20)
The polarizability α of the nanoparticle can be deduced from P NP = α 0 m E 0 [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]

α = 3V -m -2 m , ( 1.21)
where V = 1 6 πD 3 is the volume of the sphere. As a first approximation, the particle response to an incident field can thus be identified with the response of a dipole placed in its center. Analytical expressions for σ ext and σ sca can then be obtained as a function of the absolute value of polarizability and its imaginary part [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF][START_REF] Sattler | Handbook of Nanophysics: Nanoelectronics and Nanophotonics[END_REF]

σ ext (ω) = k Im{α} = 9V ω 3/2 m c 2 (ω) ( 1 (ω) + 2 m ) 2 + 2 2 (ω)
, (1.22)

σ sca (ω) = k 4 6π |α| 2 = 3V 2 ω 4 2 m 2πc 4 ( 1 (ω) -m ) 2 + 2 2 (ω) ( 1 (ω) + 2 m ) 2 + 2 2 (ω)
.

(1.23)

The scattering cross section σ sca varies with λ as 1/λ 4 (Rayleigh scattering) (λ = 2πc/ω being the wavelength in vacuum) and the ratio between scattering and extinction cross sections is proportional to (D/λ) 3 . In the approximation D λ, extinction is therefore dominated by absorption σ ext ≈ σ abs σ sca .

(1.24)

The extinction cross section is enhanced at the frequency ω = ω R minimizing the denominator in Eq. (1.22). Supposing a relatively small and weakly dispersed imaginary part 2 of the metal dielectric constant in this optical region, the resonant condition can be simplified as

1 (ω R ) + 2 m = 0 . ( 1.25)
The occurrence of this condition is associated with the appearance of a resonance in the extinction optical spectrum of the metal nanoparticle: the surface plasmon resonance (SPR). Accordingly to Eq. (1.20), the SPR corresponds to an enhancement of the optical response of the confined nanosphere resulting from the exaltation of the internal field E int as compared to the incident field E 0 , phenomenon customarily called dielectric confinement effect. The SPR can be classically associated to a collective oscillation of the metal electrons relative to the lattice, resonantly driven by the external electromagnetic field. In the frame of the quasi-static approximation, the resonant frequency ω R is given by the expression

ω R = ω p ib 1 (ω R ) + 2 m , ( 1.26)
obtained by injecting Eq. (1.12) and (1.13) in Eq. (1.25). When the SPR lies far from the interband transitions threshold, as in the simple case of a silver sphere, it can be spectrally described by a quasi-Lorentzian profile centered around ω R .

The extinction spectrum of a silver nanosphere embedded in water ( m = 1.33

Optical response of a nanosphere beyond the quasistatic approximation: Mie theory

For large nanosphere sizes (D ≥ λ), the dipolar approximation is not justified, since the spatial heterogeneity of the field within the particle and the consequent appearance of delay effects cannot be disregarded [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. Exact calculation of the optical response of a sphere has been developed by G. Mie [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF][START_REF] Horvath | Gustav Mie and the scattering and absorption of light by particles: Historic developments and basics[END_REF]. In the case of a non-absorbing medium (real m ), the cross-sections of a sphere of radius R write as

σ ext = 2πR 2 u 2 ∞ n=1 (2n + 1) Re{a n + b n } , ( 1.27
)

σ sca = 2πR 2 u 2 ∞ n=1 (2n + 1) |a n | 2 + |b n | 2 , σ abs = σ ext -σ sca , (1.28)
with The optical response of a nano-object is also strongly influenced by its shape [START_REF] Grzelczak | Colloidal nanoplasmonics: from building blocks to sensing devices[END_REF]. In most cases, no analytical solution is achievable, exception made for high symmetric configurations, such as core-shell spheres, ellipsoids and infinite cylinders. For such an elongated particle, absorption cross-section strongly depends on the incoming light polarization. In the frame of the quasi-static approximation, for a light polarization along the principal axes i = a, b, c, the extinction-cross section has been expressed by R. Gans as [START_REF] Bohren | Extinction by a spherical particle in an absorbing medium[END_REF][START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF][START_REF] Gans | Über die Form ultramikroskopischer Goldteilchen[END_REF][START_REF] Gans | Über die Form ultramikroskopischer Silberteilchen[END_REF]]

a n = mψ n (mu)ψ n (u) -ψ n (u)ψ n (mu) mψ n (mu)ξ n (u) -ξ n (u)ψ n (mu) , b n = ψ n (mu)ψ (u) -mψ n (u)ψ n (mu) ψ n (mu)ξ n (u) -mξ n (u)ψ n (mu) , ( 1
σ i ext = V (H i [η]+1) 2 ω 3/2 m c 2 (ω) ( 1 (ω) + H i [η] m ) 2 + 2 2 (ω) , ( 1.30)
where V = 4/3πabc is the volume of the ellipsoid, while H i [η] is a geometrical factor dependent on the aspect ratio η = a/b accordingly to

H i = 1 -L i L i (i = a, b, c) , (1.31) L a = 1 -e 2 e 2 1 2e ln 1 + e 1 -e -1 , L b = L c = 1 -L a 2 , e = 1 -η 2 (1.32)
Except for the additional factor H i , this relation is similar to the one obtained for a sphere (Eq. (1.22) ) and the resonance condition may be similarly determined

1 (ω R [i]) + H i m = 0 , (1.33) if 2 is weakly dispersed around ω R [i].
Considering Eq. (1.12) and (1.13)

ω R [i] = ω p ib 1 (ω R [i]) + H i m (1.34)
σ ext (ω) = V (H+1) 2 ω 3/2 m c 2 (ω) ∂ 1 ∂ω ω R (ω-ω R ) 2 + 2 2 (ω)
, (1.35) where H is the geometrical factor associated to the long axis (H = H a ). In experiments described in this thesis, the longitudinal SPR position is always located far from interband transitions, so that ∂ 1 /∂ω| ω R ∼ 2ω 2 p /ω 3 R using the Drude dielectric function of Eq. (1.13). This leads to a Lorentzian extinction profile of the type

σ ext (ω) = A π Γ 2 ( ω-ω R ) 2 + Γ 2 2 . (1.36)
Such a profile is fully characterized by three independent parameters: peak position ω R , integrated area A and spectral linewidth Γ. The energy peak position

ω R = ω p ib 1 (ω R ) + H m (1.37)
depends on nano-object shape, composition and local environment (Eq. (1.34)

with ω R [i] = ω R [a]
). The spectral area

A = π(1+H) 2 3/2 m V ω 4 R 2 c ω 2 p (1.38)
is proportional to the particle total number of electrons through the volume V and also depending on its shape, environment and SPR peak position. The SPR linewidth

Γ = γ + ω 3 R ω 2 p ib 2 ≈ γ (1.39)
is the electronic damping rate in Drude model (see Eq. (1.13) ) extended to the confined nano-object case. The discussion here does not include the radiative (Γ rad ), chemical (Γ chem ) and electron-surface scattering (γ s ) contributions to Γ: they will be introduced in Section (2.2.3) of Chapter (2) in order to discuss about the SPR broadening of gold nanoparticles with bipyramidal shape induced by pressure.

The previous calculations demonstrate that ω R and A are affected by physical parameters different from those influencing Γ. In particular, plasmon damping rate controls Γ but does not affect ω R and A, which is of key importance as Γ can be significantly altered by surface effects. These parameters can be extracted from the experimental data by fitting the SPR line, as shown in is done in the following chapters.

The optical response of elongated nano-objects strongly depends on the polarization of the incident light. This dependence, which is partially lost by orientational averaging in ensemble measurements, can be highlighted in single particle experiments. The extinction cross-section of a nanoparticle illuminated by a linearly polarized plane wave can be generally written as [START_REF] Muskens | Quantitative absorption spectroscopy of a single gold nanorod[END_REF]

σ ext (θ, ω) = σ (ω) cos 2 (θ -θ 0 ) + σ ⊥ (ω) sin 2 (θ -θ 0 ) (1.40)
where θ is the incident light polarization angle and θ 0 the particle long axis angle, with respect to a reference laboratory frame. The polarization response measured in an elongated bipyramid is well reproduced by Eq. (1.40), showing a very high contrast ratio σ ⊥ /σ ∼ 0 between the maximum amplitude corresponding to polarization along the nanoparticle long axis, and the minimum for a perpendicular

Optical and electronic properties of metal nano-objects polarization (Fig. (1.13b) ). These measurements provide the absolute orientation of the nano-object on the substrate.

In the general case of an arbitrary shape, such as penta-twinned bipyramids (which will be studied in Chapter 2) or finite cylinder (analyzed in Chapter 3) the optical response can only be computed numerically by different methods, as discrete dipole approximation (DDA), or finite element methods (FEM). In this thesis work, FEM has been consistently used to interpret experimental data and their operating principles will be explained later, in Section (2.2.1) of Chapter 2) (optical domain simulations) and in Section (3.1.4) of Chapter 3 (acoustic domain simulations).

Optical detection of single nano-objects: spatial modulation spectroscopy (SMS)

Nanoparticles have been largely investigated over the last decades because of their distinctive properties and potential use in technological (nonlinear photonic devices [START_REF] Kauranen | Nonlinear plasmonics[END_REF], data storage [START_REF] Zijlstra | Five-dimensional optical recording mediated by surface plasmons in gold nanorods[END_REF], energy harvesting [START_REF] Atwater | Plasmonics for improved photovoltaic devices[END_REF]) and biomedical applications (photothermal therapy [START_REF] El-Sayed | Surface Plasmon Resonance Scattering and Absorption of anti-EGFR Antibody Conjugated Gold Nanoparticles in Cancer Diagnostics: Applications in Oral Cancer[END_REF], biological imaging [START_REF] Dahan | Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking[END_REF] and sensing [START_REF] Willets | Localized surface plasmon resonance spectroscopy and sensing[END_REF]). In this context, optical techniques are powerful tools, yielding information on the intrinsic features of a nanoparticle (size, shape, and composition) and on its local environment. As these parameters can strongly fluctuate from object to object, conventional ensemble experiments prevent obtaining quantitative information. In this perspective, the development of optical techniques for detecting and characterizing single nanoobjects has greatly increased the knowledge in this field over the past few years.

In the first part of this section, the principal methods for the detection of single nano-objects are presented. The following part will focus on the description of the spatial modulation spectroscopy (SMS) technique exploited in this thesis.

Overview of single-particle characterization techniques

In these last years, several experimental techniques [START_REF] Ringe | Single nanoparticle plasmonics[END_REF][START_REF] Crut | Optical absorption and scattering spectroscopies of single nano-objects[END_REF] have been developed with the aim of investigating optical properties of single metallic nano-objects. These can be classified into two categories: far-field (where sources and detectors are far from the nanoparticle with respect to the wavelength) and near-field methods (based on illumination or detection typically closer than one wavelength from the nanoparticle). A summary of the relevant techniques, with relative advantages and disadvantages, can be found in Tab. (1.3). Historically, scanning near-field optical microscopy (SNOM) was one of the first methods to be introduced [START_REF] Synge | A suggested method for extending microscopic resolution into the ultra-microscopic region[END_REF]. Based on a nanometric tip maintained at a few nanometers from the sample, the technique provides a high spatial resolution (∼ 10 nm, depending on the tip [START_REF] Oshikane | Observation of nanostructure by scanning near-field optical microscope with small sphere probe[END_REF]), however the signal strongly depends on the tip-nanoparticle interaction and it interpretation requires a complex modeling of the near-field around the tip [START_REF] Liu | Modeling illumination-mode near-field optical microscopy of Au nanoparticles[END_REF]. Far-field techniques are characterized by a limited spatial resolution, being intrinsically diffraction-limited, but are free from the spurious influence of the observation apparatus. The most commonly used far-field techniques are dark-field microscopy, photothermal imaging (PHI) and spatial modulation spectroscopy (SMS). Dark-field microscopy is based on the selective collection of the scattered light from an illuminated nano-object [START_REF] Hu | Dark-field microscopy studies of single metal nanoparticles: understanding the factors that influence the linewidth of the localized surface plasmon resonance[END_REF][START_REF] Sönnichsen | Drastic reduction of plasmon damping in gold nanorods[END_REF][START_REF] Sönnichsen | Spectroscopy of single metallic nanoparticles using total internal reflection microscopy[END_REF]. Simple to apply, it does not provide a quantitative measurement of σ sca (since only a fraction of the scattered light is detected) and is limited to relatively large nanoparticle sizes (typically > 40 nm, being σ sca /σ ext ∼ V accordingly to Eq. (1.22), (1.23) and (1.25). PHI is based on the local heating of the nano-object environment through the absorption of a modulated "pump" laser beam that modifies its refractive index and on the propagation of a "probe" beam [START_REF] Berciaud | Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals[END_REF][START_REF] Boyer | Photothermal imaging of nanometer-sized metal particles among scatterers[END_REF][START_REF] Berciaud | Photothermal heterodyne imaging of individual metallic nanoparticles: theory versus experiment[END_REF]. In absorbing environment this high sensitive technique allows the detection of small nanoparticle sizes (∼ 1 nm), although a quantitative determination of σ abs remains difficult.

Optical and electronic properties of metal nano-objects

Conversely, SMS permits a quantitative measurement of σ ext in nano-objects sizes larger than ∼ 5 nm (Au) or ∼ 2 nm (Ag) [START_REF] Arbouet | Direct measurement of the single-metalcluster optical absorption[END_REF] deposited on transparent or absorbing substrates. This technique, employed in this thesis, will be explained in detail in the next section.

Principle of SMS

When a laser beam is focused on a nano-object, a part of the incident power W 0 is lost through absorption and scattering (see Section (1.3.1) ). Consider a punctiform particle of cross-section σ ext , expressed by Eq. (1.22), located in (x 0 , y 0 ) and shined by a beam with full-width-at-half-maximum FWHM and intensity profile I(x, y) in the nanoparticle plane, as illustrated in Fig. (1.14). The transmitted power W t writes as

W t = W 0 -σ ext I(x 0 , y 0 ) , (1.41) meaning the incident beam attenuation ΔW = W 0 -W t is ΔW = σ ext I(x 0 , y 0 ) . (1.42)

Optical detection of single nano-objects: spatial modulation spectroscopy (SMS) 23

For small nanoparticle sizes direct measure of ΔW is very challenging. For instance, in a sphere of diameter 20 nm illuminated by a diffraction-limited Gaussian beam,

ΔW W 0 ∼ 10 -3 , (1.43)
the signal being dominated by the power fluctuations of the laser source.

To overcome this limitation, a general solution consists in purposely introducing a periodic modulation of ΔW through the modulation of the laser wavelength [START_REF] Moerner | Optical detection and spectroscopy of single molecules in a solid[END_REF], the light polarization [START_REF] Carey | Imaging and absolute extinction cross-section measurements of nanorods and nanowires through polarization modulation microscopy[END_REF], or the illuminated nano-object position [START_REF] Arbouet | Direct measurement of the single-metalcluster optical absorption[END_REF][START_REF] Husnik | Quantitative experimental determination of scattering and absorption crosssection spectra of individual optical metallic nanoantennas[END_REF], and detecting the oscillating component of ΔW at the same frequency. The latter approach has been developed by the team. Typically, the sample is mounted on a piezoelectric shaker which modulates the nanoparticle y position according to y(t) = y 0 +δ sin(2πf t), where δ is the oscillation amplitude and f the frequency. Thus, the power loss becomes time-dependent

ΔW(t) = σ ext I(x 0 , y 0 +δ sin(2πf t)) . (1.44)
Assuming the oscillation amplitude is much smaller than the beam size, δ FWHM, Eq. (1.44) can be developed

ΔW(t) = ΔW 0 + ΔW | f sin(2πf t) + ΔW | 2f cos(4πf t) + o(δ 2 ) , (1.45)
where

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ΔW 0 = σ ext I(x 0 , y 0 ) + δ 2 4 ΔW | f = σ ext δ ∂I ∂y x 0 ,y 0 ΔW | 2f = - 1 4 σ ext δ 2 ∂ 2 I ∂y 2 x 0 ,y 0 . (1.46)
Therefore, the power modulation ΔW(t) contains constant and sinusoidal terms, all proportional to σ ext . Furthermore, the components oscillating at f and 2f are respectively proportional to the first and second derivative of the beam profile I(x, y) with respect to the coordinate y along the vibrational axis and evaluated in (x 0 , y 0 ).

Practically, power modulations ΔW | f,2f are detected as a voltage ΔV | f,2f using a photodiode followed by a lock-in amplifier (LIA) to measure relative power changes. The detection of the LIA allows to select the component of ΔW(t) at a specific frequency (f, 2f ), i.e. filtering out all the other frequencies and increasing the signal/noise ratio. The same setup is used to measure, for normalization, the power Optical and electronic properties of metal nano-objects W 0 of the incident beam when modulated as a square wave at the same frequency (f, 2f ) by a mechanical chopper, producing V 0 | f,2f (alternatively a digital multimeter can be used to continuously monitor the incident power). The SMS signal S f,2f is thus

S f,2f = ΔV | f, 2f V 0 | f,2f = π 2 ΔW | f, 2f W 0 with W 0 = R 2
I(x, y) dxdy , (1.47) where the factor π/2 takes into account the temporal shape of the normalization signal (square wave). This allows to achieve a high sensitivity (min{S f,2f } ∼ 10 -5 ), permitting the detection of nano-objects with cross-section of the order of a few nm 2 [START_REF] Arbouet | Direct measurement of the single-metalcluster optical absorption[END_REF].

In the general case, beyond the condition δ FWHM, Eq. (1.44) can be developed using the Fourier theory

S f (x 0 , y 0 ) = π 2 ΔW | f W 0 (x 0 , y 0 ) = πσ ext W 0 1 T T /2 -T /2 I(x 0 , y 0 +δ y sin(2πf t)) sin(2πf t) dt , (1.48)
S 2f (x 0 , y 0 ) = π 2 ΔW | 2f W 0 (x 0 , y 0 ) = πσ ext W 0 1 T T /2 -T /2
I(x 0 , y 0 +δ y sin(2πf t)) cos(2π(2ft)) dt .

(1.49) with FWHM= √ 2 ln2w = 0.7λ (typical experimental condition in the wavelength range 500-1100 nm). The nanoparticle position is moved through the windows 1×1 µm 2 , being spatially modulated with δ = 325 nm and f = 1500 Hz (typical experimental parameters). The extinction cross-section has been arbitrary fixed at σ ext = 10 3 nm 2 . In Fig. (1.15d,e) it should be noticed that along the vibrational axis y the detected profile at f (2f ) is similar, but not coincident, to the first (second) derivative of the intensity profile, accordingly with Eq. (1.46), being δ ≈FWHM. In the case of a detection at f the signal is zero when the particle is in the center of the beam, i.e. at (x 0 , y 0 ) = (0, 0), while at 2f the signal is maximized. Additionally, the position of the maximum does not change with the wavelength. For these reasons the 2f detection is experimentally preferred. A comparison with the first and second derivatives of the Gaussian beam (dashed lines) shows that a broadening appears when the amplitude of oscillation δ is not negligible as compared to the FWHM.

ABSOLUTE EXTINCTION CROSS-SECTION MEASUREMENT

The maximal signal amplitude S 2f (x 0 , y 0 ) can be optimized by varying w and δ. For a particle in the center of the Gaussian beam (x 0 , y 0 ) = (0, 0), the analytical solution of Eq. (1.49) writes as

S 2f (0, 0) = σ ext 2 w 2 b j1 δ 2 w 2 exp - δ 2 w 2 = σ ext R 2f (δ, w) , (1.50)
where b j1 is the modified Bessel function of first kind and first order. The proportionality coefficient R 2f relating the raw SMS signal S 2f to the absolute extinction cross-section σ ext is completely determined by the Gaussian beam waist w (or FWHM ) and the vibrational amplitude δ (experimentally tunable parameter). Fig. (1.16) shows the effect of δ and FWHM on R 2f . Note that it exists an optimal oscillating amplitude maximizing R 2f , occurring at about δ ∼FWHM. 1.4 Optical detection of single nano-objects: spatial modulation spectroscopy (SMS) 27

EXPERIMENTAL SETUP

The SMS microscope setup, used in the work of Chapter 3, has been introduced in Fig. (1.14). In standard conditions, it consists of apochromatic objectives of high magnification 100× and numerical aperture N.A. = 0.75 which focus and recollect a tunable laser beam on an individual nanoparticle deposited on a substrate. Using a piezo shaker, the sample is forced to oscillate along the y direction, typically with frequency f = 1500 Hz and amplitude δ = 325 nm. A second piezo is employed to perform XYZ movements, allowing to move single nano-objects within the beam focus and realize sample cartographies of 100×100 µm range.

The beam FWHM can be directly measured considering a linecut along x of the SMS signal map, as the modulation is absent along this direction, and thus, the signal x profile will reproduce the beam intensity profile, accordingly to Eq. (1.42).

It exhibits an Airy intensity profile close to the diffraction limit

FWHM = 0.515 N.A. λ ≈ 0.7 λ
similar, in first approximation, to a Gaussian profile (see Appendix (A) ). However, in all the reported experiments, the solution of the integral of Eq. (1.49), that for a Gaussian spot yields the analytical expression of Eq. (1.50), has been numerically computed for the Airy profile in order to obtain a more accurate conversion from SMS raw signal to absolute extinction cross-section.

In the following chapter it will be presented in detail the developments of the SMS microscope setup that were necessary to realize high-pressure measurements.

Chapter 2

Optics of a single metal nano-object under high pressure

Up to now, nanoparticle properties alterations as a function of the pressure have been studied only in ensemble experiments [START_REF] Christofilos | High pressure study of the surface plasmon resonance in Ag nanoparticles[END_REF][START_REF] Christofilos | Femtosecond Nonlinear Optical Spectroscopy of the Acoustic Vibration of Metal Nanoparticles Under High Pressure[END_REF][START_REF] Bao | The redshift of surface plasmon resonance of colloidal gold nanoparticles induced by pressure with diamond anvil cell[END_REF]. However, in order to obtain more reliable and meaningful results, an individual nano-object characterization is required. In this context, a setup to analyze single nano-objects under extreme conditions has been conceived, implemented and improved. This new apparatus is tailored not only to investigate optical properties modifications of metal nanoparticles under high pressure, but can also be used in other contexts (e.g. to study the evolution and collapse of an isolated carbon nano-tube (CNT) ). In this chapter are firstly described all the steps necessary to realize and characterize the optical microscopy measurements on individual nanostructures under high pressure.

Afterwards, the main experimental results obtained on a single elongated gold nanoparticle, as well as the modeling studies are reported. Such a setup is based on the SMS technique combined with a diamond anvil cell (here abbreviated to the acronym "DAC"), a device that makes possible the generation of a hydrostatatic high-pressure environment around the sample. These are domains of expertise of our FemtoNanoOptics group, and of the group of prof. A. San Miguel at iLM, in collaboration with whom this work has been realized.

High-pressure microscopy setup

The setup for optical characterization on individual nanostructures under high pressure is based on the SMS technique which allows to locate a single nanoobject and study its optical properties, such as the extinction cross-section, as a function of wavelength and polarization (see Section 1.4.2 of Chapter 1). Moreover, it exploits the diamond anvil cell, a structure that permits the generation Optics of a single metal nano-object under high pressure of a uniform and controlled high-pressure surrounding. As we are interested in a setup designed to investigate an individual nanostructure under high pressure, the combination of these two experimental methods is required. The far-field nature of the SMS technique is suitable for the adaptation of a DAC, but two relevant and challenging problems were encountered (and solved) during the setup realization: optical focusing of the beam and mechanical vibration of the DAC.

Generalities on the diamond anvil cell (DAC)

The DAC is a high-pressure apparatus consisting of two opposing diamonds with a sample compressed between two plane surfaces, the polished culets (diamond tips). This enables the compression of sub-millimeter sized samples to extreme pressures, typically around 100 GPa, although pressures close to 800 GPa can be reached [START_REF] Dubrovinsky | Implementation of micro-ball nanodiamond anvils for high-pressure studies above 6 Mbar[END_REF][START_REF] Dubrovinsky | The most incompressible metal osmium at static pressures above 750 gigapascals[END_REF]. An example of laser-heated DAC device is shown in Fig. (2.1).

The combination of high pressure and high temperature (generated by IR lasers penetrating the diamonds) has allowed the simulation of the planetary interiors, [START_REF] Boehler | Temperatures in the Earth's core from melting-point measurements of iron at high static pressures[END_REF][START_REF] Fei | Simulation of the Planetary Interior Differentiation Processes in the Laboratory[END_REF][START_REF] Kraus | Formation of diamonds in laser-compressed hydrocarbons at planetary interior conditions[END_REF] the discovery of new structures and phases in elements and the synthesis of novel hard materials [START_REF] Boehler | Diamond cells and new materials[END_REF][START_REF] Mcmillan | New materials from high-pressure experiments[END_REF]. What makes the DAC so suitable is that the new materials can be identified and characterized in situ. Remarkable examples includes the non-molecular ice X, [START_REF] Goncharov | Compression of Ice to 210 Gigapascals: Infrared Evidence for a Symmetric Hydrogen-Bonded Phase[END_REF] polymeric nitrogen, [START_REF] Eremets | Semiconducting nonmolecular nitrogen up to 240 GPa and its low-pressure stability[END_REF] metallic phases of xenon, [START_REF] Caldvvell | Structure, bonding and geochemistry of xenon at high pressures[END_REF] and potentially hydrogen [START_REF] Castelvecchi | Physicists doubt bold report of metallic hydrogen[END_REF].

In a DAC, two diamond anvils are used to compress the pressure chamber (a sub-millimeter thickness metallic gasket with a hole drilled in its center by spark erosion [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF]). Diamonds have a truncated conical shape with large external surfaces typically 3-4 mm in diameter, 16 facets, with 1/8 to 1/3 carat weight and cost about $1000. The diamond culets face each other, and must be polished and perfectly parallel in order to produce uniform pressure and to prevent dangerous strains. For this reason a proper alignment must be accomplished, procedure based on the removal of Newton interference fringes produced by the relative tilt of the anvils [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF].

In order to bring the diamonds closer, an external force-generating device must be introduced. This can rely on a lever arm, tightening screws, pneumatic or hydraulic pressure applied to a membrane [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF]. In all cases the force is uniaxial and is applied to the external sides of the two diamond anvils. The operation of DAC is based on the principle p = F/A where p is the pressure, F the applied force and A the area. Typical culet sizes for diamond anvils are 100-400 µm, such that a very high pressure is achieved by applying a moderate force on the much larger external diamond surfaces. Diamond is a very hard and virtually incompressible material even at high temperatures [START_REF] Nazaré | Properties, growth and applications of diamond[END_REF] and thus it minimizes the deformation and breakup of the apparatus.

The pressure chamber contains a pressure-transmitting medium (abbreviated with "PTM"), a high-pressure sensor and the object under investigation. The uniaxial pressure provided by the DAC may be transformed into uniform hydrostatic pressure using a fluid PTM, such as helium, neon, argon, nitrogen, paraffin oil or a mixture of methanol and ethanol [START_REF] Jayaraman | Ultrahigh pressures[END_REF]. The PTM must be chemically inert, as to not interact with the sample and must be stable under high pressures, in order to stay hydrostatic and ensure uniform compressibility of the sample. If the sample is heated with a laser, the PTM should not absorb the laser radiation and should have low thermal conductivity in order to keep the temperature constant (diamond anvils possess high thermal conductivity). In this case, ideal PTM are noble gases. Pressure are measured by a high-pressure sensor, a reference material whose behavior under high pressure is known. This can be a structurally simple material with a known equation of state, such as copper or platinum, monitored by X-ray diffraction [START_REF] Kinslow | High-Velocity Impact Phenomena[END_REF] or an unheated ruby ball, where its fluorescence lines are tracked [START_REF] Forman | Pressure Measurement Made by the Utilization of Ruby Sharp-Line Luminescence[END_REF][START_REF] Mao | Specific volume measurements of Cu, Mo, Pd, and Ag and calibration of the ruby R1 fluorescence pressure gauge from 0.06 to 1 Mbar[END_REF]. More details on the DAC and its alignment can be found in [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF].

Employed diamond anvil cell and pressure handling

The DAC employed in this work, reproduced in Fig. (2.2), was conceived to allow increasing the pressure environment up to 10 GPa and to permit the optical characterization of an individual nanostructure using strongly focusing microscope objectives (magnification 100x and numerical aperture 0.7). The two parts are Boehler-Almax design anvils made from rough diamonds. This geometry was introduced in 2004 by R. Boehler and compared to conventional anvils offers a superior alignment stability, larger aperture, and smaller size [START_REF] Boehler | New anvil designs in diamond-cells[END_REF]. The external diameter measures 3.30 mm, while the unpolished thickness is 1.44 mm, the smallest available. A small diamonds thickness is required in optical microscopy to minimize beam absorption. In order to produce the culets, they are polished following the [START_REF] Ruijgrok | Damping of acoustic vibrations of single gold nanoparticles optically trapped in water[END_REF] crystal orientation for the highest strength, which allows to tolerate a maximal pressure of 50 GPa. The culet has no bavels and measure 0.4 mm in diameter, with a thickness after polishing of 1.385 mm. The anvils geometry is reproduced in Fig. (2.3).

The two diamonds differ in quality: the upper one is a type IIas (∼ 1 ppm of nitrogen defects), while the lower one is a type Ia (∼ 2×10 3 ppm of nitrogen defects). The upper diamond must be purer because the excitation of the sample and the corresponding optical processes (Raman spectroscopy, fluorescence, SMS) sensitively dependent on the quality of the exciting beam. Another difference consists in a more accurate polishing necessary on the upper diamond culet to remove nano-scratches that can emerge during a long testing period. Those nano-flaws were extending a few tens nanometers into the diamond and were identified by looking at the secondary electrons of a scanning electron microscope (SEM), as one can see in Fig. (2.4). Their effect was a slight, but intolerable modification of the SMS signal from an individual nanoparticle. The practical result of this polishing is a newfound thickness of 1.385 mm for the upper anvil.

The external thickness of each anvils is glued to tungsten-carbide seats with large aperture (85 • ) allowing optical spectroscopy measurements (as well as Xray diffraction). A stretchy membrane, screwed on the external support of the superior diamond, pushes down the upper diamond by swelling up. This process is reversible and controlled by a pneumatic inflator that pumps helium towards the membrane. An example of how the external membrane pressure influences the sample chamber pressure is illustrated in Fig. (2.5). Note that the pressure within the chamber is about 10 3 times higher than the external pressure applied with the inflator.

A pierced Cu-Be gasket is placed in between the diamonds. The Cu-Be alloy was preferred to other stiffer material like steel, because, although it can stand relatively modest pressures (up to 10 GPa), its malleability permits to seal the DAC at very low pressures (in practice, to characterize the sample at 0 GPa). To

Optics of a single metal nano-object under high pressure avoid leaking at high pressure the gasket hole must be drilled regular and circular. On the other hand, in order to avoid leaking at very low pressure, seems to be essential the removal of debris around the perforation by acetone. Before drilling the hole, the Cu-Be plate must be indented using the DAC. This allows to choose the starting thickness for the pressure chamber and to make the gaskets perfectly fit the space between the anvils. The real hole is drilled in the center of this indentation by spark erosion, and its diameter will be determined by the size of the tip used. Typical sample space is given by the gasket thickness (typically 50 m) and the central hole diameter (routinely 130 m). The chamber thickness is selected on the basis of the sample and sensor size and of the maximal pressure desired, while the diameter should be around the 30-50% of the culet extension for avoiding leaking and leaving enough space for the sample. It is vital that the chamber thickness remains always larger than the pressure sensor, otherwise an uniaxial pressure is applied on the sensor giving falsified results.

In case one wants to work with nano-objects deposited on a diamond substrate, they can be deposited on one of the anvil culets via spin-coating method. It is preferable to use the upper diamond, because optical modeling as well as ex- perimental calibrations are more reliable. Since diamond surfaces are naturally lipophilic and hydrophobic, it is important to gently clean the culets before realize the spin-coating. This can be done using an optical paper soaked in methanol (the acetone is to avoid because it dissolves the glue that keep the diamond attached to the seat). Alternatively, one can introduce in the pressure chamber an external substrate with nanostructures on it. This enables to change the substrate material provided that it supports the high pressure and also to realize electron microscopy measurements on the sample before and after the experiment. Diamond or silica substrates with a few m of thickness are currently in phase of realization (by A. San Miguel group and our collaborator L. Marty in Grenoble at Institut N ÉEL).

The PTM composition is 4Me:1Et, an alcoholic and transparent mixture which remains quasi-hydrostatic until 10.4 GPa, where the solidification, displayed in Fig. (2.6), occurs [START_REF] Klotz | Hydrostatic limits of 11 pressure transmitting media[END_REF]. This liquid is volatile, therefore it must be added quickly before sealing the DAC. In order to avoid air bubbles formation inside the pressure chamber it is useful to prepare the PTM a few hours earlier, letting it some time to rest; filling the chamber with one droplet that should be gently deposed on the gasket beside its hole, instead of make it fall directly into it; closing the cell in a smooth and continuous manner, without low-and-overs.

Optics of a single metal nano-object under high pressure

The pressure inside the cell is directly measurable by the ruby fluorescence method (see the end of the section), introducing in the pressure chamber one or several spherical ruby balls with diameter typically below 10 m. To avoid contaminations, the ruby balls should be placed on the anvil culet without the sample. As shown in Fig. (2.6), the ruby balls should preferably be placed close to the gasket edge in order be far from the center of the sample. Even if ruby and nano-objects are placed on different culets, but are not well separated in the XY direction, a distortion in the SMS signal will occur. Once one adds the PTM droplet to fill the chamber, the nano-objects are not moving due to van der Waals forces, whereas the ruby balls are easily dragged away. To avoid this, it is extremely useful to introduce a first PTM droplet on the cleaned culet, then adding the rubies on this wet surface and immediately after them adding a new PTM droplet on it. With this procedure the chances of a successful sealing are much higher.

Closing the DAC produces a typical pressurization of 0.02 -0.05 GPa. In order to measure this starting pressure, it is essential to measure the reference ruby flu- orescence with the DAC open (1 Atm ≈ 1 bar ≈ 10 -4 GPa ∼ 0 GPa), before or after the experiment1 . This starting value can be stabilized by swelling the membrane a little (usually introducing a quantity of He generating 4 bars of pressure into the membrane and a pressure of the order of 0.05 GPa inside the chamber). In this low-pressure configuration the DAC may be affected by a slow leaking which empties the chamber in a few hours. For this reason the time available for the low pressure measurement is limited. However, once the pressure approaches 0.5 GPa (applying 15 bars into the membrane), the apparatus becomes very stable and it can remain pressurized for days. This steadiness persists at high pressure, until the vitrification of the 4Me:1Et. In this regime of pressure close to 10 GPa (40-45 bar in the membrane) there is a high risk of leaking due to an asymmetrical deformation of the gasket hole. Consequently, it is useful to have an optical system which permits to monitor the gasket shape and thickness during the experiment (in addition, measuring the gasket thickness allows to verify that the ruby ball is not crushed between the two approaching anvils).

The pressure can be released by gently emptying the membrane. In this phase the pressurization is stable, permitting experiments to investigate reversibility. Unfortunately, going back to the initial pressure is not possible because of a considerable hysteresis in the relation between the applied load (membrane) and the pressure obtained. An example of the phenomenon is reported in Fig. (2.5). The residual pressure is usually around 1-2 GPa, is stable, and mainly depends on the starting Cu-Be gasket thickness and on the maximal pressure reached.

In order to monitor the gasket behavior during the experiment, images of the pressure chamber at different spatial coordinates must be acquired. In the setup, the DAC is mounted on a XYZ piezo nanopositioning stage, with a close loop travel of 100×100×100 µm, a close loop resolution of 0.8 nm due to capacitive integrated sensors2 . The piezo stage, in turn, is placed on a manual XY micrometer stage for coarser movements. To monitor the gasket, the DAC must be moved under a focusing objective, which focuses an incoherent white light in the pressure chamber, passing through the upper diamond anvil. When the focal plane falls on an internal surface, like the gasket or a culet plateau, the backscattered light is recollected by the same objective and redirected to a CCD camera, reproducing the image of the sample plane on a TV screen. A schematic of the setup is displayed in Fig. appears asymmetrical, which indicates that a higher pressure may have induced the breakdown. By this imaging, one can check the surface state before and after nanoparticles deposition, verify the vitrification absence of Fig.

(2.6), the circular shape of the gasket, measure its diameter along the X and Y directions by moving the micrometric screws, as well as its thickness by looking at the Z coordinates corresponding to the two culets well-focused.

Typically, until 3 GPa, the gasket evolution consists in an initial diameter decrease. This is due to the malleability of the Cu-Be, forced to penetrate into the chamber. Beyond this point, the pressure inside the hole is enough to cause an overall increase in diameter. During this widening, it is important to check the integrity and the shape of the gasket: an excessive asymmetry induces a structural weakening and a subsequent leaking. An example of gasket evolution until its breakdown, occurred at 6 GPa, is shown in Fig. (2.9). With the pressure rising, the chamber thickness experiences a linear reduction until the breakdown, where it keeps dropping despite the internal pressure remains constant. Therefore, it is important to monitor the gap between the culets and release the pressure before the breakdown occurs. In this phase, the gasket does not change significantly its dimension, although a modest decrease in thickness could be noticed. Consequently, a new pressure ramp up with the same gasket can not reach the same high pressures anymore. The gasket should be replaced after a high-pressure cycle.

In order to measure the chamber pressure a ruby fluorescence calibration system must be implemented. As illustrated in Fig. a laser source to excite the fluorescent radiation and a spectrometer for spectral analysis. The exciting light is generated by a 531.92 nm continuous wave solid state laser (Nd:YAG doubled), with a full output power of 50 mW. The use of a long working-length objective is essential to focus the laser on a micron size spot within the pressure chamber, after passing through the diamond anvil [START_REF] Takemura | A diamond anvil cell for advanced microscopic observations and its application to the study of crystal growth under pressure[END_REF].

The excited fluorescent radiation is collected by the same objective and separated from the backscattered laser light introducing a 532 nm ultrasteep long-pass edge filter. In this way a spectrometer can be used to detect the fluorescence signal. The spectrometer is a Horiba i HR320 with a grating of 1800 grooves and blazed wavelength of 400 nm. The pressure determination requires the measurement of the fluorescence spectrum at ambient pressure (∼ 0 GPa) and at high pressure.

The correlation of the measured wavelength shift Δλ (or frequency change Δν) with applied pressure is the basis for pressure determination.

The ruby, Cr 3+ doped α-Al 2 O 3 , is a well established high-pressure sensor and does not undergo phase transitions up to 100 GPa. Suitable ruby balls contain 3000 -5500 ppm of Cr 3+ and are around 10 m in diameter. Ruby fluorescence under blue-green excitation is dominated by an intense doublet with sharp band components centered at 694.2 nm (14402 cm -1 ) and 692.8 nm (14432 cm -1 ), respectively named R1 and R2 line [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF][START_REF] Sato-Sorensen | Measurements of the lifetime of the ruby R1 line and its application to high-temperature and high-pressure calibration in the diamondanvil cell[END_REF][START_REF] Eggert | High pressure refractive index measurements of 4:1 methanol:ethanol[END_REF]. The lines exhibit a pronounced redshift with the applied pressure, making the ruby fluorescence a good pressure gauge. [START_REF] Piermarini | Hydrostatic limits in liquid and solids to 100 kbar[END_REF] and Piermarini (1975) Later, [START_REF] Mao | Specific volume measurements of Cu, Mo, Pd, and Ag and calibration of the ruby R1 fluorescence pressure gauge from 0.06 to 1 Mbar[END_REF]Mao ( , 1986) ) and Bell (1986) improved this result by a calibration through the equation of state of several metals under quasi-hydrostatic and non- Optics of a single metal nano-object under high pressure hydrostatic pressure condition up to 180 GPa and, for the R1 line, found the empirical quasi-linear relation

P = A B λ λ 0 B -1 , (2.2)
where A = 1904 and B = 7.665 for quasi-hydrostatic and B = 5 for non-hydrostatic conditions. The incertitude on the calibration introduces an error on the pressure of 4%×P [START_REF] Chijioke | The ruby pressure standard to 150 GPa[END_REF]. An example of the ruby's fluorescence evolution with the pressure is reported in Fig. (2.11). To extract the R1 line position a double-Lorentzian fit should be performed.

As shown in Fig.

(2.12), the R1-R2 lines splitting remains almost constant in a hydrostatic pressure environment. It was found that the R1 line shifts remarkably in a non-hydrostatic environment, while the R2 line shifts negligibly [START_REF] Gupta | Potential use of the ruby R2 line shift for static high-pressure calibration[END_REF][START_REF] Adams | Spectroscopy at very high pressures. X. Use of ruby R-lines in the estimation of pressure at ambient and at low temperatures[END_REF]. Furthermore, the widths of the fluorescence lines increase with increasing nonhydrostaticity [START_REF] Forman | Pressure Measurement Made by the Utilization of Ruby Sharp-Line Luminescence[END_REF]65]. Therefore the R1-R2 line separation and their linewidths can be monitored to verify the hydrostaticity of the pressure environment. The ruby lines positions are however influenced by the temperature. Specifically, Figure 2.12: R1-R2 lines quasi-linear shifts observed in the setup. The continuous lines are fits of the experimental points. At ambient pressure, the R1-R2 lines are located at 694.2 nm and 692.8 nm respectively. The pressure dependency is in accordance with Eq. (2.1) and the distance between the lines remains constant with the pressure (1.42 nm = 29.46 cm -1 ), indicating a hydrostatic environment.

the R1 line follows the empirical third-order polynomial [START_REF] Vos | On the temperature correction to the ruby pressure scale[END_REF][START_REF] Ragan | Calibration of the ruby R1, and R2 fluorescence shifts as a function of temperature from 0 to 600 K[END_REF]:

Δλ = 6.591×10 -3 ΔT + 7.624×10 -6 ΔT 2 -1.733×10 -8 ΔT 3 , (2.3)
where Δλ and T are expressed in nm and K respectively. By linearizing this formula to Δλ/ΔT ≈ 6.2×10 -3 nm/K and comparing with Eq. (2.1), it results that a temperature change of 6 K is equivalent to a pressure variation of 0.1 GPa. Therefore the heating effect due to the laser absorption in the ruby ball can be a source of systematic error in the pressure determination, mainly for powers greater than 40 mW [START_REF] Miletich | High-Pressure Single-Crystal Techniques[END_REF], and should be minimized using low power (∼ 1 µW in this work).

SMS and DAC combination: optical development

The setup for optical characterization on individual nano-objects under high pressure requires the application of the SMS technique to the DAC system mentioned above. This means: focusing at the diffraction-limit inside the DAC and kHz mechanical shaking of the whole DAC. This was achieved by carefully developing both optical and mechanical aspects.

From an optical point of view, an efficient SMS system requires a tunable laser source that must be focused on an individual nanostructure at the diffraction limit.

Concerning the laser source, a mode-locked Ti:Sa laser is employed enabling quick wavelength changes in the range 680-1080 nm 3 . The main obstacle to focus the laser light at the diffraction limit is the presence of the diamond. This is connected to the diamond's high index of refraction of 2.4 [START_REF] Eremets | Refractive index of diamond under pressure[END_REF][START_REF] Balzarzetti | Pressure dependence of the refractive index of diamond, cubic silicon carbide and cubic boron nitride[END_REF]. The problem can be easily explained treating the diamond as a plane-parallel plate. In case it is crossed by a collimated beam, no aberrations are introduced into the system. On the other hand, when the same parallel plate is placed in a converging or diverging beam, a series of aberrations including spherical and astigmatism are introduced. Since the SMS technique is performed with a single laser wavelength at the time (quasi-monochromatic regime), the chromatic aberration is not relevant. On the other hand, the spherical aberration shall be corrected, as discussed here. When a plane-parallel plate is traversed by a converging beam, its focal plane is longitudinally retarded. The displacement Δz, illustrated in Fig. (2.13), is due to the change in refractive index and in the paraxial region is given by [START_REF] Smith | Modern Optical Engineering: The Design of Optical Systems[END_REF]:

Δp = n -1 n h , (2.4)
where n is the diamond refractive index and h is the parallel plate thickness.

Considering the thickness of the used top diamond anvil (1.385 mm after the pol- ishing) the longitudinal displacement measures 0.8 mm. Eq. (2.4) applies to the paraxial region only. Outside this region, a more complex relation applies [START_REF] Riedl | Optical Design Fundamentals for Infrared Systems[END_REF]:

Δm = ⎡ ⎣ 1 - 1 -sin 2 α n 2 -sin 2 α ⎤ ⎦ h = 1 - 1 -N.A. 2 n 2 -N.A. 2 h , ( 2.5)
where N.A. is the numerical aperture. Assuming an objective with a numerical aperture in air of N.A. = sinα = 0.7, Δm results 0.9 mm. Therefore, the longitudinal spherical aberration, defined as the difference between Δm and Δp, is 0.1 mm. Moreover, the blur diameter for the spherical aberration is given by [START_REF] Riedl | Optical Design Fundamentals for Infrared Systems[END_REF]:

B spher = n 2 -1 32n 3 N 3 h . (2.6)
For the considered objective, B spher = 0.029 mm, which is too high as the necessary condition to investigate single nanoparticles by SMS is the focusing of the beam down to diffraction limit (≤ 1 m). Typically, spherical aberration are compensated by using a commercial focusing objective with a correction collar. Unfortunately, for the DAC case, the aberration is too important to be adjusted in such a way, posing a considerable challenge in individual nano-object imaging. In order to overcome these aberrations a custom made pre-compensated microscope objective has been conceived on purpose for the specific diamond thickness. It was realized by Mitutoyo (Japan), it has a 100x magnification, 0.7 as numerical aperture and 30 mm working distance, high enough to be compatible with the DAC geometry constrains.

Introducing the tunable laser source and the pre-compensated objective in the setup of Fig. In this configuration the laser source is polarized and focused on a culet surface, where the nanostructures are typically deposed. The DAC is screwed to an aluminum holder, which in turn is fixed on an Y electric piezo shaker, mounted on the top of the XYZ piezo stage. The XYZ stage allows XY cartographies, while keeping the laser beam focused. The Y piezo shaker (with a capacitive integrated sensor) produces a sinusoidal oscillation in the nano-object Y-coordinate. In order to perform spatial modulation, amplitude and frequency of vibration are controlled by the function generator of a lock-in amplifier (LIA). After transmission through the lower anvil, the modulated transmitted light beam is detected by a silicon photodiode. The photodiode signal is read in parallel by a DMM (digital multimeter), which monitors the average transmitted power in absence of the nano-object W 0 , and to the LIA, that detects the modulated transmitted power ΔW at twice the frequency of oscillation. These two instruments are connected to a computer allowing to map the "SMS signal", proportional to the relative variation in transmitted power ΔW/W 0 , over the spatial XYZ coordinates. The SMS signal can be easily converted in absolute extinction cross-section σ by knowing the amplitude of oscillation and the beam shape and size. In general, the absolute extinction crosssection σ of a nano-object is a function of the light wavelength and polarization. These tance. As clearly seen in the image, the spot corresponding to the pre-compensated objective is narrower and has a Gaussian profile, while the one generated by the standard objective is wider and presents strong aberrations, making it inappropriate for focusing purposes. A standard objective has been used to collect the transmitted light, although a second pre-compensated objective would be the best solution.

A quantitative beam characterization has been performed once the mechanical development (see Section (2.1.4) ) was carried out and the "Transmission SMS setup" of Fig. Optics of a single metal nano-object under high pressure pendix (A)). This is not surprising as the Gaussian beam is deliberately strongly truncated by the pupil of the pre-compensated objective, the incoming light being well-approximated by a plane wave. The truncation effects in the focusing and the quantitative treatment of the diffraction limit can be found in Appendix (A). In an Airy pattern profile, the FWHM of the focused spot is limited by

FWHM = Kλ N.A.
with K= 0.515 .

(2.7)

When λ = 800 nm and N.A. = 0.7, Eq. (2.7) gives FWHM (800 nm) = 590 nm, while the Airy fit of Fig. (2.17) gives FWHM (800 nm) = (625±20) nm, which is slightly larger than the diffraction limit. According to Eq. (A.6) of Appendix (A), fitting an Airy-like spot profile with a Gaussian (max-free) function, rather than a Bessel function, underestimates the FWHM (reducing the constant K of Eq. (2.7) giving diffraction limit to K Gauss = 0.496). This is due to the fact that the Gaussian fit of a Bessel distribution is too large at the base and consequently narrower at the the FWHM to compensate, as shown in Fig. (2.17). This is consistent with the experimental case, as K Gauss = 0.496 yields a theoretical diffraction limit FWHM (800 nm) = 570 nm and the Gaussian fit of Fig. (2.17) gives FWHM (800 nm) = 605 nm (again 35 nm larger than theoretical limit). This suggest that the image spot profile is the Airy pattern main lobe, despite the fact that outer rings are not visible. Their absence is due to some residual spherical aberration that generates some spikes, clearly visible for x < 0.5 µm in Fig. (2.17). The spot size larger than the diffraction limit and the presence of some aberrations are limited inconveniences and not an obstacle to perform SMS experiments. These imperfections are probably due to the difference between the diamond thickness after the polishing (1.385 mm) and the nominal thickness compensation of the objective (1.5 mm).

In the available range of laser wavelength, 680-1080 nm, the FWHM is proportional to λ, in accordance with Eq. (2.7). This means that the objective correctly focuses the beam and no significant aberration are occurring in this wavelengths range. Repeating the analysis of Fig. (2.17) for several wavelengths at a specific pressure as illustrated in Fig. (2.18a) yields a more reliable estimation of K exp for the experimental FWHM formula

FWHM exp = K exp λ N.A. (2.8)
Experimentally K exp is also a function of the pressure P , as the latter affects the diamond refractive index and thickness, as well as the PTM refractive index. Indeed the beam is focused and studied at the nano-particle level, whose center is located in the pressure chamber, 30-40 nm under the culet of the superior diamond anvil. The PTM refractive index n P T M grows with the pressure, as illustrated in Figure 2.17: Quantitative image spot analysis realized through the SMS technique performed with a 2f detection. If the oscillating particle is smaller than the beam, then the X profile of the signal corresponds to the beam intensity profile; if the amplitude of oscillation is much smaller than the beam size, then the Y profile approaches the second derivative of the beam intensity profile. The X profile is well fitted by a Gaussian function (although the base is too large and the linewidth too narrow), and even better through a Bessel function (first kind, order one) corresponding to an Airy pattern.

The key parameter is the fitted FWHM (W 0 identifies the total beam power and c is an experimental parameter, converting the arbitrary SMS signal into absolute crosssection).

Optics of a single metal nano-object under high pressure where the pressure P is in GPa. In conclusion, the FWHM is simply given taking Eq. (2.8) with N.A. = 0.7

FWHM exp = 0.8 + 3×10 -4 P 2 λ , (2.10)
where P is expressed in GPa.

In the "Transmission SMS setup" of Fig.

52

Optics of a single metal nano-object under high pressure For a gasket thickness h gasket = 30 µm and N.A. = 0.7, this results in SMS ∼ 70 µm.

Another source of SMS signal distortion comes from the ruby on the opposite culet.

The contaminated area is a circle of diameter 20-30 µm, as shown in Fig. (2.19).

In conclusion, only the nano-objects far enough from the contamination sources should be studied.

From an optical point of view, the knowledge of the image spot profile and size are the keys elements to realize a proper quantitative SMS experiment. Using the pre-compensated objective, the spot shape is well described by an Airy pattern whose size is governed by Eq. (2.10). The optical part characterized, in order to realize SMS experiments, an appropriate mechanical oscillation must now be applied and quantified.

SMS and DAC combination: mechanical development

From a mechanical point of view, an efficient SMS system should be able to precisely control in time the XYZ coordinates of an individual nano-object. The particle must be kept into the focused laser beam, while it oscillates along the Y-direction with a proper amplitude and frequency. The vibration amplitude should be comparable to the beam spot size in order to have a strong SMS signal, while the working frequency should be high enough to reduce the 1/f noise coming from laser power fluctuations and electronics, but not too elevated either, otherwise mechanical coupling will appear and odd XYZ vibration could occur. To this respect, for a sample of a few tens of grams the operating working frequency lies around 1500 Hz. Unfortunately diamond anvil cells are pretty heavy and, even after important improvements, the employed DAC still weights 0.4 Kg. This strongly reduces the amplitude of oscillations and generates mechanical resonances at relatively low frequencies. In order to characterize and improve the mechanical apparatus, a simple model based on an interferometric study has been realized.

In the "Transmission SMS setup" of Fig

v LIA (t) = V LIA sin(2πf t + φ LIA ) , (2.12)
forcing a periodic expansion in the Y piezo shaker that can be followed through its capacitive integrated sensor on an oscilloscope

v Y (t) = V Y sin(2πf t + φ Y ) , (2.13)
where the prime symbol ( ) indicates quantities measured through the capacitive integrated sensors. The voltage amplitude V Y (measured in Volt) can be directly converted in spatial amplitude A Y (in µm) using the calibration factor

C Y = 12 µm/V of the Y piezo shaker: Y Y (t) = A Y sin(2πf t + φ Y ) . (2.14)
Ideally, the Y piezo shaker and the DAC (with the nano-objects contained in the pressure chamber) should move as one. If this is the case, the system can be modeled by a forced harmonic oscillator, where the only limitation in frequency would come from the Y piezo shaker push-pull force capacity, that is equal to F = 20 N. According to this, M DAC ŸY (t) < F ∀t with M DAC = 0.6 Kg, the maximal available amplitude of oscillation being inversely proportional to the frequency square and quantified in Fig. (2.20). The graph highlights the need of choosing a frequency below 1.5 kHz, where the vibration amplitudes can attain at least half of the beam spot FWHM in the infrared, and hence yield an intense SMS signal.

Unfortunately the system oscillates in such an ideal movement only for low frequencies, beneath 100 Hz, where the noise from laser fluctuations is way too large for the SMS technique. In fact, the DAC important weight causes some mechanical couplings between the DAC, the Y piezo shaker and the XYZ piezo stage, making the apparatus more akin to a forced system of three coupled harmonic oscillators, rather then a single one. As a consequence, the real amplitude of oscillation of the DAC is insufficient to generate a perceptible SMS signal for most of the accessible frequencies, whereas for three specific frequencies dangerous resonances show up. Therefore, in order to perform the SMS technique, an optimal working frequency range must be found, hence requiring a mechanical system characterization. The capacitative sensors integrated in the piezos allow to monitor the motion of the system components with respect to the element they are attached to, and what discussed for the Y piezo shaker applies to the XYZ piezo stage too. In the Y direction a periodic movement shows up due to the coupling with the Y shaker

Optics of a single metal nano-object under high pressure

Y XY Z (t) = A XY Z sin(2πf t + φ XY Z ) , (2.15)
while along the X and Z direction the movement is always negligible (unless in the case of a strong resonance).

To study the absolute movement of the system components (with respect to the optical table) a home-made interferometer has been assembled. Outlined in Fig. and of the micrometric stage

Y XY Z (t) = A XY Z sin(2πf t + φ XY Z ) (2.
Y base (t) = A base sin(2πf t + φ base ) ∼ 0 . (2.18)
The latter resulted negligible, yielding an equivalence between relative and absolute movement of the XYZ piezo stage (Eq. (2.15) and Eq. (2.17) ). Absolute (measured by the interferometer) and relative (sensor) movement of the system components are summarized in Fig. (2.22).

When amplitudes and phases are deduced from piezo sensors, they are obtained directly by the sinusoidal signal in the oscilloscope, proportional to the internal mouvement. However, when they are measured through the interferometer, a more complex signal occurs. Basically, the more fringes are detected per period, the greatest the oscillation amplitude is, but the sinusoidal motion makes the fringes denser in time when the oscillation speed is maximal. To be more quantitative, the intensity of the interference between the path 1 (reflected by the fixed mirror) Figure 2.22: Sketch of the system components (movable piezoelectric parts in yellow).

Table : absolute and relative movements along Y are summarized in column "Y movements". Absolute movements (i.e. with respect to the optical table) are measured by the interferometer ("INTERF"). Relative movements (i.e. movement of the piezo component with respect to its base) are measured by piezo internal sensor ("SENSOR").

and the path 2 (reflected by the oscillating component) takes the form

I int (t) = I 1 + I 2 + 2 I 1 I 2 cos 4π λ (L 2 -L 1 + A cos(2πf + φ)) , (2.19)
where λ is the laser wavelength, I 1 and I 2 are the intensities along the path 1 and 2 respectively, L 1 is the distance between the beam-splitter and the reference mirror, L 2 + A cos(2πf + φ) is the distance between the beam splitter and the mirror on the moving part, A, f and φ are the amplitude, frequency and phase of oscillation. The intensity I int (t) is detected through a photodiode, measured on an oscilloscope and fitted using Eq. (2. [START_REF] Gans | Über die Form ultramikroskopischer Goldteilchen[END_REF]). An example of the interference signal and its fit is reported in Fig. (2.23).

The main results on the mechanical vibration characterization are presented in ). This fit curve slightly deviates from the experimental one, as the photodiode acts like a low-pass frequency filter, reducing the amplitudes of the signal when it varies faster. However, it was verified that this signal distortion does not affects the relevant fit parameters (amplitude and phase).

not supposed to move (Y base = Y XY Z = 0). As illustrated in Fig. (2.22), the DAC holder was characterized via interferometer only, the Y piezo shaker through the capacitative sensor, the XYZ piezo stage with the two methods, albeit only the capacitative sensor data are reported (the other data being similar). Three resonances dominate: "R 3 " ∼ 100 Hz, "R 1 " ∼ 770 Hz and "R 2 " ∼ 1770 Hz, each one being attributed to the proper frequency of the three equivalent oscillator. The XYZ piezo stage introduces the resonance "R 3 ". Indeed, its nominal unloaded resonant frequency is 190 Hz(±20%) that becomes 140 Hz(±20%) with a load of 0.33 Kg (data from manufacturer). Thus, it can be associated to an equivalent oscillator with unloaded mass 0. and "R 2 " at higher frequencies. Indeed, the Y shaker nominal unloaded resonant frequency is 5600 Hz(±20%) and its nominal stiffness 45×10 6 N/m(±20%). Therefore, it can be associated to an equivalent oscillator with unloaded mass 0.036 Kg and spring constant 45×10 6 N/m. A mass load of 0.6 Kg (DAC and holder) would produce a resonant frequency at about 1350 Hz.

The amplitudes reported in Fig. (2.24) show that in the frequency range [1000-1500 Hz] the Y shaker is oscillating, but the XYZ piezo stage and the DAC holder are not moving. Therefore, this range can not be used for SMS experiments, as the sample would not vibrate. A possible working frequency could be the nodal point at 825 Hz, indicated as "N" in Fig. (2.24), where A Y ∼ 0. Here A holder ∼ A Y and therefore relatively high oscillation amplitudes of the DAC can be produced. However, "N" is located in the proximity of "R 1 ", where the dephasing are strongly varying with the frequency. For stability reasons, it is preferable to work at frequencies where all the system components are in phase with the driving force. In order to better understand the behavior of the dephasing around "N", the relative movements of the DAC holder (Y holder -Y XY Z ) and Y shaker (Y Y -Y XY Z ) with respect to the XYZ piezo stage has been modeled through a forced two-spring system along the Y direction

M 1 Ÿ1 = -k 1 Y 1 + k 2 (Y 2 -Y 1 ) -β 1 Ẏ1 + F 1 sin(2πf t) M 2 Ÿ2 = -k 2 (Y 2 -Y 1 ) -β 2 Ẏ2 , ( 2.20)
where a sinusoidal force of amplitude F 1 is applied to the first oscillator tied to a wall, while the second oscillator is connected to the first one. The first Optics of a single metal nano-object under high pressure resonator represents the Y piezo shaker, while the second one the load (DAC plus holder), as illustrated in Fig. (2.25). The system of Eq. (2.20) can be analytically solved assuming that the solutions are sinusoidal functions with frequency f . The solutions are:

Y 1 = A 1 sin(2πf t + φ 1 ) Y 2 = A 2 sin(2πf t + φ 2 ) , ( 2.21)
where

A 1 = F 1 M 1 (2πf ) 4 + [(β 2 /M 2 ) 2 -2k 2 /M 2 ] (2πf ) 2 + (k 2 /M 2 ) 2 p 2 + q 2 , ( 2.22
)

A 2 = F 1 M 1 k 2 /M 2 p 2 + q 2 , ( 2.23
)

φ 1 = -arctan q -rp p + rq , ( 2.24
)

φ 2 = -arctan q p , (2.25
)

p = (2πf ) 4 - k 1 + k 2 + β 1 M 1 + k 2 + β 2 M 2 (2πf ) 2 + k 1 k 2 M 1 M 2 , ( 2.26
)

q = - β 1 M 1 + β 2 M 2 (2πf ) 3 + k 1 β 2 + k 2 β 2 + k 2 β 1 M 1 M 2 (2πf ) , (2.27) r = β 2 (2πf ) k 2 -M 2 (2πf ) 2 .
(2.28)

In Fig.

(F 1 , M 1 , M 2 , k 1 , k 2 , β 1 , β 2 )
. The model reproduces the main features of the experimental data "R 1 ", "R 2 ", "N", showing that around 1500 Hz the amplitude of oscillation of the DAC holder is smaller than the Y shaker one. Varying the model parameters pointed out that the nodal point "N" occurs every time that an oscillator of mass M 1 is placed between two elements with higher masses (M 2 and the wall). More importantly, comparison of fitted curves (Fig. (2.27) ), enlightens that the Y shaker is in phase with the driving force in the interval ["N", "R 2 "], while they are in antiphase in the regions ["R 1 ", "N"], making the frequencies higher [START_REF] Dahan | Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking[END_REF]. Notice that at frequencies slightly lower than "N" the oscillators are not in phase, while at slightly higher frequencies they are both in phase with the driving force.

than "N" more appropriate to work with.

In conclusion, it is necessary to work at a frequency in the range 700 Hz < f < 1500 Hz to have a reasonable signal-over-noise ratio (large frequencies) and to avoid mechanical breakdowns (occurring at too high frequencies). In this range, the DAC can oscillate with an amplitude comparable to the size of the beam spot, the dephasing between the oscillators and the applied force is small and has not discontinuities.

A good choice would be to work close to the nodal point "N" (825 Hz), where the internal movement of the Y piezo shaker is negligible (Y Y ∼ 0) and the amplitude of its movement with respect to the optical table is comparable to the DAC holder one Y Y ∼ Y holder . For this reason the choice f = 825 Hz was initially made. Even better is to work at slight higher frequencies than "N" (∼ 840 Hz) in order to have all the components almost in phase with the driving force.

In order to improve the signal-over-noise ratio, the working frequency should be increased. In the model, this can be done by changing the parameters M 1 , M 2 , k 1 , k 2 . Indeed, decreasing M 1 would mainly increase the frequency for "R 1 "; increasing k 1 would increase "R 1 ", "R 2 " and slightly "N"; decreasing M 2 or increasing k 2 would increase "R 1 ", "R 2 " and "N".

In practice, a modification of the connection (removal of part of a mechanical adapter mass) between the base of the Y piezo shaker and the top of the XYZ piezo stage (corresponding to a modification of M 1 and K 1 ) has enabled to shift the working frequency at f = 940 Hz, which is the one used in the pressure experiments presented in this chapter.

The application of the SMS technique requires the determination of the DAC amplitude of oscillation at f = 940 Hz. This oscillation amplitude is a linear function of the amplitude of the applied sinusoidal voltage, then can be written in the form

δ(940 Hz) = c • V LIA . (2.29)
The coefficient c, measured in µm/V, has been determined applying the SMS technique on a gold nanoparticle deposed on the culet of the superior diamond. If the beam shape and size are well known this kind of measurement is more reliable than the interferometric one, because it gives the movement of the nanoparticle ot the center of the DAC, while the interferometer measures the vibrations of the outer part of the holder (indeed even the holder and the DAC are not completely moving as one). It results c = (0.52±0.02) µm/V. To have a strong SMS signal, the LIA voltage was tuned yielding δ(940 Hz) of the order of the beam size described by Eq. (2.10). The following experiments has been performed with V LIA = 1.13 V corresponding to a spatial modulation of δ(940 Hz) = (0.52 ± 0.02)µm V -1 ×1.13 V = (590 ± 25) nm .

(2.30)

From a mechanical point of view, determination and optimization of the sample oscillation amplitude is the key element to realize a proper SMS experiment. The apparatus improvement and characterization have made it possible to work with the desired amplitude, at a reasonably high frequency. At this point, the SMS technique was operational and high-pressure experiments could be carried out.

A single metal nanoparticle under high pressure

The implementation and characterization of a setup for optical microscopy measurement on individual nano-objects under high pressure allowed to realize the first quantitative experiments, which are presented in this section.

Up to now, the studied nanostructures are elongated gold nanoparticles: nanobipyramids (BPs) directly deposed on a diamond culet and then immersed in hydrostatic 4Me:1Et (properties of this PTM can be found in Appendix C). The choice of working with this kind of nanoparticles was made on the basis of the solid knowledge of their optical properties (experiments and modeling) already established at standard pressure [START_REF] Lombardi | Surface Plasmon Resonance Properties of Single Elongated Nanoobjects: Gold Nanobipyramids and Nanorods[END_REF][START_REF] Pertreux | Surface Plasmon Resonance of an Individual Nano-Object on an Absorbing Substrate: Quantitative Effects of Distance and 3D Orientation[END_REF]. Examples of BPs experimental extinction cross-section spectra, σ(ω), dominated by a quasi-Lorentzian SPR are illustrated in Fig. (2.28). In the current section are reported the successful measurements realized on gold BPs, starting from the most recent one ("BP1") and ending with the first measured ("BP5"). In order to describe them, it is convenient to introduce the following terminology: "ramp-up", which indicates the experiment part where the pressure is increased; "ramp-down", which corresponds to the phase where the pressure is decreased and "pressure cycle", hence the union of a ramp-up with the following ramp-down. Furthermore, when the word "complete" is added, it means that the whole available pressure range [0-10 GPa] was investigated. In Tab. 2.2 on page 79 are summarized the working conditions and the main features of each experiment (from the most recent "BP1" to the first attempts "BP5"). Here, the experiment "BP1" is discussed in detail, while for the others only the final results and some peculiarities are reported. Optics of a single metal nano-object under high pressure

BP1

During experiment "BP1", the particle under investigation was an individual gold BP from sample 1 placed on the superior diamond anvil. Typically, after the spincoating, isolated BPs are searched on the culet at 0 GPa, without PTM. This is accomplished through the SMS technique, which allows to measure the extinction spectrum dominated by the SPR and the polarization response of each detected particle. The longitudinal SPR of a single BP shows up as a Lorentzian shape, while the light polarization dependence is sinusoidal and drops to zero for light polarized orthogonally to the long axis. These are key points to distinguish an isolated BP from a spherical particle or a cluster, which present several resonances or a polarization response not going to zero. As this mapping and selection of the nano-object may take some hours, it is realized without PTM (avoiding evaporation). The characterization of an analyzed BP in air is presented in Fig. (2.31).

The ramp-up begins after the introduction of a ruby ball on the clean culet (the one with no nanoparticles deposited) and of the PTM in the pressure chamber. The pressure is already increased by a small amount when loading the DAC. For the studied BP, the starting point was 0.055 GPa and the maximal value 9.72 GPa. Afterwards, the ramp-down was started, dropping the chamber pressure until a Optics of a single metal nano-object under high pressure residual value of 1.82 GPa. During the whole pressure cycle, the longitudinal SPR of the BP was measured and its polarization response verified. Lastly, the BP is characterized again in air, at 0 GPa. The beam spot FWHM was also characterized as a function of wavelength and pressure, yielding the pressure and wavelength dependence of Eq. (2.10) and allowing to convert the raw SMS signals in absolute cross-section, measured in nm 2 .

Measuring the polarization response enables to monitor the orientation of the BP long axis. As illustrated in Fig. Optics of a single metal nano-object under high pressure Optics of a single metal nano-object under high pressure

BP2

The studied "BP2" particle was again an individual gold BP from sample 1 placed on the superior diamond anvil. Firstly, the BP was characterized in air. The rampup began at 0.058 GPa and ended at 9.52 GPa. Afterwards, a complete ramp-down took place, dropping the chamber pressure until the residual 1.59 GPa. Lastly, the BP was characterized in air, at 0 GPa. During this experiment, the beam spot FWHM was not directly characterized, but, as the working conditions were the same as "BP1", Eq. (2. reversibility of this parameter. The peak position shows a non-reversibility only when measured in air, but this is probably due to a change in the environment refractive index after putting the BP back to air at the end of the cycle, as SPR is Optics of a single metal nano-object under high pressure very sensitive to residual liquid or water vapor. Linewidth and area undergo a soft augmentation throughout the complete ramp-up. During the complete ramp-down these parameters seem to decrease and come back. However, between these two measurements an additional broadening shows up. This unexpected behavior was observed only in "BP2" and its origins are unknown. One possibility is a systematic error introduced through a wrong focusing of the beam during the ramp-down experiments. The amplitudes of the induced pressure changes are very similar to those measured for "BP1": red shift ΔE ∼ -0.15 eV (with ΔE/E ∼ -10%), broadening Δ( Γ) ∼ 15 meV (with ΔΓ/Γ ∼ 10%), and enhancement area ΔA ∼ 1000 eV nm 2 (with ΔA/A ∼ 18%), the three presenting a non-linear dependence on pressure.

BP3

During experiment "BP3", the studied object was a cluster composed by two aligned nano-objects from sample 1, placed on the superior diamond anvil. Firstly, the BPs were characterized in air. At that time, the ruby ball was not characterized in air, at 0 GPa. Therefore the first pressure point of the ramp-up was not precisely measured. Supposing that the DAC loading produces a starting pressure similar to the one of "BP1" and "BP2", the ramp-up then started at 0.05 GPa and ended at 8.54 GPa. Afterwards, a complete ramp-down took place, dropping the chamber pressure until the residual 0.33 GPa. Lastly, the BP was studied again in air, at 0 GPa. Over the course of the experiment "BP3", the beam spot FWHM was not directly characterized. However, as the working conditions were the same as "BP1", Eq.

(2.10) was exploited to convert the raw SMS signals to absolute extinction crosssection.

The measured extinction spectra present two SPR peaks and are well fitted by a double-Lorentzian distribution. Supposing that the two nano-objects are two BPs subjected to the same broadening, the two linewidths are fitted with the same value, while the two peak positions and areas are kept independent. As illustrated in Fig. (2.38), this kind of fit is satisfactory until the end of the ramp-down, where the SPR second peak is not recognizable anymore. During the complete pressure cycle, the polarization response was repeatedly measured and the signal is always minimized at the same angle, as showed in Fig. (2.39). The peak position of the two resonances undergo a redshift that resembles to the one seen in "BP1" and "BP2", despite a very little discontinuity can be noticed about 7 GPa (see Fig. linewidth and the area should not be examined, though a general trend similar to the previous ones seems to occur.

A single metal nanoparticle under high pressure

BP4

During experiment "BP4", the studied particle was a gold BP from sample 2, placed on the inferior diamond anvil. The ruby ball was not characterized in air (0 GPa) and, as in the case of "BP3", the ramp-up is supposed to start at 0.05 GPa, ending at 7.61 GPa. Afterwards, a complete ramp-down took place, dropping the chamber pressure until the residual 1.75 GPa. During the complete pressure cycle, the polarization response was repeatedly measured and the signal goes to zero always at the same angle. During the experiment "BP4", the beam spot size and the BP amplitude of oscillation were unknown, preventing from a conversion of the raw SMS signal to an absolute cross-section quantitative value. Even without conversion, the observed SPRs can be satisfactorily reproduced by a Lorentzian distribution. Note that, when the raw SMS signal is directly fitted (instead of absolute extinction cross-section), peak position and linewidth are still reliable (however with introduction of a modest systematic error that can be estimated of the order of -0.01 eV and therefore can be corrected). On the other hand the area, measured in eV • nm 2 , cannot be quantified. Original and retrieved data are illustrated in Fig. (2.41). The typical redshift is dominating, while a small broadening shows up. Data indicate that a pressure change of 10 GPa induces a redshift ΔE ∼ -0.12 eV (with ΔE/E ∼ -9%) and a broadening Δ( Γ) ∼ 30 meV (with ΔΓ/Γ ∼ 16%). The complete ramp-down of "BP4" counts several pressure points, because the aim was to verify the reversibility of the SPR. During the complete pressure cycle, no discontinuity was observed.

BP5

During experiment "BP5", the studied particle was a gold BP from sample 2, placed on the inferior diamond anvil. The ramp-up began at 0.5 GPa and ended at 7.5 GPa. Afterwards, the gasket hole became asymmetrical (as seen in Fig. (2.6) ) and a PTM leaking forced to abort the experiment. During experiment "BP5", beam spot size and amplitude of oscillation were unknown, making impossible to convert the raw SMS signal to absolute extinction cross-section. Over the course of the ramp-up, the polarization response was repeatedly verified. When the raw SMS signal is directly fitted, peak position and linewidth are still reliable, despite the introduction of a modest systematic error, that has been corrected. This is illustrated in Fig. ( Table 2.2: Brief summary of the characteristics of the various experiments, identified by the columns "BP1", "BP2", "BP3", "BP4", "BP5". The line "SAMPLE" introduces the studied nanostructures and their environment; "SMS" reports the main working parameters and specifies if the optical study was "quantitative" or "qualitative", i.e. if the absolute cross-section in nm 2 was retrieved or not; "PRESSURE" describes the range of pressure investigated; "RESULTS" contains the main effects of the pressure ramp-up and states their eventual reversibility; the line "OTHER EFFECTS" reports additional observations in the SPR position or linewidth while the pressure was changing.

Pressure dependence of the SPR spectral position

As seen in the previous section, the pressure ramp-up produces a redshift in the SPR position which is modeled in the current section. Firstly, complete numerical simulations based on the finite elements modeling (FEM) and taking into account the real geometry of the experiment are presented. Afterwards, a simplified analytical model is developed with the aim of enlightening the main physical origins of the observed evolution.

The experimental SPR energy data of all BPs are summarized in the upper graph of Fig. (2.42), while the lower one directly shows the SPR energy shifts with respect to initial position. This peak translation is not linear with the pressure, and is modeled in the current section. Moreover, until 5 GPa the curves are overlapped, while they separate at higher pressures. In particular, "BP1" present a strong shift at 5 GPa that can not be explained through a simple model. The reasons of these deviations at high pressure will be discussed later (Section (2.2.3) ).

NUMERICAL MODEL

The FEM has been realized in the configurations reported in Fig. (2.43), where an individual Au-BP of volume V , aspect-ratio η and 30 nm short axis, is separated from a diamond substrate by distance d, which could be introduced to take into account the presence of surfactants. The nanoparticle is characterized volumetrically by the compressibility of bulk gold and optically through the pressure dependent dielectric functions of gold, whose real and imaginary parts are Optics of a single metal nano-object under high pressure

1,2 (ω) = D 1,2 + ib 1,2
, where D 1,2 indicates the Drude components and ib 1,2 the interband contributions. The BP environment consists of the diamond substrate and the 4Me:1Et as surrounding liquid, which are respectively modeled by the real dielectric functions d (ω) = n 2 d and P T M (ω) = n 2 P T M , whose imaginary parts are negligible for red and near-IR photons, as explained in Appendix (B) and (C). A polarized and monochromatic Gaussian wave of intensity I 0 , passes through the diamond and illuminates the BP, exciting its longitudinal SPR at the photon frequency ω. Computations in this geometry yield the spatial distribution of the electromagnetic field from which absorption and scattering cross-section are deduced. The absorption component is calculated integrating the differential form of the Joule heating equation over the nano-object volume

σ abs = 1 I 0 Re{ j • E * }dV , (2.31)
where j is the current density and E is the electric field. The scattering part is computed from the flux of the Poynting vector of the scattered field over a virtual sphere surrounding the nanoparticle

σ sca = 1 I 0 1 2 Re{ E sca • H * sca }dS . ( 2.32)
In order to avoid artifacts coming from the boundaries back-reflected radiation, a perfectly-matched layer (PML) has been introduced between two wider spherical boundaries, of radii 200 and 260 nm. At a fixed frequency ω, the computed extinction cross-section is σ ext = σ abs + σ sca .

(2.33)

By varying ω, the entire σ ext spectrum can be calculated. A change in pressure P affects both the volume of the gold nanoparticle and the optical properties of the nanoparticle and environment ( D 1,2 , d and P T M , through relations: Eq. (1.13) of Chapter 1 , Eq. (B.4) of Appendix (B) and Eq. (C.4) of Appendix (C). These effects are taken into account in the numerical simulations, which provide the longitudinal SPR spectrum for each desired pressure.

As illustrated in Fig. (2.44), the FEM simulations with horizontal BPs in contact with the diamond substrate (d = 0 nm) show an excellent agreement with the experimental SPR shift observed in all the BPs, meaning that the considered ingredients (pressure dependence of dielectric functions and volume) are responsible for the observed response. The aspect-ratio η of each nanoparticle has been chosen to reproduce the SPR of the measured cross-section σ at the initial lowest pressure value. This aspect-ratio, coherent with TEM measurements of the initial solution, is kept constant for all the applied pressures. The numerical simulations Optics of a single metal nano-object under high pressure also allows to take into account different configurations, such as the presence of the surfactant layer around the nanoparticle, introducing a distance d between particle and substrate from 0 to 2 nm [START_REF] Pertreux | Surface Plasmon Resonance of an Individual Nano-Object on an Absorbing Substrate: Quantitative Effects of Distance and 3D Orientation[END_REF]. As displayed in Fig. (2.44), considering d = 0 does not change the curvature of the simulated shift. The main consequence is a uniform blueshift (+0.01 eV if d = 2 nm), which can be compensated by slightly increasing η (from 3 to 3.05, consistent with the uncertainty of the morphological characterization), thus recovering a good fit of the low pressure spectrum. The simulation on experiment "BP5" has not been reported as experimental data overlap with the ones of "BP4".

SEMI-ANALYTICAL MODEL

The semi-analytical model is a powerful tool to provide a physical insight into the observed pressured-induced effects. In dipolar approximation, when the SPR lies far from the interband transition threshold, as in the case of an elongated Au-BP, the SPR is spectrally described by a quasi-Lorentzian profiled centered around

ω R = ω p ib 1 + H P T M , ( 2.34)
where

ω p = Ne 2 V m 0 (2.35)
is the plasma frequency of gold. Eq. (2.34) is an extension of Eq. (1.26) and (1.34) of Chapter 1, valid for a sphere and elongated ellipsoid in an homogeneous environment, for an arbitrary nano-object in a generic environment. Here, H is a geometric factor depending on the BP aspect-ratio η and on the configuration considered (shape, size, environment, substrate distance). For an ellipsoidal shape, analytical expressions can be used to calculate H. For the general case, as the present one, numerical FEM simulations have been performed for a BP of aspectratio η deposited on diamond and embedded in PTM (see Fig. (2.43) ). Once the aspect-ratio identified for each BP, its extinction spectrum is computed for different values of P T M . From Eq. (2.34) it follows

(ω R /ω p ) -2 = H P T M + ib 1 (2.36)
which has been exploited to retrieve H, as illustrated in Fig. (2.45).

The following approximations are made in the semi-analytical model presented here: the dipolar regime applies (i.e. the nanoparticle sizes are small with respect to the light wavelengths); the SPR is located far away from the gold interband transition frequencies (ω R ω ib ), being a quasi-Lorentzian with position independent from 2 ; the interband component of the gold real dielectric function has negligible dispersion in frequency ib 1 ≈ ib 1 (ω R ); the nanoparticle has an elongated shape with aspect-ratio η (but it is not an ellipsoid) and H(η) can be numerically determined performing FEM simulations; the product H P T M allows to take into account the impact of the global environment (including both PTM and diamond substrate) on the BP resonance position ω R at P = 0 GPa; only the pressure dependence of the PTM is considered because the pressure changes of the diamond refractive index are 60 times smaller than the PTM ones in the 0-10 GPa range (Appendix (B) and (C) ); the frequency dependence of P T M can be neglected (as shown Fig. (C.2) of Appendix (C)); in the interval 0-10 GPa, the BP volumetric compression is governed by the linear relation

V = V 0 (1 -k Au P ) , (2.37)
where V is the BP volume at the pressure P , V 0 is the BP volume at 0 GPa and k Au = 5.77×10 -3 GPa -1 is the bulk gold isothermal compressibility at 300 K [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

Considering Eq. (2.34), the pressure shift of ω R may be seen as a consequence of the pressure variations of ω p , P T M and ib

1 dω R dP = ∂ω R ∂ω p ∂ω p ∂P + ∂ω R ∂ P T M ∂ P T M ∂P + ∂ω R ∂ ib 1 ∂ ib 1 ∂P (2.38)
which can be developed as follows

dω R dP = ω R 2 k Au - ω R 2 H ω R ω p 2 ∂ P T M ∂P - ω R 2 ω R ω p 2 ∂ ib 1 ∂P . ( 2

.39)

The first term has been rearranged considering ∂ω R /∂ω p = ω R /ω p (from Eq. (2.34)) and ∂ω p /∂P = ∂ω p /∂V ×∂V /∂P = -ω p /(2V )×(-k Au V ), following from the plasma frequency dependence on volume (Eq. (2.35)) and from Eq. (2.36). Transformation of the second and third terms follows from Eq. (2.34).

Considering ω R ∼ 1.55 eV, it results (ω R /ω 0 p ) 2 ∼ 3×10 -2 (ω 0 p being the gold plasma frequency calculated at V 0 : ω 0 p = 9 eV), while H ∼ 10, d P T M /dP ∼ 1×10 -1 GPa -1 and d ib 1 /dP < 4×10 -2 GPa -1 , meaning the last term of Eq. (2.39) is negligible compared to the first two, yielding the non-linear differential equation in

ω R = ω R (P ) dω R dP - k Au 2 ω R + H 2 d P T M dP ω 3 R ω 2 p = 0 , ( 2.40)
where ω p depends on P according to Eq. (2.35) and (2.37). Its solution depends on the value of H, which is determined numerically, as discussed above. Importantly, for both full-numerical (FEM) and semi-analytical modeling, an initial numerical computation is needed to determine aspect ratio η, or equivalently H, by comparison of numerically generated and experimental spectra for a specific pressure. Note that the differential equation (Eq. (2.40) ) can also be obtained starting from ω R general dependency on environment index ( m ) and gold dielectric functions ( D 1,2 and ib 1,2 ) under the same assumptions made here, as detailed in Appendix (D). Eq. (2.40) has been numerically solved using, as boundary condition, the experimental point at the lowest measured pressure for "BP1" and "BP2", and the second lowest measured pressure for "BP3" and "BP4". Results of this simplified approach, reported together with the experimental data in Fig. (2.46), are almost identical to the ones obtained through the complete FEM simulations. This proves that the main effects responsible for the observed SPR shift are the pressure induced modifications of the environment refractive index (especially of the PTM one) and of the metal (real part of its dielectric function).

Optics of a single metal nano-object under high pressure

In order to go further into the physical mechanisms behind the SPR shift, an analytical study of the non-linear Eq. (2.40) is proposed, based on an iterative approach. A small pressure increment from P 0 = 0 GPa to P 1 will shift the resonant frequency from

ω 0 R to ω 1 = ω 0 R +Δω 1 (ω 0 R ), where Δω 1 (ω 0 R ) = ω 0 R P 1 P 0 k Au 2 - H 2 (ω 0 R ) 2 (ω 0 p ) 2 ∂ P T M ∂P dP = ω 0 R k Au 2 ΔP - H 2 (ω 0 R ) 2 (ω 0 p ) 2 Δ 1 P T M , (2.41) with ΔP = P 1 -P 0 Δ 1 P T M = P T M (P 1 ) -P T M (P 0 ) . (2.42)
Note that the pressure dependence of 1/ω 2 p introduces a correction that will be neglected here ( ω 0 p = 9 eV). Eq. (2.41) has been evaluated at the starting frequency ω 0 , and is valid only for small increments in pressure. The new resonant frequency becomes

ω 1 = ω 0 R 1 + k Au 2 ΔP - H 2 (ω 0 R ) 2 ω 2 p Δ 1 P T M . (2.43)
Subsequently, the pressure can be raised from P 1 to P 2 = P 1 +ΔP , producing a new resonance at ω 2 = ω 1 + Δω 2 (ω 1 ) that is obtained integrating at ω 1 fixed. As the dependence of ω 1 (ω 0 R ) on ω 0 R has a known expression (Eq. (2.43) ), ω 2 (ω 1 ) can be expressed as a function of ω 0 R . After N steps, one obtains

ω R (P = N ΔP ) = ω 0 R + lim ΔP →0 N i=1 Δω i = ω 0 R + Δω 1 + ... + Δω N . (2.44)
where the expression of the frequency shift results

Δω i = ω i-1 k Au 2 ΔP - H 2 ω 2 i-1 (ω 0 p ) 2 Δ i P T M , (2.45)
with Δ i P T M = P T M (P i ) -P T M (P i-1 ) .

(2.46)

Neglecting the terms beyond the second order in ΔP (keeping ΔP 2 , ΔP Δ P T M and Δ 2 P T M ) will bring, after N integrals, to

ω N ≈ ω 0 R 1 + k Au 2 N i=1 ΔP -H 0 N i=1 Δ i P T M + H 2 0 N i=1 Δ i P T M i-1 j=1 Δ j P T M + -H 0 k Au 2 ΔP N i=1 (N -1)+2(i-1) Δ i P T M + N (N -1) 2 k Au 2 ΔP 2 ,
(2.47)

A single metal nanoparticle under high pressure 89

where

H 0 = H 2 (ω 0 R ) 2 (ω 0 p ) 2 .
(2.48)

As N is a big number, Eq. (2.47) can be rewritten as

ω N ≈ ω 0 R 1 + k Au 2 N ΔP -H 0 P T M (NP ) -P T M (0) + + H 2 0 N i=1 Δ i P T M P T M ((i -1)P ) -P T M (0) -H 0 k Au 2 ΔP N i=1 (N + 2i)Δ i P T M + 1 2 k Au 2 N ΔP 2 .
(2.49)

Considering the limit of Eq. (2.44) the sums are transformed into integrals, yielding an expression of the resonance frequency at the pressure

P ω R (P ) ≈ ω 0 R 1 + k Au 2 P -H 0 P T M (P ) -P T M (0) + + H 2 0 P 0 P T M (P ) -P T M (0) ∂ P T M ∂P dP -H 0 k Au 2 P 0 P + 2P ∂ P T M ∂P dP + 1 2 k Au 2 P 2 .
(2.50)

Finally, by defying the variation of PTM dielectric function as Δ P T M (P ) = P T M (P ) -P T M (0) , (2.51) the resonance position can be expressed as a function of P and Δ P T M as follows

ω R (P ) ≈ ω 0 R 1 + k Au 2 P -H 0 Δ P T M [P ] + H 2 0 P 0 Δ P T M (P ) ∂Δ P T M ∂P dP + -H 0 k Au 2 P Δ P T M (P ) -H 0 k Au P 0 P ∂Δ P T M ∂P dP + 1 2 k Au 2 P 2 .
(2.52)

In Eq. (2.52), the term +k Au P/2 results from the volumetric compression of gold, which increases the density of conduction electrons, and hence ω R through ω p (numerator of Eq. (2.34), the plasma frequency dependence term). It is always Comparison between simplified analytical model and complete numerical approach in the case of experiment "BP1" (b) and "BP2" (c). Numerical and semi-analytical solutions always have similar shapes. A vertical shift of the order of the experimental incertitude is visible in the case "BP2". This difference comes from errors in the numerical determination of aspect ratio by fitting the lowest pressure σ ext spectrum with FEM model. This uncertaintly translates into an offset in all the SPR energypressure curve of the order of 0.01 eV.

positive and proportional to the pressure, and therefore introduces a blueshift. The element -H 0 Δ P T M (P ) originates from the grow of PTM refractive index with pressure. It is always negative and dominates at lower pressure, where P T M changes quickly. Consequently it provokes a redshift (through the denominator of Eq. (2.34), the environment dependence term). The remaining four terms results from a retroactive process. They are introduced to take into account that, when the resonance is modified by the previous two mechanisms, the frequency ω = ω R at which ∂ω R ∂ P T M and ∂ω R ∂ ib 1 are evaluated changes, modifying in turns the SPR position. As, between these corrective contributions, +H 2 0 P 0 Δ P T M [P ] ∂Δ P T M ∂P dP is dominant and always positive, additional blueshift shows up. Altogether, at lower pressure the PTM influence strongly dominates, since P T M increases fast, while at higher pressure the change of the gold dielectric function becomes more and more important.

In Fig. (2.47a), the resonance position as a function on the pressure defined by Eq. (2.52) is plotted. The contribution from the gold compression (+k Au P/2), from the environment (-H 0 Δ P T M [P ]), and from the feedback terms (sum of the last four terms) are reported separately. Their sum is compared to the full solution of Eq. (2.40). In the lower graphs, the comparison between this solution and the complete numerical model shows a very similar dependence, confirming that the main ingredients at the origin of the SPR shift of a single elongated metal nanoparticle under high pressure are the modifications of the metal dielectric function (the real part of the Drude term) and the environment refractive index. From experiments, other characterization of the SPR can be analyzed, providing insightful views of physical phenomena. In the following, the integrated spectral area of the measured cross-section spectra and their spectral width will be discussed.

Pressure dependence of the SPR area

The spectral area under the experimental BP cross-section at different pressure is discussed here. Only experiments "BP1" and "BP2" provide quantitative values of the optical cross-section of a single BP, which is obtained from experimental data after the calibration procedure. Results are summarized in Fig. (2.48). Even if error bars are large, in the two situations the spectral area slightly increases with pressure, in a nonlinear way. To understand the possible physical origin of this effect, one can analyze the optical response of a model elongated nanoparticle, i.e. a prolate spheroid of volume V , aspect-ratio η, embedded in a medium of dielectric function m (ω). As already mentioned in the previous section, an ellip- soid in an homogeneous environment is well-adapted to mimic the configuration of the horizontal BP seen in Fig. (2.43), where the exact geometry is taken into account through the H parameter. In the frame of the quasi-static approximation, where the optical response is dominated by absorption, Eq. (2.34) applies and the spheroid cross-section σ(ω) becomes a Lorentzian distribution. For light polarized along the longer ellipsoid axis, the area under σ(ω) writes as

A = π(1+H) 2 3/2 m V ω 4 R 2 c ω 2 p = π(1+H) 2 m 0 2 c N e 3/2 m V 2 ω 4 R . (2.53)
Here, the elements affected by the pressure are m , V and ω R . Therefore, for small changes, the relative variation of the area is expected to be where the zeros enlightens the quantities evaluated at standard pressure. One obtains

ΔA A 0 = (A-A 0 ) A 0 ≈ 3 2 Δ m m (0) + 2 ΔV V(0) + 4 Δω R ω R (0) , ( 2
A(P ) = A 0 1 + 3 2 Δ P T M P T M (0) + 4 Δω R ω R (0) -2k Au P , (2.55)
where Δω R (P ) = ω R (P )-ω R (0) is given by the numerical solution of Eq. (2.40) computed in the precedent section for "BP1" and "BP2". Note that at the zero order approximation ( Eq. (2.43) ) the volumetric contribution vanishes, giving

A(P ) ≈ A 0 1 + 3 2 - 2Hω 2 R m (0) ω 2 p Δ m m (0) (2.56)
which is consistent with the fact that the total absorption in a metal nanoparticle is proportional to the number of its gold electrons, that remains constant at different pressures (only their density is modified). Therefore, the main area modifications come, once again, from the change with pressure of the environment, affecting the nanoparticle polarizability. This analysis explains the non-linear increase of the area with pressure, being directly proportional to the pressure dependence of the PTM dielectric function Δ P T M (P ).

Figure 2.49: Fits of the experimental area of "BP1" and "BP2" optical cross-section using Eq. (2.55). The only free parameter is D eq , the ellipsoid equivalent short-axis.

Optics of a single metal nano-object under high pressure

More quantitatively, Eq. (2.55) allows to calculate the spectral area of the ellipsoid cross-section. Considering the values of H and η used in the previous section, the only free parameter is the initial volume of the equivalent ellipsoid: V = π 6 ηD 3 eq . The choice of a good value for D eq , the ellipsoid short-axis, permits to reproduce the experimental areas evolution and also to reproduce their absolute values, as displayed in Fig. (2.49).

Perspectives: SPR linewidth and discontinuities

The SPR linewidth dependence on the pressure has been quantitatively studied for single BP in the experiments "BP1", "BP2", "BP4" and "BP5", all showing a broadening of the order of 10 -15% in the range 0 -10 GPa (see Fig. (2.35) and (2.37) ). The analysis of the SPR spectral linewidth provides useful informations on the electronic processes. The SPR linewidth Γ is the sum of indipendent contributions

Γ(ω R ) = ω 3 R ω 2 p ib 2 (ω R ) + Γ rad (ω R ) + Γ chem + γ e (ω R ) + γ ph (ω R ) + γ s (ω R ) , (2.57)
including an interband (

ω 3 R ω 2 p ib
2 ), a radiative (Γ rad ), a chemical (Γ chem ), an electronelectron (γ e ), an electron-phonon (γ ph ) and an electron-surface scattering (γ s ) term. The interband contribution to damping is here absent ( ib 2 (ω R ) = 0), as the considered spectral region is far from interband transitions (ω R ω ib ).

The radiative term at the frequency ω R scales as

Γ rad = 2 k rad (ω R )V (2.58)
and is important only at large sizes (typically > 30 nm). Previous measurements on lithographed gold nanodisks and gold-silver nanoboxes, as well as discretedipole approximation calculations, show that k rad (ω R ) ∼ n m [START_REF] Hartland | Optical Studies of Dynamics in Noble Metal Nanostructures[END_REF]. Consequently, Γ rad ∼ n m V and its relative variation in 10 GPa should be ΔΓ rad Γ rad ≈ Δnm nm + ΔV V ≈ +22% -6% = +16% .

For surfactant stabilized nanoparticles, the presence of molecules at the matrix interface introduces the so called chemical damping Γ chem . Increasing the pressure could progressively densify the surfactant layer in contact with the nanoparticle, modifying Γ chem . This term is difficult to quantify and model, however it only weakly affects the SPR linewidth in BPs. The main contribution to the linewidth of the SPR optical absorption comes from the conduction band electronic scattering rates, where the absorption of a photon at ω R is assisted by unklapp electronelectron, electron-phonon and electron-surface scattering processes.

The scattering between umklapp electrons at optical frequencies yields [START_REF] Gurzhi | Surface effects in infrared optics[END_REF]

γ e (ω R ) = ω p K B T e ω p 2 1 + ω R 2πK B T e 2 ,
(2.59) that, in the domain

K B T e ω R , becomes γ e (ω R ) ≈ ω p ω R 2πω p 2 .
(2.60)

Consequently, γ e (ω R ) ∼ V 1 2 ω 2 R and its relative variation of γ e (ω R ) in 10 GPa should be Δγe(ω R ) γe(ω R ) = 1 2 ΔV V + 2 Δω R ω R ≈-3%-22%=-25%
. However, the term γ e (ω R ) only weakly contributes to the linewidth of the SPR, as γ e (ω R ) γ ph (ω R ). The two following terms, γ ph (ω R ) and γ s (ω R ), are the dominant ones for small size nanoparticles. Their quantitative pressure dependence is not known, and these first studies may contribute to sheld light on such processes at nanoscale. Even if further investigations will be required for a complete analysis, the following qualitative elements can already be discussed.

The absorption of a photon of energy ω R mediated by electron-phonon scattering generates [START_REF] Tsai | Theoretical model for intravalley and intervalley free-carrier absorption in semiconductor lasers: beyond the classical Drude model[END_REF][START_REF] Mongin | Spectroscopie ultrarapide de nanoparticules métalliques et hybrides[END_REF][START_REF] Fatti | Dynamique électronique femtoseconde dans les systèmes métalliques massifs et confinés[END_REF][START_REF] Del Fatti | Nonequilibrium electron dynamics in noble metals[END_REF][START_REF] Del Fatti | Electron dynamics and surface plasmon resonance nonlinearities in metal nanoparticles[END_REF]

γ ph (ω R ) = G e-ph ω R ∞ 0 √ E E + ω R f F D (E) (1 -f F D (E + ω R )) dE , (2.61)
for parabolic energy dispersion and ω R K B T e . Consequently, γ ph (ω R ) is related to the electron-phonon coupling, G e-ph , which could be affected by the pressure.

Ultrafast pump&probe experiments performed by the team on silver nanoparticles embedded in glass at high pressure have demonstrated that the electron-phonon energy relation time γ ph is weakly modified by the pressure [START_REF] Christofilos | Femtosecond Nonlinear Optical Spectroscopy of the Acoustic Vibration of Metal Nanoparticles Under High Pressure[END_REF]. Although the optical γ ph (ω R ) and the ultrafast γ ph are not the same physical quantity, they are related to G e-ph , and thus, one can exploit this dependence to get an idea of the possible variation in 10 GPa range

Δγ ph (ω R ) γ ph (ω R ) ≤10% .
The electron-surface scattering term writes as [START_REF] Kawabata | Electronic properties of fine metallic particles[END_REF]

γ s (ω R ) = g s (ω R ) v F L ef f . (2.62)
The confinement factor g s (ω R ) has been recently quantitatively investigated at single particle level, both experimentally [START_REF] Baida | Quantitative Determination of the Size Dependence of Surface Plasmon Resonance Damping in Single Ag@SiO2 Nanoparticles[END_REF][START_REF] Baida | Surface plasmon resonance spectroscopy of single surfactant-stabilized gold nanoparticles[END_REF][START_REF] Juvé | Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods[END_REF] and theoretically [START_REF] Lermé | Size Evolution of the Surface Plasmon Resonance Damping in Silver Nanoparticles: Confinement and Dielectric Effects[END_REF][START_REF] Lermé | Size Dependence of the Surface Plasmon Resonance Damping in Metal Nanospheres[END_REF].

It has been correlated with the extent of the electronic spillout tail and a detailed recent study [START_REF] Lermé | Size Evolution of the Surface Plasmon Resonance Damping in Silver Nanoparticles: Confinement and Dielectric Effects[END_REF] has shown that it is proportional to the refractive index of the environment n m (ω R ). No quantitative experimental evidence of this dependence has been proved, and such pressure investigations, where this refractive index can be modified in a controlled way by pressure could bring the first quantitative demonstration of this quantum effect. The effective confinement length L ef f is tied to the nanoparticle size and geometry. In the case of a nanorod

L ef f = √ LD ∼ η 1 2 D ∼ η 1 2 V 1 3
, where L and D are its long and short axis, while η = L/D is the aspect-ratio [START_REF] Juvé | Size-Dependent Surface Plasmon Resonance Broadening in Nonspherical Nanoparticles: Single Gold Nanorods[END_REF]. Considering the volumetric dependence of the Fermi velocity,

v F ∼ V -1 3 , it results γ s ∼ n m η -1 2 V -2 3
and its relative variation in 10 GPa should be Δγs γs

≈ Δnm nm -2 3 ΔV V ≈ +22% + 4% = +26% .
This simple analysis is qualitatively in line with the observed broadening. In fact, the only negative variation comes from γ e (ω R ), which gives a marginal contribution to the total Γ, as [START_REF] Crut | Linear and ultrafast nonlinear plasmonics of single nano-objects[END_REF]. Furthermore, as in the case of SPR shift, the environment matrix change plays here an important role. In order to be more quantitative, the knowledge of the percentage contribution of each term in Eq. (2.57) is needed. This has been previously investigated in the group [START_REF] Lombardi | Surface Plasmon Resonance Properties of Single Elongated Nanoobjects: Gold Nanobipyramids and Nanorods[END_REF] for smaller Au-BP (76 nm×24 nm). The pressure induced modifications of the SPR linewidth therefore write as γ e ≈ 6%Γ, that, in the case ΔΓ chem Γ chem = 0 leads to a total relative variation at 10 GPa between +6 and +12% . As expected, the relevant contribution comes from γ ph and γ s , that would introduce a broadening of ≤ +6% and +6% respectively. Considering that the present Au-BP are slightly larger, one would expect that the weight of Δγ rad γ rad is larger at the expense of Δγs γs , so the global broadening should be little modified. Overall, these consideration agrees well with the variation observed in the experimental linewidth. This open the ways to more studies in order to better understand and quantify the physical mechanisms at the origin of the SPR broadening at different pressures. For instance a single particle pump&probe investigation under high pressure could help to understand the behavior of γ ph (ω R ) as a function of the pressure [START_REF] Christofilos | Femtosecond Nonlinear Optical Spectroscopy of the Acoustic Vibration of Metal Nanoparticles Under High Pressure[END_REF]. Changing the size and aspect-ratios will help to investigate γ s (ω R ) and distinguish its contribution from the electron-phonon bulk one. Moreover, performing experiments on single gold BP with different environments (e.g. changing PTM or using surfactant-free encapsulated in SiO 2 BPs) will possibly shed light on this quantum 2.2 A single metal nanoparticle under high pressure 97 confinement effect and, for the first time, its dependence on the environment.

γe(ω R ) γe(ω R )+γ ph (ω R ) ≈ 10%
γ ph ≈ 56%Γ, γ s ≈ 22%Γ, Γ rad ≈ 2.4%Γ
Another aspect to investigate is the presence of some discontinuities in the optical behaviors ot high pressure (> 5 GPa), as for instance the jump in peak position or linewidth occurring in some experiments. The most notable cases are "BP1" at 5 GPa for the peak position and "BP2" at about 10 GPa for the broadening. This may come from exceeding the elastic limit of the nanoparticle metal that could abruptly modify the particle lattice (changing γ ph (ω R ) ) and aspect-ratio (modifying ω R (η) and γ s (ω R ) ). The gold elastic limit measures only 0.2 GPa in the bulk case [START_REF] Ogata | Ideal shear strain of metals and ceramics[END_REF], but it increases reducing the size [START_REF] Kim | Tensile and compressive behavior of gold and molybdenum single crystals at the nano-scale[END_REF]. The threshold for the investigated Au-BPs is unknown, but plastic deformation could be a source of discontinuity and irreversibly. To shed light on this point, the particles should be deposed on adequate substrates, in order to perform SEM or TEM microscopy before and especially after a complete pressure cycle.

Later on, these experiments will be extended to isolated carbon nanotubes. An in-depth experimental analysis (that will rely on combinations of different techniques at high pressure: SMS, Raman spectroscopy and pump&probe) will give an enhanced understanding of the CNTs properties evolutions under extreme conditions, with special regard to the collapse phenomenology.

Chapter 3

Mechanical properties of a single nano-resonator: damping of acoustic vibrations

The vibrational properties of nano-objects can be experimentally addressed using optics-based methods in the frequency (e.g. Raman spectroscopy) or time (e.g. pump&probe spectroscopy) domains and have been the object of many investigations in the last 20 years. The main outcome of such experiments are specific vibrational modes which are detected as damped oscillations with period T and damping time τ . The way vibrational periods depend on nano-object size, shape, environment and encapsulation, has now been clarified [START_REF] Crut | Acoustic vibrations of metal nano-objects: Time-domain investigations[END_REF]. In particular, it has been demonstrated that they are accurately described by continuum mechanics even for ultrasmall (∼ 1 nm) nanoparticles [START_REF] Juvé | Probing Elasticity at the Nanoscale: Terahertz Acoustic Vibration of Small Metal Nanoparticles[END_REF][START_REF] Sauceda | Vibrational Properties of Metal Nanoparticles: Atomistic Simulation and Comparison with Time-Resolved Investigation[END_REF]. However, many questions still remain open on the damping mechanisms, which are subtler to determine since small structural, morphological and environmental changes provoke dramatical alterations of the vibrational modes lifetime. The greater complexity of their study as compared to that of frequencies has two origins. First, in contrast with vibrational periods which are predominantly determined by intrinsic nano-object properties (e.g., composition, morphology and crystallinity) in most experimental cases [START_REF] Crut | Acoustic vibrations of metal nano-objects: Time-domain investigations[END_REF], damping times are very sensitive to the properties of the nanoobject/environment interface, which determine the efficiency with which acoustic energy is transmitted from the former to the latter. Additionally, damping investigations by ensemble experiments is precluded by the size and shape dispersion of the nanoparticles, meaning that experimenal techniques with single-particle sensitivity have to be used for such studies. In this regard, ensemble studies do not allow a precise investigation of the relaxation processes, due to the unavoidable inhomogeneous effects (dephasing between the vibrations of nanoparticles induced by small differences in their morphology) that they introduce. As illustrated in acoustic vibrations , even studies on single nanoparticles produce an important dispersion in the modes damping times. In fact, while the periods present a weak dispersion (mostly reflecting the size dispersion in the samples used), their lifetimes strongly fluctuate from a particle to another, an effect that was initially ascribed only to large variations of nanoparticle-substrate coupling [START_REF] Burgin | Timeresolved investigation of the acoustic vibration of a single gold nanoprism pair[END_REF][START_REF] Ruijgrok | Damping of acoustic vibrations of single gold nanoparticles optically trapped in water[END_REF][START_REF] Staleva | Vibrational dynamics of silver nanocubes and nanowires studied by single-particle transient absorption spectroscopy[END_REF] (note that the impact of nanoparticle geometry, which constitutes the main finding of this chapter, had not been invoked). In order to shed light on these aspects, and in particular to investigate a possi-

Fundamentals on vibrational properties of single metal nanoparticles 101

ble dependence of damping rate on nano-objects morphology, a systematic study on individual nanodisks produced by electron lithography was carried out. The choice of this kind of geometry relies on the simple shape characterized by two dimensions only (diameter and thickness) and on the relatively large contact with the substrate. Furthermore, lithography avoids the presence of ligand molecules and residual solvent occurring for deposed, chemically synthesized nano-objects. Making use of the experimental pump&probe (P&P) technique, the damping times were successfully correlated with the disks aspect ratio, while simulations based on finite elements method (FEM) helped to clarify the general physical origin of the observed behavior. This chapter opens with some elements of the elasticity theory and a description of the numerical simulation modus operandi. Afterwards, the analyzed samples are described in details and the experimental investigations, as well as their analysis, are illustrated. The experimental results are compared with the outcomes of the numerical modeling and the consequential conclusions are drawn. In order to describe a material deformation, it is convenient to consider the differential form Δ = dl( L, t) Considering a three dimensional coordinate system with axis i = x, y, z, vectors u i , and small displacements, so that r ≈ L ≈ l, the deformation Δ can be expressed through the strain matrix S as

Fundamentals on vibrational properties of single metal nanoparticles

Δ = 3 i,j=1
S i,j dr i dr j , (

where S, in linear approximation, consists in a symmetric tensor

S i,j = ∂u i ∂r j + ∂u j ∂r i
; written in compact form as:

S = ∇ sym u . ( 3.4)
A deformation is produced by some forces which can be of volume ( F ), like the gravity, or of surface ( T ) that are transmitted in the volume through elastic forces.

The traction forces acting on the surface of normal vector n are described introducing the stress matrix T, defined as

T n = T • n . ( 3.5)
The Newton principle allows to relate the force fields ( F , T ) with the displacement field ( u) as follows

ρ ∂ 2 u ∂t 2 = ∇ • T + F , (3.6)
where ρ is the material density. In the regime of linear elastic deformation the Hooke law applies, giving rise to the elastic constitutive equation

T i,j = k,l c ijkl S kl ;
written in compact form as: T = c : S , (

where the stiffness tensor c has 3 4 components, of whom only 21 are independent. 1 For cubic crystals the components of the stress tensor c can be summarized through a 6×6 matrix containing c 11 , c 12 and c 44 . Besides, if there is elastic isotropy (polycrystalline case),

c 11 = c 12 +2c 44 (3.8)
and the material elasticity is described by only two independent elastic constants, e.g. Lamé constants

λ = c 12 μ = c 44 , ( 3.9
)

1 c ijkl = c jikl = c ijlk = c jilk and c ijkl = c klij .

Fundamentals on vibrational properties of single metal nanoparticles 103

connected to the longitudinal (c L ) and transversal (c T ) speed of sound via

c L = λ + 2μ ρ and c T = μ ρ , ( 3.10)
or to the Young modulus (E) and Poisson's ratio (ν) speed of sound via

E = μ 3λ + 2μ λ + μ and ν = λ 2(λ + μ) . ( 3.11)
Combining Eq. (3.4), (3.6) and (3.7) for an elastically isotropic material, in the case F = 0, yields the Navier equation

ρ ∂ 2 u ∂t 2 = (λ + 2μ)∇(∇ • u) -μ∇ × (∇ × u) , (3.12)
which is used in the following to describe the vibrations of metal nanoparticles. Despite neglecting the atomistic character of matter, this law has proved to be valid down to nanoparticles sizes of ∼ 1 nm [START_REF] Juvé | Probing Elasticity at the Nanoscale: Terahertz Acoustic Vibration of Small Metal Nanoparticles[END_REF][START_REF] Sauceda | Vibrational Properties of Metal Nanoparticles: Atomistic Simulation and Comparison with Time-Resolved Investigation[END_REF].

In addition to the elastic properties, a viscous damping can also be phenomenologically taken into account adopting the viscosity tensor q T = c : S + q : ∂S ∂t , (

whose effects consist in the introduction of a frequency-dependent mechanical damping. At room temperature, two important physical mechanisms contributing to this viscous damping are the thermoelastic attenuation (due to the irreversible heat conduction from compressed to rarefied regions through longitudinal waves) and the Akhieser mechanism (lying on the passage of the coherently-excited acoustic wave that interacts with the equilibrium phonon distribution) [START_REF] Auld | Acoustic Fields and Waves in Solids[END_REF]. These viscous effects introduce attenuations with a quadratic frequency dependence α ∼ f2 . However, other dissipation mechanisms may occur, introducing a wide variety in the attenuation-frequency curve. For instance, in polycrystalline materials, randomly oriented grains provoke scattering of acoustic waves. All these sources of attenuation are referred to as intrinsic damping, since they are independent on the particular geometry and are present in bulk systems. On the other hand, the configuration and composition considered could include other sources of relaxation (extrinsic damping). That is the case of the dissipation occurring at the interface between two different materials, introduced later through the infinite film example or when investigating confined nano-objects at nanoscale. acoustic vibrations Eq. (3.4), (3.6) and (3.13), results in a system of three equations and three parameters ( u, S, T). Formally, the acoustic theory is very similar to the electromagnetic one, and by introducing the particle velocity v = ∂ u ∂t and the momentum density p = ρ v, a parallelism can be established with the Maxwell's equations, considering

T ⇐⇒ E S ⇐⇒ D p ⇐⇒ B v ⇐⇒ H .
This comparison leads to the acoustic Poynting theorem: in any volume V with closed surface S 2 , the energy balance is described by

S P • n dS = - V v • ∂ p ∂t dV - V T • ∂S ∂t dV - V v • F dV , ( 3.14)
where the tensor quantity

P = -v • T (3.15)
is called the acoustic Poynting vector. 3Therefore, the total power flow outward through the surface S is the force applied times the velocity of the application point.

When a three dimensional linear elastic solid is subjected to point forces causing local displacements, the produced work is stored in the form of strain energy E (in absence of energy dissipation). The incremental strain energy, dE, for the elementary volume dV can be written

dE = 1 2 T • S dV , ( 3.16)
that, integrated over the entire volume V , yields the total strain energy

E E = 1 2 V T • S dV .
(3.17)

Eigenmodes and Q-factor

The vibrational eigenmodes of a nano-object are the harmonic solutions of Eq. (3.12) with complex frequency f , of the form [START_REF] Crut | Acoustic vibrations of metal nano-objects: Time-domain investigations[END_REF] u( r, t) = u( r) exp i2π ft + c.c.

(3.18)
It is possible to distinguish two important configurations satisfying the boundary conditions of the problem. For nano-objects in vacuum, these correspond to

Fundamentals on vibrational properties of single metal nanoparticles 105

vanishing stress on the nanoparticle surface, yielding undamped vibrations modes, characterized by real frequencies f . For matrix-embedded nano-objects, the displacement field has to obey Navier equation in both the nanoparticle and its environment (where it should correspond to a purely diverging acoustic wave) with continuity of displacement and stress at their interface. Damped modes characterized by complex frequencies ( f ) are obtained in this case, with damping associated to the emission of acoustic waves in the environment.

The frequency f has real part f = 1/T , where T is the oscillation period, and imaginary part α = 1/τ , where τ is the lifetime of the vibration. Acoustic vibration associated to a given mode it thus characterized by this complex frequency ( fn = f n + iα n /(2π) ), the normalized displacement field ( U n ( r)) and its complex amplitude ( Ãn ).

u n ( r, t) = 1 2 Ãn U n ( r) exp (i2πf n t -α n t) + c.c. ( 3.19)
A convenient parameter to characterize the damping of an acoustic mode is its quality factor Q n , defined as

Q n = π f n α n , (3.20)
with α n /π also corresponding to the bandwidth Δf n of the n-th mode in the spectral domain

Q n = f n Δf n . ( 3.21)
Over time, an oscillating object returns to equilibrium thanks to extrinsic (emission of acoustic waves in the environment, see Sections (3.1.2,3.1.3) ) and intrinsic (see discussion in Section Section (3.1) ) damping mechanisms. Since these damping channels are independent, the total quality factor can be decomposed as

1 Q = 1 Q extrinsic + 1 Q intrinsic . ( 3.22)
where Q extrinsic and Q intrinsic are the quality factors associated to extrinsic and intrinsic damping mechanisms, respectively. Q extrinsic is discussed below in two simple cases: a film on an infinite substrate and a nanosphere in homogeneous environment.

Film on substrate: interface contribution to damping

Consider an infinite film of thickness h, perfectly attached to an infinite extended substrate. The geometry is outlined in Fig. (3.3). Analyzing the longitudinal acoustic vibrations waves along the vertical direction ( u(z, t) = u(z, t) z ), Eq. (3.12) reduces to the wave equation

∂ 2 u ∂t 2 = c 2 L ∂ 2 u ∂z 2 . (3.23)
The perfect contact film/substrate translates in two boundary condition at the interface: continuity of the acoustic displacement and acoustic pressure. This implies at the interface

u[F] = u[S] ρ[F] c 2 L [F] ∂u ∂z [F] = ρ[S] c 2 L [S] ∂u ∂z [S] , ( 3.24)
where ρ is the density, and the letters "F" and "S" indicate the film and the substrate, respectively. Assuming a free boundary condition (i.e. vanishing pressure) at z = h, one gets

∂ u ∂z z=h = 0 (3.25)
and imposing propagation through the substrate in the -z direction only (without reflection components) complex frequencies of oscillation ( fn ) are obtained. fn depend on the acoustic impedance of the materials (Z = ρ c L ), and in the

Z F > Z S case it results [104] fn = c L [F] 2 h n + i 2π c L [F] 4π h ln Z F -Z S Z F + Z S with n=1,2,3 ... . ( 3.26)
This results in Q-factors

Q n = 2π 2 n ln Z F -Z S Z F +Z S (3.27)

Fundamentals on vibrational properties of single metal nanoparticles 107

independent from h, the geometrical parameter characterizing this mono-dimensional system.

Spherical nano-objects embedded in an homogeneous environment

The acoustic modes of an elastic homogeneous and isotropic sphere can be obtained from the resolution of Eq. (3.12) in spherical coordinates. They were originally computed by Lamb [START_REF] Lamb | On the vibrations of an elastic sphere[END_REF] in the case of a sphere in vacuum, before being extended to spheres embedded in an elastic homogeneous and isotropic environment [START_REF] Dubrovskiy | Natural vibrations of a spherical inhomogeneity in an elastic medium[END_REF]. In the case of a sphere of radius R in vacuum, undamped modes are obtained with frequencies

f n = c L ξ n 2πR ≈ c L (n+1) 2R with n=0,1,2 ... , (3.28)
where the coefficients ξ n are the solutions of

tan(ξ n ) = ξ n 1 -c L ξn 2c T 2 . (3.29) 108
Mechanical properties of a single nano-resonator: damping of acoustic vibrations

In the case of an elastic homogeneous and isotropic infinite matrix, stress-free boundary has to be replaced by the imposition of a purely diverging acoustic wave in the environment, yielding to the complex fn

fn = c L 2πR Re{ ξn } + i 2π c L R Im{ ξn } ≈ c L (n+1) 2R + i 2π c L R Z s Z m (3.30)
with Z s and Z m the acoustic impedances of the sphere and of the matrix respectively (the approximation of Eq. (3.30) is valid when the values of impedances are not too close.).

Remarkably, this mono-dimensional system defined by the radius R, has Q-factors

Q n = π 2 Z m Z s (n+1) (3.31)
independent from R (periods and damping time being both proportional to R), a result thus similar to the previously considered nanofilm case.

The goal of experiments performed in this thesis will be to investigate more complex systems, where the acoustic damping through the environment could be tuned by modifying the morphology of the nano-object (e.g. its aspect ratio).

Numerical modeling

The acoustic modes can be analytically calculated only for isotropic highly symmetric geometries (sphere or infinite cylinder in a uniform matrix or infinitely extended film attached on substrate). In all the other cases, their numerical computation is required. In this work, the finite-element method (FEM) was adopted to simulate a nanodisk (ND) on an infinite substrate. It has been realized in the experimentally relevant configuration reported in Fig. (3.5), where an individual Au-ND of diameter D, thickness h and aspect ratio η = D/h, is perfectly attached to a sapphire substrate, much thicker than the ND. This translates in two boundary conditions at the gold/sapphire interface: continuity of displacement and stress. The system is characterized by the densities ρ and the Lamé constant of gold and sapphire.

Numerical simulation were performed using a frequency-domain approach consisting in calculating the forced vibrations of the ND induced by a given periodical excitation. Vibrational spectra were then obtained by plotting physically relevant quantities (such as the strain energy stored in the ND) as a function of excitation frequency, leading to resonances each associated to a specific excited vibrational . This feature was exploited in the simulations, where only a section of the system was actually considered, reducing the computational time (b). A PML layer was introduced between two virtual spheres of fixed radii (120 and 160 nm).

mode and yielding its frequency and damping time (see pages 126-130 ). To avoid contamination effects associated to acoustic wave reflection at the domain border, a perfectly matched layer (PML) was introduced, an approach also used for optical simulations. In this work, ND excitation consists of uniform stress with the aim of simulate the effect of a femtosecond pulse laser (namely a uniform dilatation) used in the experiences illustrated later on.

Simulations show the general result that rescaling both the dimensions of the ND (D, h) by a factor M (η being unchanged) will not affect the nature of the modes, but simply reduce by M their complex frequency (the Q-factor remains unchanged, as it depends only on the specific considered mode and on the aspect ratio η). Such rescaling will be exploited later to compare frequencies of the same mode from ND of different sizes.

Experimental measurements on single gold nanodisks

The previous section has introduced two nano-resonators (film, sphere), showing that the Q-factor of their acoustic modes does not depend on the system dimension (thickness, radius). In order to understand whether this is also the case for more complex nanostructures with a geometry characterized by two or more parameters, an in-depth study on cylindrical nano-resonators was carried out. These acoustic vibrations oscillators consist in isolated gold nanodisks (ND), geometrically characterized by their thickness (h) and diameter (D) and coupled with a thick sapphire substrate.

Investigated samples

Two ND samples, referred to as S1 and S2 in the following, were realized via electron lithography (collaboration with F. Banfi in Cattolica univeristy of Brescia and F. Rossella in Pisa university). Their layout is illustrated in Fig. (3.6). The

SAMPLES S1 S2

NOMINAL THICKNESS (nm) 30 15 AFM THICKNESS "h" (nm)

POLARIZATION RESPONSE

Polarization measurements are extremely sensitive to shape anisotropy, allowing to identify and discard elliptical NDs. Indeed σ ext is independent on the laser polarization for a circular symmetric ND, while a longitudinal and a transversal SPR emerge if the symmetry is broken, as in the case of the ellipsoid described in Section (1.3.4) of Chapter 1. A comparison between the response of an elliptical and a circular ND is proposed in Fig. (3.9). The polarization selection is complementary to the SEM screening, since the latter cannot discern between small ellipticity or mis-calibrations. Note that for strong deviations from the circular shape, however, the polarization response may remain constant. For instance, in equilateral nanotriangles σ ext is independent on the polarization [START_REF] Burgin | Time-resolved investigation of the acoustic vibration of a single gold nanoprism pair[END_REF], meaning that non-circular NDs with triangular shape cannot be distinguished from circular ones.

FURTHER APPLICATIONS OF SMS EXPERIMENTS

The linear optical characterization of the NDs by SMS provides other benefits when pump&probe (P&P) experiments are subsequently performed. In P&P spectroscopy, two pulsed laser beams of different colors are focused on the nano-object under investigation. Therefore, the SMS technique helps superposing spatially the two laser beams, by overlapping their SMS images from the same nano-object. Furthermore, the knowledge of the extinction spectrum of NDs allows to select an appropriate probe frequency maximizing the detection efficiency of an acoustic mode in ultrafast experiments. Finally, after a P&P experiment, where a ND has been repeatedly excited, a novel SPR measurement permits to verify the nanoparticle integrity.

Excitation and detection of acoustic modes in a single gold nanodisk by time-resolved spectroscopy

Acoustic vibrations of nano-objects can be detected through spectroscopy techniques, such as Raman and P&P. The latter is the most suitable choice for this work, since allows to study single (higher signal over noise ratio) and relatively large nanoparticles (Raman spectroscopy is well adapted to detect vibrational modes of small size objects, typically less than 20 nm, as larger sizes are associated to lower frequency vibrations difficult to detect [START_REF] Portales | Resonant Raman scattering by quadrupolar vibrations of Ni-Ag core-shell nanoparticles[END_REF]).

TIME-RESOLVED PUMP-PROBE EXPERIMENTS

In optical time-resolved P&P experiments, energy is deposited into a nano-object via a "pump" light pulse and its ultrafast response is monitored via a time-delayed "probe" pulse. The two-colors P&P setup used during this study is sketched in Fig. (3.10). It is based on a Ti:Sa laser source delivering 100 fs pulses at a repetition rate of 80 MHz in the frequency range [690-1040 nm]. The output pulse train is split in two parts to generate the pump and the probe beams, a different wavelength being generated for one of them through second harmonic generation [390-520 nm] or with an optical parameter oscillator [540-700 nm]. In the context of the experiments reported here, the two beams are focused on the single ND using an apochromatic objective, by combining the P&P setup with the SMS microscope. The time delay between the pump and the probe pulses is changed by means of a mechanical delay stage. The high sensitivity detection of the pumpinduced time-dependent relative change in transmission ΔT /T of the probe beam through the sample is achieved by chopping the pump beam at 30 kHz and detecting the induced probe beam transmission change using a lock-in amplifier.

In the context of noble metal nanoparticles, the pump absorption initially gener- ates non-thermal electronic distributions [START_REF] Voisin | Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles[END_REF]. Electron-electron scattering leads to electronic thermalization within a few hundred of fs [START_REF] Voisin | Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles[END_REF][START_REF] Voisin | Size-dependent electron-electron interactions in metal nanoparticles[END_REF], while electronphonon coupling leads to complete nanoparticle thermalization in a ∼ 1 ps timescale [START_REF] Hartland | Optical Studies of Dynamics in Noble Metal Nanostructures[END_REF][START_REF] Arbouet | Electron-phonon scattering in metal clusters[END_REF]. This is followed by the cooling of the excited nanoparticle, i.e. the transfer of energy injected by the pump beam towards the surrounding. This process takes several ps-ns depending on the nanoparticle size, the composition and geometry of the interface nanoparticle/matrix, the thermal resistance at the interface (Kapitza resistance) and on the environment thermal conductivity [START_REF] Crut | Vibrations of spherical core-shell nanoparticles[END_REF][START_REF] Cahill | Nanoscale thermal transport[END_REF][START_REF] Ge | AuPd metal nanoparticles as probes of nanoscale thermal transport in aqueous solution[END_REF].

More precisely, the coupling between hot electrons and phonon modes of the nanoparticle is described by the two-temperature model (TTM), which describes acoustic vibrations the rate of energy exchange between electrons and phonons in the system [START_REF] Hartland | Optical Studies of Dynamics in Noble Metal Nanostructures[END_REF] C e (T e ) dTe dt = -g(T e -T l ) C l dT l dt = g(T e -T l ) , (3.32) where T e and T l are the electronic and lattice temperatures, C e (T e ) = cT e is the temperature dependent electronic volumetric heat capacity, C l is the lattice volumetric heat capacity and g is the electron-phonon coupling constant. Calling T 0 the ambient temperature and ΔT e the increase in the electronic temperature due to the pump pulse, in the low excitation limit (ΔT e T 0 ) the relaxation time is given by τ B = c(T 0 +ΔT e )/g. P&P measurements leads to τ B ∼ 1 ps for bulk gold. At 300 K, C e C l , meaning the initial electronic temperature after laser excitation is much higher than the equilibrium temperature of whole nanoparticle after internal thermalization. The increase in lattice temperature from optical excitation is typically a few K to tens of K. Higher lattice heating is avoided in the following single particle experiments as it may cause nanoparticle melting.

The global cooling of the thermalized nanoparticle at temperature T s = T e = T l > T 0 (and T m being the surrounding matrix temperature) is then ruled by two main mechanisms: the heat transfer across the particle/matrix interface and the heat diffusion in the matrix itself 4 . Therefore, for a nanosphere of radius R embedded in a matrix of thermal diffusivity α m (α m = Λ m /C m considering the environment thermal conductivity Λ m and volumetric heat capacity C m ) one has

∂Ts ∂t = -3G RC l (T s (t) -T m (R, t)) ∂ 2 (rTm) ∂r 2 = 1 αm ∂(rTm) ∂t , ( 3.33)
where T s and T m are the temperatures of the nanosphere and of the matrix and G is the interface thermal conductance (typical order of magnitude 100 MW m -2 K -1 ).

The system can be solved using Laplace transform techniques to determine the rate of cooling of the nanosphere, treating G as an adjustable parameter [START_REF] Hartland | Optical Studies of Dynamics in Noble Metal Nanostructures[END_REF]. In the limiting case where the interface conductance mechanism is slow compared to the heat dissipation in the surrounding, T s decays exponentially with a characteristic time of τ C = C l R/3G. Considering a small gold nanosphere with R = 10 nm and G = 150 MW m -2 K -1 , it results τ C ≈ 60 ps. On the other hand, if heat dissipation in the matrix is the slowest process, then the timescale is given by

τ D = (C l R) 2 /(9C m Λ m )
, that results in τ D ≈ 50 ps for glass [START_REF] Juvé | Cooling dynamics and thermal interface resistance of glassembedded metal nanoparticles[END_REF]. The similar magnitudes show that, generally, both interface resistance and environment diffusion have to be taken into account. However, considering a sapphire matrix and R = 10 nm, the diffusion timescale becomes much smaller, τ D ≈ 1 ps, due to the high diffusivity of sapphire, nanoparticle cooling being then mostly limited by interfacial thermal transfer. Note that, due to the different scaling with nanoparticle size of the relaxation times associated to diffusion and interface, heat diffusion in sapphire becomes again relevant for larger nanoparticles. For instance, a nanosphere with R = 100 nm embedded in sapphire presents τ C ≈ 600 ps and τ D ≈ 100 ps. This thermal relation induces a long-time ΔT /T signal which will be subtracted, as well as the electronic contributes, in order to focus the attention on the acoustic component of the signal investigated in this work.

In this frame, acoustic vibrations are launched via lattice dilation upon heating (heating modifying the equilibrium configuration of the nanoparticle), and thus, acoustic modes with displacement field similar to the particle dilatation can be ). The P&P signal is usually described through several time domain events: particle equilibrium (1); electron heating by a femtosecond pump pulse and thermalization of the electron gas (2); electron-lattice thermalization (3); damped acoustic vibrations (4); cooling [START_REF] Rangel | Band structure of gold from many-body perturbation theory[END_REF]. acoustic vibrations excited. They are damped through intrinsic processes discussed in section (3.1) [START_REF] Pelton | Damping of acoustic vibrations in gold nanoparticles[END_REF][START_REF] Ruijgrok | Damping of acoustic vibrations of single gold nanoparticles optically trapped in water[END_REF][START_REF] Major | Damping of the acoustic vibrations of a suspended gold nanowire in air and water environments[END_REF] and by mechanical energy dissipation towards the environment [START_REF] Del Fatti | Coherent acoustic mode oscillation and damping in silver nanoparticles[END_REF][START_REF] Voisin | Environment effect on the acoustic vibration of metal nanoparticles[END_REF], with a rate dependent on the nanoparticle size and on the composition and geometry of the nanoparticle/matrix interface.

All the mechanisms described above temporally modify the dielectric function of the nanoparticle ( = 1 + i 2 ), affecting the observed probe transmission. In particular, an acoustic vibration corresponds to a specific harmonic displacement of the nanoparticle interatomic distances. This locally induces a periodic variation of that reflects into a periodical SPR shift for metal nanoparticles. When the probe frequency ω pr is chosen in the flanks of the SPR, the spectral oscillations of σ ext induced by a vibration mode can be detected at fixed ω pr , as it manifests with a temporal modulation of the observed probe transmission (its complex frequency f directly reflecting that of the related acoustic mode [START_REF] Del Fatti | Coherent acoustic mode oscillation and damping in silver nanoparticles[END_REF]). An example of the transmission change ΔT /T as a function of the P&P time delay is reported in Fig. (3.11). The experimental signal is firstly characterized by a sharp peak close to the origin that corresponds to the electronic excitation and to the relaxation via electron-electron and election-phonon coupling. This ultrafast response can be well fitted through the convolution between a Gaussian and a decaying exponential, whose lifetime τ B is the characteristic time for the electronphonon scattering. Afterwards, the signal returns globally to the zero because of the cooling mechanisms. In most of the reported cases, this trend is empirically well reproduced by the sum of one (or two) decaying exponential with lifetime τ C (and τ D ). When acoustic vibrations are detected, sinusoidal damped oscillations are superposed to the quasi-decaying exponentials. Globally, the decaying part of the experimental signal can be fitted using

ΔT T = N i=1 A i e -t τ i cos (2πf i t + φ i ) + Be -t τ B + Ce -t τ C + De -t τ D , (3.34)
where A i , τ i , f i , φ i are fit parameters setting out the i-th mode oscillation amplitude, lifetime, frequency and phase; B, C, D, τ B , τ C , τ D are fit parameters describing the decaying exponentials amplitudes and lifetimes.

DATA ANALYSIS

The experimental P&P signal of each investigated ND has been analyzed following the protocol described in Fig. (3.12). In this way, the i-th acoustic mode of a ND is characterized by its frequency f i = 1/T i , attenuation α i = 1/τ i , phase φ 1 and amplitude A i (note that the two former are intrinsic to vibrational modes, while the two later depend on the way they are excited, and, for A i , detected). To count the number of oscillations carried out by a mode before its attenuation, the Q-factor is also considered following the definition of Eq. (3.20): Q i = πf i /α i . Furthermore, for each ND the cooling dynamics has been monitored measuring the lifetime τ C (and, in some cases, τ D if another decaying exponential is needed to properly fit the data).

It might be pointed out that after 1 ns (the maximal P&P delay available in ex-acoustic vibrations periments previously performed in our group on single nanoparticles, including those on the S2 sample) some experimental signals have not relaxed to their initial values (Fig. (3.11) ). This usually does not affect much frequency determination. However, in order to achieve more reliable values of Q i and τ C it was worthwhile to extend the P&P time delay further during this thesis. This was accomplished by extending the delay line maximum range from 1 ns to 3 ns, introducing some back and forward paths that make the probe beam of , where another elliptical ND (with yet same nominal dimensions), shows a dominant frequency close to f 1 , but with Q-factor 40% higher. Therefore, reliable Q estimation is challenging for non-circular NDs. The detection of several vibration modes with close frequencies and similar lifetime generally induces a beating pattern which can be fitted by several damped sinusoids with exchangeable damping times, precluding from a precise attribution of lifetime for each mode. As the primary goal of this investigation was the determination of the Q-factor dependence on the aspect ratio, the subsequent investigations were therefore limited to circular NDs, identified from their quasi-polarization independent σ ext in SMS measurements.

Some representative oscillating P&P components of circular NDs from samples S1 and S2 are presented in Fig. (3.15). As described in detail in its caption, every oscillation trace has been centered on the η of the related ND (vertical axis) and compressed along the time delay axis getting unitary periods (horizontal axis). This normalization allows to visually compare the Q-factors of the long-lasting mode for several η. Since Q is roughly the number of visible oscillations, its typical value is about 15, but an enhancement can be noticed at η = 2.5 (Q ∼ 50), while a second broader and weaker enhancement seems to occur around η = 6 (Q ∼ 30).

Acoustic P&P traces associated to η in the range [3.2 -8.5] are accurately fitted with a single damped vibrational mode, which is referred to as mode1. It is characterized by its frequency f 1 , phase φ 1 , lifetime τ 1 and amplitude A 1 . An example is reported in the graph (a) of Fig. (3.16). f 1 decreases with the ND di- In order to compare the different curves, some changes were made to the original data. Firstly, they have been vertically translated in order to match with the aspect ratio of the related ND (vertical axis), and then colored and rescaled to make them visible. Period T and amplitude A reported beside each curve correspond to the mode with higher Q-factor (identified by two-sinusoidal fits realized on every acoustic trace). The delay time axis of each acoustic trace was normalized over the period T yielding the number of oscillations (horizontal axis). This allows to visually compare the Q-factors of the most long-lasting mode for several aspect ratios η. acoustic vibrations

Mechanical properties of a single nano-resonator: damping of acoustic vibrations

given that its frequency does not change with the diameter and it is in antiphase with mode1 and mode2 (a thickness increase inducing a SPR blueshift, opposite to the redshift induced by a diameter increase).

As already mentioned, frequencies f i and Q-factors Q i are intrinsic to the i-th vibrational mode, and therefore independent on the excitation and detection conditions. This aspect is a key point that allows to compare frequencies and (comparisons with frequencies, Q-factors and amplitudes computed in FEM simulations, reported later in Fig. (3.22), also helped to distinguish the nature of these three modes). A salient result of the experiments reported here is the enhancement of Q 1 and Q 2 around specific ND aspect ratios, with Q 2 presenting a sharp peak near η = 2.5 (max{Q 2 } = 66) and Q 1 a broader one about η = 6 (max{Q 1 } = 31). In order to clarify the nature of the detected modes, to model their frequencies and damping rates and to understand the origin of these Q-factor amplifications, a comparison with FEM simulations were carried out.

COMPARISON WITH NUMERIC MODELING

To better understand the origin of the observed acoustic modes, FEM simulations (Section (3.1.4) ) were realized on single NDs of several η (ranging from 1.4 to 9) and h = 40 nm, in the configuration shown in Fig. (3.5). The ND excitation consists in a uniform and periodic stress at a frequency f (chosen in the interval 0 -25 GHz). For each η and f , the ND strain energy was deduced from the computed displacement field (accordingly with Eq. (3.16) of Section (3.1) ). In Fig. (3.19) the ND strain energy is plotted as a function of η and f (and normalized to 0.5 for visualization purposes). The Lorentzian peaks appearing in the graphs correspond to excitation-frequencies for which the ND reacts with an acoustic eigenmode. The Q-factor of each mode is directly evaluated as defined in Eq. (3.21), considering the frequency over bandwidth ratio Q = f n /Δf n , where f n is the central frequency and Δf n is its linewidth. The computed excitation spectra exhibit vibrational modes at frequencies close to those measured experimentally. For instance, the graph (a) and (b) on the right of Fig. (3.19) focuses on the modes observed in P&P experiments: in graph (a) mode1 exhibit a high Q-factor at η = 6 (with f 1 ≈ 8 GHz); in graph (b) mode1 is crossing mode2 about η = 2.5, which in turn present a maximal Q-factor at η = 2.2 (with f 1 ≈ 16 GHz). In the 10 GHz range, mode3 may also be noticed for aspect ratios between 2 and 2.5 (its computed excitation amplitude becomes negligible for larger aspect ratios).

The frequencies f n are rearranged in a more convenient way on the graph of Fig. (3.20). A visual comparison with the top graph of Fig. (3.18) is now straightforward. Frequencies corresponding to mode1, mode2 and mode3 have been enlightened using gray, red and blue lines respectively. mode1 and mode2 are crossing each other at η = 2.7, where only one frequency is observed. In order to individually understand the nature of mode1 and mode2, their associated displacement fields and averaged energy fluxes were computed for η = 4, where the two modes are spectrally well separated (i.e., they do not significantly hybridize, as seen Fig. (3.19b) ). As shown in the lower panel of Fig. (3.20), mode1 exhibits a radial-like displacement. Its energy flux is mostly vertical and unidirectional and intense at the ND/substrate interface, associated to an efficient energy transfert to the environment (poor Q-factor). mode2 presents a torsional-like displacement. The acoustic Poynting vector shows vortex patterns at the interface, meaning that the emission of acoustic waves in the sapphire substrate can be partly compensated by the reentrance of some of the emitted energy in the ND, suggesting possible higher quality factors for this mode. mode3 has been characterized at η = 2. It is dominated by a thickness deformation and the energy flux is mainly vertical and intense at the interface, accounting for the poor quality factor of this mode.

According to the simulations, a maximal Q-factor enhancement for mode2 occurs at η = 2.2. At this aspect ratio, relative displacement field and energy flux have been computed for mode1 and mode2 and the results are illustrated in Fig. (3.21). At η = 2.2 the modes are close in frequency and a hybridization of their displacement fields takes place. In particular, mode1 presents some torsional deformations characteristic of mode2. On the other hand, mode2 does not exhibit displacement in the region close to the interface, in contrast to what seen in Fig. (3.20). This lack of deformation can be visually obtained superposing the relative displacements fields of mode1 and mode2 from Fig. (3.20). As a results, the vortex patterns present in the energy flux of mode2 are getting further apart from the interface, being higher into the ND. This effect reduces even more the energy flux intensity through the interface, and thus, strongly increases the Q-factor of mode2. It can be therefore concluded that the Q-factor enhancement is mainly caused by a mode hybridization that reduces the energy loss towards the substrate for one of the hybridized modes. This mode hybridization occurs when two modes intersect in frequency. The mode crossing thus requires a 2D system with two different geometrical dimensions (or higher), since in 1D systems all the mode frequencies scale with the size parameter in the same way. Following this argument, the Q-factor enhancement observed in mode1 at η = 6 could be attributed to an hybridization between mode1 and mode3. Indeed, ob-Figure 3.20: In the graph: ND frequencies calculated via FEM. Frequencies corresponding to mode1, mode2 and mode3 have been enlightened using gray, red and blue lines respectively. mode1 and mode2 are crossing each other at η = 2.7. Below: mode1, mode2 and mode3 example of normalized displacements field and energy flux (evaluated at η far away from the modes interception). mode1 is radial-like, mode2 is torsional-like and mode3 is thickness-like.

serving the graph of Fig. (3.20), a prolongation of f 3 would cross f 1 . The weak displacement of mode3 (difficult to detect when other modes are present) together with the fact that f 1 and f 3 have similar slopes at η = 6 would explain why the The comparison between the measured (a) and computed (b) Q-factors, shows a general agreement, although in the experimental case the Q-factors enhancement are lower, and the peak of η = 2.2 is shifted towards η = 2.5 (which could be ascribed to a slight deviation in the effective aspect ratios). The frequencies have similar arrangements, despite the intersection is not clearly visible experimentally. Another difference is that when the experimental Q amplification occurs at η = 2.5, mode2 has higher frequency than mode1, while in the simulations the opposite is expected.

The comparison between the measured (c) and computed (d) normalized squared amplitudes cannot be performed quantitatively, since experimentally the amplitudes depends on the detection configuration. However, observing numerical amplitudes and Q-factors allows to understand that at certain η some modes are not detectable, since both their Q-factor and amplitude are negligible (dark blue). This is the case for mode1 at η <2.

Conclusions and perspectives 135

probe frequencies ω pr could introduce additional fluctuations (the time constant τ C may be influenced by the detection frequency, because, depending on ω pr , the sensibility to the variations of the ND and environment dielectric functions changes [START_REF] Stoll | Advances in femto-nanooptics: ultrafast nonlinearity of metal nanoparticles[END_REF]).

Conclusions and perspectives

By combining the spatial modulation microscopy with the pump&probe utrafast spectroscopy technique, the linear and ultrafast properties of single gold NDs deposited on a sapphire substrate have been investigated, with a focus on their acoustic properties. A campaign of time-resolved measurements on Au-ND of various aspect ratio η has been accomplished. Three different acoustic modes have been detected and called mode1, mode2 and mode3. The former two exhibit a Qfactor enhancement, reaching Q 1 = 66 at η = 2.5 and Q 2 = 31 at η = 6 respectively. Conversely, the analysis of the slow relaxation of the signal shows that the thermal energy transfer at the interface (involving all vibrational modes of the ND) is not influenced by the phenomenon, being independent from η. In order to understand the physical origin of the Q-factors enhancements, numerical simulation based on FEM computations has been achieved. In this way the NDs modes activated from a uniform and periodic stress has been traced and characterized in terms of Q-factor, amplitude, relative displacement field and energy flux. The numerical simulations confirm en enhancement of Q-factor at similar η, but with higher intensity, strongly resulting in maximal values of Q 1 ≥ 2250 and Q 2 ≈ 120. This difference probably originates from the lack of intrinsic damping in the FEM computation and/or from the presence of geometrical imperfections and uneven contacts in the lithographed NDs. The relative displacement field shows that the mode1 is radial-like, mode2 has some torsional components and mode3 has a thickness nature. Typically, their energy flux are maximal at the ND/sapphire interface. When approaching the Q 2 resonance, i.e. η = 2.5, the relative displacement shows an hybridization between mode1 and mode2, reducing the energy losses of mode2 throughout the interface. Therefore, Q-factor enhancements can be obtained by exploiting such mode hybridization that reduces the energy loss towards the substrate. This mode hybridization occur when two modes intersect in frequency. In practice, one can make two modes cross each other by tuning the morphology of the nano-object, as the aspect ratio of the ND. This phenomenon takes place for complex geometries, which involves at least two dimensional parameters, and could be exploited to optimize the geometry for high quality resonator or nanobalances. [START_REF] Chaste | A nanomechanical mass sensor with yoctogram resolution[END_REF] acoustic vibrations Extension of these studies to similar samples with different interfaces, including adhesion layers between NDs and substrate, could allow to improve the ND/substrate contact and investigate configurations with different interface properties. An attempt with chrome adhesion layer NDs was unsuccessful, because its presence broadens the NDs SPRs [START_REF] Debu | Surface plasmon damping effects due to Ti adhesion layer in individual gold nanodisks[END_REF], reducing the P&P signal amplitude of one order of magnitude (compensating with higher pump powers caused NDs irreversible damage) and because at optical frequencies the chrome light absorption is one order of magnitude higher than the gold one, meaning the excitation mechanism is radically different and difficult to compare with the former samples results. In this contest, polymeric adhesion layers should be considered, having the advantage of avoiding SPR broadening [START_REF] Habteyes | Metallic Adhesion Layer Induced Plasmon Damping and Molecular Linker as a Nondamping Alternative[END_REF].

In order to be quantitative, a study on the differences observed between measured and simulated Q-factors is necessary. A possible model system to study Q intrinsic would be a free-standing nano-object, as long nanowires (high aspect ratio) deposed on large holes, where the contact with the substrate occurs only in two remote regions. This kind of configuration has already been experimentally studied via P&P spectroscopy [START_REF] Jean | Direct Observation of Gigahertz Coherent Guided Acoustic Phonons in Free Standing Single Copper Nanowires[END_REF][START_REF] Major | Damping of the acoustic vibrations of a suspended gold nanowire in air and water environments[END_REF] and a Q-factor of the order of 100 was observed. However, in such measurements the pump beam was focused on relatively little fractions of the suspended nanowire, exciting acoustic propagating longitudinal waves that contributed to the acoustic damping. In order to reduce this contribution, the system will be further investigated exciting the entire nanowire, providing values for Q intrinsic and a more complete insight of the damping processes at the nanoscale.

Conclusions and perspectives

In this thesis, the optical and acoustic properties of a single metal nanoparticle have been tuned by altering its environment and morphology.

In a first part, the impact of a pressurized environment on the linear optical response of individual gold bipyramids has been quantitatively studied. In order to follow the evolution of the single-particle extinction spectrum under high pressure, a challenging combination of a spatial modulation spectroscopy (SMS) microscope with a diamond anvil cell has been achieved. The characteristic longitudinal surface plasmon resonance (SPR), dominating the optical response of single gold bipyramids, has been experimentally monitored by measuring its energy position, spectral linewidth and area. Rising the pressure results in nonlinear red-shift, broadening and area increases. The SPR red-shift has been successfully modeled numerically, using the finiteelement method (FEM), and then analytically, showing that this behavior is essentially the result of two phenomena: the change in environment refractive index, which causes a strong nonlinear red-shift, and the metal compression, which translates in a weaker linear blue-shift. The spectral area dependence on pressure has also been explained through a simple model for elongated morphologies and, again, the observed increases are mainly attributed to the environment refractive index changes. The spectral linewidth rise with pressure contains multiple contributions, including modifications of the electronic scattering processes. An important term affected by pressure is the electron-surface scattering contribution which, according to merely theoretical quantum models, is sensitive to the environment refractive index. In that respect, pressure measurements on single nanoparticles can be considered as a first experimental evidence supporting these theories. Precise interpretation of the physical mechanisms at the origin of this phenomenon requires further investigations at different sizes and environments, that will be performed in the future. The high-pressure SMS microscope also opens many perspectives for investigating other nanomaterials at single particle level under extreme conditions (e.g. nanotubes). acoustic vibrations

In the second part of this thesis, the morphology of a single nanoparticle on a substrate has been correlated to its acoustic properties. Specifically, two lithographed samples composed of gold nanodisks of various aspect ratios on a sapphire substrate were produced and characterized. SMS has been consistently exploited to select the most circular nanodisks. The combination of the SMS microscope with an ultrafast pump&probe (P&P) setup allowed to realize time-resolved measurements on single nanodisks. Considering the acoustic components of the ultrafast responses, a few dominant vibrational modes could be optically excited and detected for each single nanodisk, and experimentally characterized providing access to their frequency, damping rate and quality factor (Q-factor). A particular attention was devoted to the dependence of these parameters on the aspect ratio. In the investigated region, three different modes were detected in total, two of them exhibiting a Q-factor enhancement at specific aspect ratios. Acoustic FEM simulations has allowed to precisely identify the three modes, confirming the presence of these Q-factor enhancements. The origin of the phenomenon has been clarified by looking at the displacement and energy flux fields computed in the nanodisk-substrate system. Q-factor enhancements are induced by a mode hybridization that results in a reduction of the energy loss to the environment (through the interface with the substrate). Such hybridization can be achieved by tuning the nanodisk aspect ratio until two acoustic modes cross each other, and can be exploited for applications requiring high Q-factor acoustic resonators. This is not possible in monodimensional systems, such as films or spheres, where all the mode periods identically rescale with the object dimension. Q-factor enhancements expected from FEM simulations occur for similar nanodisks aspect ratios, however they are much larger than the experimental ones. This difference is attributed to intrinsic vibrational damping and possibly to defects in the nanodisk morphology, not considered in computations. For a quantitative comparison with theory and a complete understanding of the physical phenomena at the origin of the acoustic damping, further investigations focused on the characterization of the intrinsic damping are required, and will be developed in the future (e.g. ideally studying quasi-free standing nano-objects). Conversely to acoustic damping rates, the long time thermal components of the ultrafast relaxation were found to be independent from the aspect ratio of this specific system. A complete thermo-optical modeling is currently under development in the group, and more detailed investigations will be performed on the thermal relaxation of nano-objects, including morphological and environment-dependent effects and any possible connection with acoustic relaxation. The optimal choice for truncation ratio depends on the relative importance of spot size, shape and power loss ratio that, for a truncated Gaussian beam, decreases exponentially with T : w L = e -2/T 2 .

(A.7)

Typically, in SMS experiments the required powers are very modest and it is therefore possible to work in the T ≥ 2 regime, where the incoming laser Gaussian beam is strongly truncated, the pupil is uniformly illuminated and the image spot approaches the ultimate focusing limit due to Airy pattern. In reverse, during pump&probe measurements on single nanoparticles the required pump power can be higher and, for certain wavelengths, w L must be kept small (low truncation). However, in these kinds of experiments, reducing the size of the pump beam spot close to the diffraction limit is less critical, allowing a choice of smaller T . The thickness h of a diamond slab contained in a DAC pressure chamber can be studied using the experimental diamond equation of state [START_REF] Edwards | Infrared refractive index of diamond[END_REF] h(P ) = h 0 1 + 4.28 λ(P ) λ 0 -1 Alternatively, the thickness h can be calculated directly from the diamond elastic moduli (experimentally determined using ultrasonics, X-rays and Brillouin scattering). Considering one of the most reliable value, c 11 = 1080.4 ± 0.5 GPa, [START_REF] Nazaré | Properties, Growh and Applications of Diamond[END_REF] 4Me:1Et properties under high pressure

Appendix D SPR optical modifications under high pressure: from numerical to semi-analytical model

Each small isotropic change in pressure dP modifies the dielectric functions of the metal and the environment, as well as the volume of the object. This yields a variation in the BP cross-section σ(ω), where ω = 2πc λ is the photon frequency. Therefore, its peak position, which occurs at the resonant frequency ω R , will undergo a shift induced by these variations

dω R = ∂ω R ∂ m ∂ m ∂P dP + ∂ω R ∂ 1 ∂ D 1 ∂P + ∂ ib 1 ∂P dP + ∂ω R ∂ 2 ∂ D 2 ∂P + ∂ ib 2 ∂P dP , (D.1)
where 1,2 (ω) = D 1,2 + ib 1,2 are respectively the real and imaginary parts of the gold dielectric function which must be evaluated at ω R , D 1,2 indicates the Drude components and ib 1,2 the interband contributions. m (ω) is the environment dielectric function (including both the PTM and the diamond substrate), which is a real number and weakly dispersed in the SPR region (see Appendix (B) and (C)). In order to evaluate the peak position change dω R induced by a pressure variation dP , all the partial derivatives in Eq. (D.1) shall be determined. The term ∂ m /∂P can be calculated using Eq. (C.4) of Appendix (C), which defines the pressure and frequency dependence of the PTM refractive index. The diamond contribution, determined by Eq. (B.6), can be neglected as in the 0-10 GPa range the variation of its refractive index is 60 times smaller. Moreover, the frequency dependence of where k Au = 5.77×10 -3 GPa -1 is the bulk gold isothermal compressibility at 300 K [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. By taking into account the modifications of the plasma frequency ω p , due to a change of the nano-object volume V (ω p ∼ V -0.5 ) and therefore of the conduction electron density. At 1.55 eV For elongated gold BPs the frequencies of resonance (ω R < 1.5 eV) are far from the gold interband transition region (ω ib > 2.5 eV): ω R ω ib . As a consequence, ib 2 (ω R ) = 0 and ib 1 (ω) is weakly dispersed, their derivatives with respect to P will be negligible. To be more quantitative and estimate the order of magnitude of these derivatives one can consider the measurements of Garfinkel et al. [START_REF] Garfinkel | Piezoreflectivity of the Noble Metals[END_REF] on the bulk gold reflectivity subjected to biaxial pressure, or the temperature dependent ellipsometric measurements in bulk gold of Winsemius et al. [START_REF] Winsemius | Splitting of the interband absorption edge in Au: Temperature dependence[END_REF], analyzed by Stoll et al. [START_REF] Stoll | Advances in femto-nanooptics: ultrafast nonlinearity of metal nanoparticles[END_REF], using

V ∂ D 1 ∂V = ω 2 p ω 2 V ∂ D 2 ∂V = γ ω V ∂ D 1 ∂V ≈ 1 10 V ∂ D 1 ∂V . (D.
∂ ib 1,2 ∂P = ∂ ib 1,2 ∂T ∂T ∂V ∂V ∂P = - k Au α Au ∂ ib 1,2 ∂T , (D.7)
where α Au = 4.3×10 -5 K -1 is the bulk gold volumetric thermal expansion coefficient at 0 GPa [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. The two approaches gives similar results and in the domain 1. with its two main contributions due to the pressure-induced changes in the metal (Drude) and environment dielectric functions.

Summary

In this thesis, the properties of a single metal nanoparticle have been tuned by altering its environment and its morphology. Specifically, in the first part of this work, the surface plasmon resonance of individual gold nanobipyramids has been experimentally studied under a pressure-adjustable surrounding, and its evolution physically interpreted through theoretical modeling. In order to access to the optical response under high-pressure at single-particle level, a challenging combination of a spatial modulation spectroscopy (SMS) microscope with a diamond anvil cell has been achieved. In the second part of the thesis, the acoustic vibrations of individual gold nanodisks on sapphire substrate have been experimentally characterized by combining the SMS microscope with an ultrafast pump&probe setup. The dependence of their damping on the disk morphology has enlightened the presence of quality-factor enhancements. Numerical modeling has provided a physical insight for the observed phenomena, showing that mode hybridizations occur at specific aspect ratios, reducing the acoustic energy loss through the nanoobject/environment interface.

Résumé

Dans cette thèse, les propriétés d'une nanoparticule métallique unique ont été contrôlées en modifiant son environnement et sa morphologie. Plus spécifiquement, dans la première partie de ce travail, la résonance plasmon de surface de nanobipyramides d'or individuelles a été étudiée expérimentalement dans un milieu sous pression contrôlable, et son évolution interprétée à l'aide de modélisations théoriques. Afin d'accéder à la réponse optique d'une nanoparticule unique à haute pression, la combinaison d'un microscope de spectroscopie à modulation spatiale (SMS) avec une cellule à enclume de diamant a été réalisée. Dans une seconde partie, les vibrations acoustiques de nanodisques d'or individuels sur un substrat en saphir ont été caractérisées en combinant le microscope SMS avec un dispositif de spectroscopie ultrarapide pompe-sonde. L'étude de leur amortissement en fonction de la morphologie des disques a permis de démontrer d'exaltations des facteurs de qualité mécaniques. Les modélisations numériques ont fourni un aperçu physique du phénomène observé, montrant qu'une hybridation entre modes se produit pour des rapports d'aspect spécifiques, réduisant ainsi les pertes en énergie acoustique à travers l'interface disque/environnement.

Figure 1 . 1 :

 11 Figure 1.1: Schematic of a FCC structure with lattice periodicity a.

5 Figure 1 . 2 :

 512 Figure 1.2: Sketch of the band structure of a metal with a low-dispersed valence band (d band) and a partially filled conduction band (s-p band). Photon absorption by a d-electron may cause a transition towards the s-p band. The electron in the conduction band can realize intraband transitions with the assistance of a phonon (dashed arrow) or another electron.

Figure 1 . 3 :

 13 Figure 1.3: Real (a) and imaginary (b) part of the dielectric function of gold and silver experimentally measured [3].

  (1.2) ). Fig.(1.4) shows the intraband transitions contribution, computed using the Drude model for the gold dielectric function. The agreement is good for large wavelengths, conversely, for photon energies E >2 eV, the interband transitions contribution becomes dominant.

Figure 1 . 4 :

 14 Figure 1.4: Real (a) and imaginary (b) part of the dielectric function of gold. The differences between measured 1,2 (black dots) [3] and the calculated contributions of the Drude model χ D 1,2 (solid lines) dielectric function allows to extract the interband contributions χ ib 1,2 (dashed lines).

Figure 1 . 5 :

 15 Figure 1.5: Nanoparticle illuminated by an incident electromagnetic wave E 0 . Interaction with the particle yields a scattered field E sca and an internal one E int .

1. 3 11 Figure 1 . 6 :

 31116 Figure 1.6: Nanosphere (yellow sphere) of dielectric function under an electric field E 0 in a medium characterized by m . E 0 induces a polarization P NP corresponding to electronic movement (blue sphere).

Figure 1 . 7 :

 17 Figure 1.7: Charge distribution in the case D λ (left) and D ∼ λ (right).

2 13 Figure 1 . 8 : 2 m

 213182 Figure 1.8: (a) Real (solid line) and imaginary (dashed line) part of the silver dielectric function. The intersection between 1 and -2 m (dash-dot) determines the SPR peak position (b), being ω R ≈ 3.3 eV for a silver nanosphere of diameter D = 20 nm in water ( m = 1.33 2 ).

Figure 1 . 9 :

 19 Figure 1.9: (a) Extinction spectrum of a gold nanosphere of diameter 20 nm in water ( m = 1.33 2 ) in quasi-static approximation (D λ). (b) Extinction spectrum of a silver nanosphere of diameter 20 nm in vacuum ( m = 1 2 ), water ( m = 1.33 2 ) and glass ( m = 1.45 2 ) in quasi-static approximation (D λ).

  .[START_REF] Crut | Optical absorption and scattering spectroscopies of single nano-objects[END_REF] where m = ( / m ) 2 , u = kR = 2πn m R/λ, ψ n and ξ n are the Riccati-Bessel and Hankel functions, respectively. The index n corresponds to the multipolar expansion order (dipolar for n = 1, quadripolar for n = 2, etc...). Fig. (1.10) shows a comparison between the extinction spectra of single gold nanospheres computed using 1.3 Optical properties of metal nano-objects the Mie theory and the quasi-static approximation. For small sphere diameters (D < 30 nm and u 1), the dipolar is a good approximation of the Mie theory (Fig. (1.10a) ). In fact, a first order development of Eq. (1.27) leads to Eq. (1.22) of the quasi-static case. Increasing the nanosphere size progressively induces a redshift and broadening of the SPR (Fig. (1.10b,c) ) and the introduction of additional resonances related to quadrupolar modes (Fig. (1.10d) )[START_REF] Billaud | Optical extinction spectroscopy of single silver nanoparticles[END_REF][START_REF] Kreibig | Anomalous frequency and temperature dependance of the optical absorption of small gold particles[END_REF][START_REF] Juvé | Spectroscopie linéaire et ultra-rapide de nanoparticules métalliques : de l'ensemble au nano-objet individuel[END_REF][START_REF] Billaud | Optical extinction spectroscopy of single silver nanoparticles[END_REF].

Figure 1 . 10 :

 110 Figure 1.10: Extinction spectra of a gold nanosphere in water ( m = 1.33 2 ) accordingly to the Mie theory (solid line) and to the quasi-static approximation of Eq. (1.22) (dashed line) for different diameters D.

Optical and electronic properties of metal nano-objects 1 . 3 . 4

 134 Optical response of an ellipsoid and more complex shapesConsider a prolate ellipsoid of semi-axes a, b and c (a > b = c), characterized by a complex dielectric function = 1 + 2 and embedded in a medium of real dielectric constant m , as the one illustrated in Fig. (1.11).

Figure 1 . 11 :

 111 Figure 1.11: Prolate ellipsoid (a > b = c) of dielectric function illuminated by an electromagnetic wave (of components E along a and E ⊥ orthogonal to a) propagating in a medium of dielectric function m .

Fig. ( 1

 1 Fig. (1.12a) shows the extinction spectra of an ellipsoid of aspect ratio η = 2 for different polarization orientations: two separate resonances appear which are referred to as longitudinal and transversal SPR. The first one is induced by an incident electric field ( E ) polarized along the major axis a, while the second one by a field ( E ⊥ ) with orthogonal polarization in respect to a. As illustrated in Fig. (1.12b), increasing the ellipsoid aspect ratio η redshift the longitudinal SPR in respect to the transversal one.

Figure 1 . 12 :

 112 Figure 1.12: (a) Extinction spectra of a prolate Au ellipsoid of aspect ratio η = a/b = 2 and long semi-axis a = 20 nm for incident waves of different polarizations. (b) Extinction spectra of a prolate ellipsoid with long semi-axis a = 20 nm and several aspect ratio for a light polarization along a. All the calculations have been realized in water ( m = 1.33 2 ).

Fig. ( 1 Figure 1 . 13 :

 1113 Figure 1.13: (a) The polarization dependence of the extinction cross-section measured for a gold bipyramid at 1.68 eV is reproduced by a sinusoidal function (Eq. (1.40) ). (b) Its absolute extinction cross-section for light polarized along the long axis of the nano-object is fitted by a Lorentzian function (Eq. (1.36) ).

Figure 1 . 14 :

 114 Figure 1.14: SMS principle: a laser beam is focused on a sample containing nanoparticles. Sample position is modulated along y at frequency f and amplitude δ.

Fig. ( 1

 1 Fig. (1.15a,b,c) presents the results of numerical computations of S f (x 0 , y 0 ) and S 2f (x 0 , y 0 ) supposing a Gaussian intensity profile centered in (0, 0) of form I(x, y) = exp -2 x 2 +y 2 w 2

1. 4 25 Figure 1 . 15 :

 425115 Figure 1.15: (a) Gaussian intensity profile (FWHM =370 nm) and spatially modulated nanoparticle (δ=325 nm and σ ext = 10 3 nm 2 ). SMS signal simulated at f (b) and 2f (c)for 1×1 m 2 scan of a nanoparticle located at (x 0 , y 0 ) = (0, 0). SMS signal profiles along the vibration direction (solid lines) at f (d) and 2f (e). A comparison with the first and second derivatives of the Gaussian beam (dashed lines) shows that a broadening appears when the amplitude of oscillation δ is not negligible as compared to the FWHM.

Figure 1 . 16 :

 116 Figure 1.16: R 2f as a function of the oscillation amplitude δ. Different curves correspond to different focal spot sizes at fixed wavelength λ = 500 nm. Black squares are the normalized experimental S 2f (0, 0) values for a small gold nanoparticle (diameter 30 nm) illuminated by a beam of λ = 500 nm and FWHM = 0.7λ ≈ 350 nm for different modulation amplitudes δ (f = 1500 Hz). Under these conditions the experimental signal is maximized by δ ≈ 350 nm (dotted line).

Figure 2 . 1 :

 21 Figure 2.1: Schematic of a DAC. The object under investigation (blue disk in the center of the inset) is surrounded by a pressure medium, contained by a metallic gasket (yellowish), and heated with an IR laser. The fluorescence (excited by a blue or green laser light) from a small ruby ball (red sphere) located in the unheated part of the pressure chamber is used for pressure measurement.

Figure 2 . 2 :

 22 Figure 2.2: Schematic of the employed DAC. Two Boehler-Almax diamond anvils are supported by a tungsten-carbide seat and separated by a Cu-Be gasket.

Figure 2 . 3 :

 23 Figure 2.3: Boehler-Almax geometry has high conical aperture and low thickness, valuable features to realize a microscopy system. In this picture the diamond tip is still uncut. The reported length are measured in mm.

Figure 2 . 4 :

 24 Figure 2.4: Secondary electron SEM scan realized on the upper diamond culet. Nanoscratches deep a few tens of nanometers are visible.

Figure 2 . 5 :

 25 Figure 2.5: Typical external-internal pressure plot, highlighting a strong hysteresis.

Figure 2 . 6 :

 26 Figure 2.6: Left: microscope image of a Cu-Be gasket indentation after a high-pressureexperiment. The shape of the hole is dangerously asymmetrical. Close to its left edge it is possible to recognize the ruby ball used as pressure probe. Right: a TV screen capture of a culet portion taken during an experiment. The picture outlines the appearance of crystals in the PTM by increasing the pressure. The phenomenon, called vitrification, is caused by an excessive pressure or by the presence of impurities in the PTM. In the reported case an incorrect ratio in the PTM preparation was the origin (1Me:1Et rather than 4Me:1Et), decreasing the solidification pressure from 10.4 to about 8 GPa.

Figure 2 . 7 :

 27 Figure 2.7: "Culet imaging setup" (front view of the setup, the instruments lying on the optical table): the white light is focused on a diamond culet by an objective; the backscattered light is detected by a CCD camera connected to a TV screen.

Figure 2 . 8 :

 28 Figure 2.8: Collages of TV screen captures reproducing the upper culet (on the left)and the lower culet (on the right). Every rectangle correspond to a screen capture. The small grey spots appearing on the surfaces are microdusts or big nanoclusters introduced during the ruby ball placement, the DAC sealing and the spin-coating process. These particles are very stable and can be exploited to orientate on the surfaces. The bright rounded border that covers the upper culet (left) corresponds to the superior part of the gasket reflecting the white light, while the circular shadow that blots out the lower culet (right) represents the bottom of the gasket obstructing the light path.

Figure 2 . 9 :

 29 Figure 2.9: Example of gasket deformation until its breakdown. In the ideal case (no breakdown), the chamber pressure is supposed to increase linearly with the external load (membrane pressure); the diameter should grow slowly beyond 3 GPa; the thickness should decrease linearly. Here all the trends are suddenly interrupted around 6 GPa.

Figure 2 . 10 :

 210 Figure 2.10: "Ruby luminescence setup": a monochromatic green laser is focused on a ruby ball placed on a diamond culet; the yielded fluorescence (red line), filtered from the backscattered green light by a long-pass filter, is analyzed with a spectrometer. Translating a pair of mirrors allows to quickly recovered the "culet imaging setup" of Fig. (2.7).

  calibrated the wavelength shift through the Decker equation of state of the NaCl up to 20 GPa and found a linear pressure dependency

Figure 2 . 11 :

 211 Figure 2.11: Typical response of a ruby ball observed in the setup. The process is exploited for pressure measurements using Eq. (2.2). On the top axis the frequency shift from the 532 nm carrier is indicated.

Figure 2 . 13 :

 213 Figure 2.13: Longitudinal displacement of image by plane-parallel plate.

  (2.10) yields the "transmission SMS setup" of Fig. (2.14) that permits to implement the SMS technique and characterize the focused beam.

Figure 2 . 14 :

 214 Figure 2.14: "Transmission SMS setup": a tunable laser beam is polarized and then focused into the pressure chamber through a pre-compensated objective; the DAC is forced to oscillate by a Y piezo shaker and the outcoming modulated power is detected using a lock-in amplifier. Moving a few mirrors on magnetic supports allows to quickly switch between the "transmission SMS setup", the "ruby luminescence setup" of Fig. (2.10) and the "culet imaging setup" of Fig. (2.7).

Figure 2 . 15 :

 215 Figure 2.15: Qualitative beam spot comparison performed using a CCD camera. Left: the standard objective introduces serious aberrations. Right: the pre-compensated objective enables to reach diffraction limit.

  (2.14) was fully operational. As explained in Section (1.4.2), a proper SMS image realized using a particle much smaller than the beam size allows a precise beam characterization. When this condition is respected the SMS image profile along the oscillation movement (Y) tends to the second derivative of the beam profile for small amplitudes (the LIA detection was always realized at the second harmonic), while across the orthogonal direction (X) it simply coincides with the beam profile. In order to produce an SMS image, a nano-bipyramid was deposed on the culet of the superior diamond anvil and the laser beam (with λ = 800 nm) focused on it. As shown in Fig.(2.16), the SMS image at the focal point may present asymmetries and rings. This is a consequence of the spot displacement and defocussing occurring when a plane-parallel plate is tilted relative to the optical axis. This effect can be minimized rotating the DAC in its holder with the aim of yielding a symmetric 3-lobe SMS image, as illustrated in Fig.(2.16).

Figure 2 . 16 :

 216 Figure 2.16: DAC tilt correction made by optimization of the SMS image map.

Figure 2 .

 2 Figure 2.18: (a) For a specific pressure, the FWHM measured via SMS (squares) shows a linear dependence with the wavelength. (b) The experimental K exp dependence on pressure (triangles) is well fitted by a parabola (solid red curve). Its increment in the 0-10 GPa range corresponds to a FWHM widening of 20-30 nm.

Fig

  Fig. (C.2) of Appendix (C)). Conversely, the refractive index of the diamond n d decreases linearly with the pressure, as shown in Fig. (B.2) of Appendix (B)). Furthermore, even the diamond thickness h decreases linearly when the pressure is incremented, as displayed in Fig. (B.3) of Appendix (B)). For a variation of 10 GPa the changes in h are negligible (-12 m over 1.385 mm), while the variation of n d (-5×10 -3 ) may have a significant impact due to the important thickness of the anvil. Overall, during the pressure ramp-up, the light path is modified (n P T M increases, n d and h decrease), eventually inducing non-corrected aberrations. The image spot FWHM has been measured at several wavelengths in air and in 4Me:1Et during an entire pressure experiment. The phenomenological relation between K exp and the pressure is quasi-parabolic as shown in Fig. (2.18b). It can be expressed

Figure 2 . 19 :

 219 Figure 2.19: SMS signal contamination induced from the gasket borders (outer green ring in the upper cartography) and from a ruby deposed on the opposite culet (circular blur in the lower cartography). The simplistic diagram explains the origin of the problem.

Figure 2 . 20 :

 220 Figure 2.20: Maximum amplitude of oscillation as a function of the frequency, accordingly to the push-pull limit of the Y piezo shaker. Amplitudes can be reached only for f <1.5 kHz, region where they are comparable to the beam spot size.

  (

  2.21), it is based on an optical mirror mounted on the mechanical component to characterize and enables to detect the absolute movements along the Y direction of the DAC holder Y holder (t) = A holder sin(2πf t + φ holder ) , (2.16) of the XYZ piezo stage

17

 17

  )

Figure 2 .

 2 Figure 2.21: "Interferometer setup": a monochromatic green laser beam is separated in two arms using a beam splitter; the first line is reflected by a mirror that is fixed with respect to the laboratory reference system; the second line reflected by a small silicon wafer (acting as a light mirror) glued on the DAC holder, on the XYZ piezo stage, or on the micrometric stage. The two beam paths are recombined and sent to a silicon photodiode connected to an oscilloscope, that allows to study their interference as a function of time. Translating a few mirrors allows to quickly recover the "Transmission SMS setup" of Fig. (2.14), the "ruby luminescence setup" of Fig. (2.10) or the "culet imaging setup" of Fig. (2.7).

Fig. ( 2 Figure 2 . 23 :

 2223 Figure 2.23: Example of signals monitored on the oscilloscope for interferometric analysis. The lower black curve is the voltage introduced by the LIA that drives the Y shaker; in red its sinusoidal fit; the upper black curve is the voltage proportional to the interference induced by the DAC holder oscillation; in red the fit based on Eq. (2.19). This fit curve slightly deviates from the experimental one, as the photodiode acts like a low-pass frequency filter, reducing the amplitudes of the signal when it varies faster. However, it was verified that this signal distortion does not affects the relevant fit parameters (amplitude and phase).

4 Figure 2 . 24 :

 4224 Figure 2.24: Main results from the system experimental characterization. Green triangles correspond to the motion of the XYZ piezo stage (measured through its integrate capacitative sensor); red squares indicate the absolute motion of the DAC holder (measured by the interferometer); black dots are the relative motion of the Y piezo shaker (sensor); blue reversed triangles represent the absolute motion of the Y piezo shaker (obtained from Y Y = Y Y +Y XY Z ). Note that the XYZ piezo stage is approximately in phase with the driving force in the intervals [0 Hz, "R 3 "] and ["R 1 ", "R 2 "], while they are in antiphase in the regions ["R 3 ", "R 1 "] and ["R 2 ", 2000 Hz].

Figure 2 . 25 :

 225 Figure 2.25:Forced two-spring model adopted to understand the dephasing behavior of the system components: a sinusoidal force of amplitude F 1 and frequency f is applied to M 1 to simulate the LIA driving force applied on the Y shaker. This mass is attached to the wall (replacing the XYZ piezo stage) through a spring of constant k 1 and dissipative parameter β 1 . On the other side, it is connected to a second oscillator (simulating the DAC and its holder) characterized by M 2 , k 2 and β 2 .

  (2.26) are plotted experimental amplitudes and dephasings of Y holder = Y holder -Y XY Z and Y Y = Y Y -Y XY Z and their fits, realized using Eq. (2.22), (2.24) and (2.23). Imposing k 1 = 4.5×10 7 N/m (so that the first spring constant is close to the nominal stiffness per unit length of the Y piezo shaker) and fitting the amplitudes and the dephasing concomitantly determines all the model parameters

Figure 2 . 26 :Figure 2 . 27 :

 226227 Figure 2.26: System simulation with the forced two-spring model and comparison to experiments. Black dots represent the movement (amplitude and phase) of the Y shaker relative to the XYZ piezo stage (Y Y ) (see also Fig. (2.24) ). Black lines are amplitude and phase (A 1 and φ 1 ) derived by the fit of all the points with the forced two-spring model. Red diamonds correspond to motion of the DAC holder relative to the XYZ piezo stage (Y holder = Y holder -Y XY Z ). Red line is amplitude A 2 from the fit. In these fits, k 1 was taken equal to the Y shaker stiffness, k 1 = 4.5×10 7 N/m; F 1 was chosen so that at low frequency the oscillation amplitudes where of 1 m/V, F 1 = 40 N; the equivalent sum of masses of M 1 + M 2 should be close to the masses sum of DAC, holder and Y shaker (0.76 Kg). Fitting yields M 1 = 0.44 Kg and M 2 = 0.25 Kg; the other parameters were completely determined from the fits, resulting k 2 = 7×10 6 N/m, β 1 = 300 Kg/s and β 2 = 60 Kg/s.

Figure 2 . 28 :

 228 Figure 2.28: Extinction cross-section of individual gold BPs deposed on a silica substrate. The red curve is associated to a BP from sample A, constituted by sharp BPs with aspect-ratio η ∼ 4. It exhibits a quasi-Lorentzian longitudinal SPR centered at 1.64 eV and a smaller peak at 2.15 eV related to the presence of sharp tips. The blue curve corresponds to a BP extracted from sample B, composed by rounded BPs with η ∼ 2. It shows a quasi-Lorentzian longitudinal SPR located at higher energy 2.04 eV. Interband transitions increase the extinction cross-section σ beyond 2.5 eV.

Figure 2 . 29 :

 229 Figure 2.29: TEM images realized on standard TEM grids. at the left is presented the sample 1 (very low size and shape dispersion); at the right sample 2.

Figure 2 . 30 :

 230 Figure 2.30: SEM images realized on the culet of the superior anvil: at the top left the diamond anvil glued on its tungsten-carbide seat; at the top right the culet plateau where the BPs are deposed via spin-coating; at the bottom left a BP from sample 1; at the bottom right a cluster from sample 2.

Figure 2 . 31 :

 231 Figure 2.31: Extinction cross-section σ of "BP1" placed on the upper culet and surrounded by air. Top: the longitudinal SPR spectrum is fitted through a Lorentzian function centered at the photon energy ω 0 , with linewidth Γ and area A (Eq. (1.36) of Section (1.3.4) ). Bottom: the polarization response at λ = 800 nm is fitted using a sinusoidal function with maximum σ(θ 0 ) at θ 0 and passing through zero at θ 0 +90 • (Eq. (1.40) of Section (1.3.4) with σ ⊥ = 0).

Figure 2 . 32 :

 232 Figure 2.32: The polarization response of "BP1" remained unchanged during the experiment.

Figure 2 . 33 :

 233 Figure 2.33:In the upper graph some of the measured extinction spectra of "BP1" are presented as well as their Lorentzian fits. In the lower graph, all experimental curves are shown, vertically shifted adding a constant proportional to the related pressure (the offsets are indicated by white lines below each curve and intersect the right axis at the corresponding pressure). The strong redshift experienced from 0 to 0.055 GPa is mainly due to the introduction of PTM. The reversibility of the process has been verified at 7.37 GPa (dashed red line).

Figure 2 . 34 : 2 . 2 A single metal nanoparticle under high pressure 71 Figure 2 . 35 :

 2342271235 Figure 2.34: Extinction spectra during ramp-up and ramp-down show perfect reversibility at 7.4 GPa for "BP1".

Figure 2 . 36 :

 236 Figure 2.36: The polarization response of "BP2" remained unchanged during the experiment.

Figure 2 . 37 :

 237 Figure 2.37: Evolution of the Lorentzian fit through the complete pressure cycle of "BP2".

( 2 .Figure 2 . 38 :

 2238 Figure 2.38: Double-Lorentzian fit curves allow to reproduce the SPR shape of "BP3" cluster until 4 GPa of the ramp-down, where only one peak remains.

Figure 2 . 39 :

 239 Figure 2.39: Polarization responses for "BP3" cluster measured at 6.2 GPa of the rampdown for two different wavelengths.

Figure 2 . 40 :

 240 Figure 2.40: Evolution of the double-Lorentzian fit through the complete pressure cycle of "BP3". Vertical error bars become larger starting from 3 GPa of the ramp-down because the second peak is very weak and its characterization becomes uncertain.

Figure 2 . 42 :

 242 Figure 2.42: Upper graph: SPR energy measured from ramp-up experiments realized on individual gold BP placed on the diamond substrate. Lower graph: SPR energy shift induced by pressure increase.

Figure 2 . 43 :

 243 Figure 2.43: Configuration adopted for the FEM.

Figure 2 . 44 :

 244 Figure 2.44: Reproduction of the experimental data through the FEM approach. Experimental data are points with error bars. Lines identify the SPR peak position obtained from the FEM realized on an horizontal BP of aspect ratio η: the violet solid line corresponds to a perfect contact between BP and diamond (d = 0 nm); gray dotted lines correspond to the configuration where the BP is separated from the diamond because of the surfactant presence (d = 2 nm).

Figure 2 . 45 :

 245 Figure 2.45: Determination of the factor H through linear fits of the numerically computed SPR position of BP in different conditions: in each curve, related to a particular aspect-ratio η, the slopes are equal to H.

Figure 2 . 46 :

 246 Figure 2.46: Reproduction of experimental data through the numerical resolution of differential equation Eq. (2.40), based on the Mie theory and the Drude model. These results are basically identical to the ones obtained through the FEM approach with d = 0 nm (Fig. (2.44) ).

Figure 2 .

 2 Figure 2.47: (a) The three components of Eq. (2.52) are separately plotted; their sum (dashed green curve) is compared to the full solution of Eq. (2.40) (solid black curve).Comparison between simplified analytical model and complete numerical approach in the case of experiment "BP1" (b) and "BP2" (c). Numerical and semi-analytical solutions always have similar shapes. A vertical shift of the order of the experimental incertitude is visible in the case "BP2". This difference comes from errors in the numerical determination of aspect ratio by fitting the lowest pressure σ ext spectrum with FEM model. This uncertaintly translates into an offset in all the SPR energypressure curve of the order of 0.01 eV.

Figure 2 . 48 :

 248 Figure 2.48: Evolution of the spectral area of the optical single nanoparticle crosssection as a function of the pressure. Note that for "BP1" a discontinuity in the SPR position occurs about 5 GPa.

Figure 3 . 1 :

 31 Figure 3.1: Statistics on periods T and damping times τ of the modes detected in previous P&P experiments on single nanoparticles [99, 100]. For each graph, the investigated particle geometry is displayed. Lithographed or deposited on a glass substrate (a) nanotriangles (aspect ratio 2.4); (b) spheres (radius 40 nm) and (c)(d) rods (axes 25×60 nm) have been studied. All the reported oscillations are associated to a breathing mode, apart from the last one (d) which is an extensional mode. Each data point correspond to a different nano-object of a given sample. Green dashed lines (b)(c) are the results of continuum mechanics computations for spheres and infinite cylinders in the indicated homogeneous environments [102]. These curves enlighten the important dispersion in τ .

Fig. ( 3

 3 Fig.(3.1), even studies on single nanoparticles produce an important dispersion in the modes damping times. In fact, while the periods present a weak dispersion (mostly reflecting the size dispersion in the samples used), their lifetimes strongly fluctuate from a particle to another, an effect that was initially ascribed only to large variations of nanoparticle-substrate coupling[START_REF] Burgin | Timeresolved investigation of the acoustic vibration of a single gold nanoprism pair[END_REF][START_REF] Ruijgrok | Damping of acoustic vibrations of single gold nanoparticles optically trapped in water[END_REF][START_REF] Staleva | Vibrational dynamics of silver nanocubes and nanowires studied by single-particle transient absorption spectroscopy[END_REF] (note that the impact of nanoparticle geometry, which constitutes the main finding of this chapter, had not been invoked). In order to shed light on these aspects, and in particular to investigate a possi-

AFigure 3 . 2 :

 32 Figure 3.2: Left: idealization of a solid as a web of points. Right: local displacement of an element.

Figure 3 . 3 :

 33 Figure 3.3: Slab of thickness h is spread over a semi-infinite substrate.

  The solutions of the Navier equation in spherical coordinates consist in two types of displacements: spheroidal, characterized by a local variation of the volume (i.e. with ∇ u = 0) and torsional, where the volume does not change (i.e. with ∇ u = 0). The radial and angular variations of displacement are respectively described by Bessel spherical functions j 1 (kr) and y 1 (kr) (for the radial component) and spherical harmonics Y m l (θ, φ) (for the angular component), with angular momentum l ≥ 0 and azimuthal number m ∈ [-l, l]. In time-resolved experiments, only the isotropic l = 0 modes, associated to purely radial displacement field are detected. A schematic of the mode displacements is displayed in Fig. (3.4).

Figure 3 . 4 :

 34 Figure 3.4: Schematic of the vibrational modes of a sphere: the blue arrows indicate the superficial deformation occurring in a torsional and radial mode.

3. 2 Experimental measurements on single gold nanodisks 109 Figure 3 . 5 :

 210935 Figure 3.5: Configuration adopted for the FEM simulations. The geometry of the problem has a circular symmetry (a). This feature was exploited in the simulations, where only a section of the system was actually considered, reducing the computational time (b). A PML layer was introduced between two virtual spheres of fixed radii (120 and 160 nm).

Figure 3 . 6 :

 36 Figure 3.6: Left: schematics of the studied samples (not in scale), consisting in NDs with same thickness and different diameters. Right: composition of the samples: Au-ND of thickness h and diameter D are attached to a sapphire substrate.

Table 3 . 1 :

 31 Geometry of the studied samples: the NDs aspect ratio η has been obtained using the thickness h, measured via AFM, and the diameter D, target of the electron lithography.

Figure 3 . 7 :

 37 Figure 3.7: AFM measurements performed on sample S2 on two ND with very different diameters and locations. The thickness h remains equal in different regions. Note that precise D measurements by AFM are hampered by tip convolution effects.

3. 2 . 2

 22 Optical characterization of a single gold nanodisk ND morphology characterizations were complemented by a linear optical characterization of single ND. Using the spatial modulation spectroscopy (SMS) technique, extensively described in Section (1.4.2) of Chapter 1, the absolute extinction spectrum of individual NDs, as well as the polarization dependence of extinction were measured. acoustic vibrations EXTINCTION SPECTRUM The extinction cross-section σ ext spectrum, dominated by the surface plasmon resonance (SPR) of the gold nano-objects, provides information on η, as discussed in detail in Fig. (3.8).

Figure 3 . 8 :

 38 Figure 3.8: Examples of extinction spectra obtained on NDs from S1, all exhibiting clear SPR peaks. The experimental cross-section σ (dots) are well fitted by Lorentzian distributions (solid lines).Increasing the diameter yields a redshift of the SPR peak, whose energies are plotted in the second graph (dots). These data were compared with the optical FEM calculations including the sapphire substrate (solid lines). FEM calculations were performed for several h values, confirming that the (40 ± 5) nm value determined by AFM also leads to a good reproduction of the measured SPR positions.

Figure 3 . 9 :

 39 Figure 3.9: SEM-SMS characterization performed on two different ND from sample S2, with same nominal thickness h and diameter D. Left: the SEM profiles show that the diameter is slightly larger in the 59 • direction; the SPR is clearly red-shifted and enhanced when measured along 59• (continuous lines are Lorentzian data fits), confirming the ND ellipticity suggested by SEM observations. Right: similar characterization with another more circular ND, that was identified as a suitable candidate for subsequent acoustic experiments. In the lower graphs, the angle-dependent and constant response as a function of the light polarization direction measured at a fixed wavelength confirm the two different geometries.

Figure 3 . 10 :

 310 Figure 3.10: Setup for P&P experiments on single nano-objects.In the configuration illustrated here, pump and probe beams are respectively generated by frequency doubling (blue beam) and optical parametric oscillator (green beam) from the red/infrared output of the tunable Ti:Sa laser source. After filtering the pump, the ultrafast response is characterized by monitoring the relative transient change of probe beam transmission ΔT /T , which is a function of the delay between the pump and probe pulses (controlled through the mechanical delay stage on the probe beam path). In order to achieve high sensitivity, the pump beam is mechanically modulated (chopper) at the frequency f , and the pump-induced probe beam power fluctuations are detected via lock-in amplification at the same frequency.

Figure 3 . 11 :

 311 Figure 3.11: Ultrafast response of the asymmetrical ND presented on the left hand side of Fig. (3.9). The P&P signal is usually described through several time domain events: particle equilibrium (1); electron heating by a femtosecond pump pulse and thermalization of the electron gas (2); electron-lattice thermalization (3); damped acoustic vibrations (4); cooling (5).

Figure 3 . 12 :

 312 Figure 3.12: Illustration of the analysis of P&P experiments realized on the NDs. In this example, the ultrafast response of Fig. (3.11), related to the asymmetrical ND characterized on the left hand side of Fig. (3.9), is the starting point. Upper panel: by subtracting the electronic and thermal exponential decays only the acoustic component remains (black) and a Fourier transform is realized to help identifying the modes frequencies (green). Central panel: oscillating part of the P&P signal is fitted using three damped sinusoids (red); above this fit, the three sinusoids are presented separately; in the table are reported the useful parameters of every mode (respectively: period, real frequency, lifetime, attenuation, Q-factor, amplitude, phase). Lower panel: the total P&P signal (black) is accurately reproduced by Eq. (3.34), using the sum of three damped sinusoidal function and two decaying exponentials (red), whose amplitudes and lifetimes are summarized in the table besides.

Fig. ( 3 . 10 )

 310 passing three times through the delay stage. This improvement has been exploited to study NDs from sample S1.EXPERIMENTAL RESULTSShapeeffects have first been investigated by measuring the acoustic response of circular and elliptical NDs, as shown in Fig. (3.13). As illustrated in Fig. (3.13a), P&P experiments on the circular ND of Fig. (3.9) enable the detection of a single acoustic mode. Similar observations were made for several NDs, showing that circular symmetry breaking leads to the replacement of the dominant mode by two or three modes with close frequencies. These results are in line with the free ND eigenmodes evolution with ellipticity deduced from the FEM study presented in in Fig. (3.14). In the specific case of the NDs from Fig. (3.12) and (3.13a), characterized by D = 70 nm and h = 19 nm, the circular ND presents a dominant mode with frequency f 1 = 23 GHz and the elliptical one shows a mode at same frequency f 1 (with similar damping) and two different modes with frequencies within the range f 1 ±3 GHz. Typically the frequencies exhibited by the circular NDs are detected also in elliptical ones, with very small deviations. However, their damping times and Q-factors present large fluctuations, as illustrated in Fig. (3.13b)

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: (a) Analysis of P&P experiments performed on the circular ND described in Fig. (3.[START_REF] Lebedev | Optical extinction by spherical particles in an absorbing medium: Application to composite absorbing films[END_REF]. Top panel: the data (black) are fitted using a simple damped sinusoid (red), whose parameters are reported in the table below the Fourier transform graph. Bottom panel: the signal (black) descent is reproduced using the sum of two decaying exponentials (red), whom amplitudes and lifetime are summarize in the table. (b) Analysis of acoustic trace obtained from P&P experiment performed on another elliptical ND of similar size. The data (black) are fitted using three damped sinusoid (red), whose parameters are reported in the table below the Fourier transform graph. This enlightens that anisotropy may affect the estimated Q-factor of similar modes (43 and 45 ps).

3. 2 Experimental measurements on single gold nanodisks 123 Figure 3 . 15 :

 2123315 Figure 3.15: Representative acoustic P&P traces of circular NDs from samples S1 (solid lines) and S2 (dotted lines).In order to compare the different curves, some changes were made to the original data. Firstly, they have been vertically translated in order to match with the aspect ratio of the related ND (vertical axis), and then colored and rescaled to make them visible. Period T and amplitude A reported beside each curve correspond to the mode with higher Q-factor (identified by two-sinusoidal fits realized on every acoustic trace). The delay time axis of each acoustic trace was normalized over the period T yielding the number of oscillations (horizontal axis). This allows to visually compare the Q-factors of the most long-lasting mode for several aspect ratios η.

Figure 3 . 16 :Figure 3 . 17 :

 316317 Figure 3.16: Four P&P acoustic traces (green and black) are fitted using one or two damped sinusoid (red). Each graph contains the Fourier transform and a table with the fitting parameters. The first table line describes the dominant mode of the Fourier transform. Overall, three different modes (i = 1, 2, 3) are detectable. Typically mode1 and mode2 are in phase φ 1 ∼ φ 2 ∼ π/2 (except when η = 2.5, i.e. Q is enhanced), while mode3 is always in antiphase φ 3 ∼ -π/2.

Figure 3 . 18 :Figure 3 . 19 :

 318319 Figure 3.18: Extracted parameters of the detected modes for all the circular investigated single NDs from samples S1 and S2. The frequencies from S2 have been multiplied by 19/40 in order to compensate the difference in h with S1. mode1 and mode2 present frequencies that decrease with the diameter, while mode3 has a constant frequency.mode3 is in antiphase with respect to mode1 and mode2. mode2 exhibits an enhancement of Q at η = 2.5, while mode1 shows a broader and weaker amplification of Q around η = 6. mode1 also presents relative strong amplitude about η = 2.5.

Figure 3 . 21 :Figure 3 . 22 :

 321322 Figure 3.21: Normalized displacement field and energy flux computed for mode1 and mode2 at η = 2.2. The modes have difference appearance in respect to the ones illustrated in Fig. (3.20) at η = 4. The proximity to the mode frequencies intersection provokes an hybridization, which results in a reduction of energy loss throughout the interface in the case of mode2.

3 Figure 3 . 23 :

 3323 Figure 3.23: Numerically computed Q-factors. Inset: zoom of the region with Q 3 .

Figure 3 . 24 :

 324 Figure 3.24: The value of τ C (black diamonds) does not shows any dependence on η and its average value is ≈1.8 ns (red line). The fluctuations are mainly attributed to the important experimental incertitudes. In the green box: τ C obtained on the same ND (D = 85 nm) from S1 repeating the experiment at a different ω pr .

Figure A. 1 :

 1 Figure A.1: Constant K as a function of the truncation ratio T .

Figure B. 1 :

 1 Figure B.1: Refractive index of the diamond at standard pressure as a function of the wavelength. The data in the far-IR region are calculated using Eq. (B.1), while the others come from the literature. The red solid curve is the decreasing three-exponential fit of Eq. (B.2) and the inset drawn the attention to the wavelengths available in the "Transmission SMS setup" of Fig. (2.14).

Figure B. 2 :

 2 Figure B.2: The refractive index of the diamond decreases linearly with the pressure through the coefficient p(λ), which is wavelength dependent. A linear fit was realized using the formula Eq. (B.5) to estimate the coefficients p(λ = ∞), p 0 and p 1 . For the wavelengths available in the "Transmission SMS setup" of Fig. (2.14) p = p(λ = ∞) = -5×10 -4 can be used.

Figure C. 1 :Figure C. 2 :

 12 Figure C.1: Modest dispersion of the 4Me:1Et refractive index in standard condition.

P 1 D 1 (

 11 T M can be neglected, as shown Fig. (C.2) of Appendix (C). The derivatives of the Drude terms D 1,2 with respect to the pressure are analytically determined from 154 SPR optical modifications under high pressure: from numerical to semi-analytical model the Eq. (1.13) of Chapter

D 2 2 ∂P

 22 is also dependent on γ which increases with the pressure as seen in the experiments reported. Keeping the volume constant V 0 = V (P = 0 GPa), an increment of γ of 15% induces ∂ D V 0 ∼ 0.04 GPa -1 . (D.6)

  Eq. (D.5), (D.8) and (D.9) to simplify Eq. (D.1) in the form dω R ≈ ∂ω R derivatives must be evaluated at ω = ω R . Taking into account that at ω = ω R ⎧

Table 1 . 2 :

 12 6 ms -1 ) Effective mass (m

	Gold (Au)	1	5.53	6.42	1.39
	Silver (Ag)	1	5.49	6.38	1.39

* ) (m * = 1 indicate a particle with mass equal to one

1.4 Optical detection of single nano-objects: spatial modulation spectroscopy (SMS) 21

		Technique	Measured Quantity	Advantages	Limitations
	Near-field	Scanning near-field optical microscopy (SNOM)	Scattering	-High resolution (∼ 10 nm) -Electric field mapping in proximity of the nano-object environment -Suitable for liquid	-Complex interpretation -Perturbation by the tip -Long scan times
					-No absolute cross-section
				-Simple setup	-Nanoparticle sizes
		Dark-field spectroscopy	Scattering	-Dynamic nanoparticle tracking	> 40 nm -Background scattering
				-Short scan times	from other objects
					-Diffraction-limited
	Far-field	Photothermal imaging (PHI)	Absorption	-Detection of very small particle sizes (∼ 1 nm) -Suitable for liquid and biological environments	-No absolute cross-section -High laser powers -Complex setup (two beam alignment) -Diffraction-limited
		Spatial modulation spectroscopy (SMS)	Extinction	-Absolute extinction cross-section -Detection of small particles sizes (≥ 5 nm) -Low laser powers	-No quantitative measurements for large nanoparticle sizes (σ sca ∼ σ abs ) -Diffraction-limited

Table 1 . 3 :

 13 Overview of the main optical single nanoparticle detection techniques[START_REF] Ringe | Single nanoparticle plasmonics[END_REF].

2.1 High-pressure microscopy setup 53 shaker

  .(2.14) the XYZ position of a nanoobject can be manually controlled via a micrometric stage which is fixed to the optical table through 4 posts. For a finer positioning, a XYZ piezo nanopositioning stage with a capacitive integrated sensor is mounted at the top of the micrometric stage. In order to generate the sinusoidal oscillation, a Y piezo shaker is installed on the XYZ piezo nanopositioning stage (mass of piezo 0.16 Kg, maximum displacement 12 µm, presence of a capacitive integrated sensor). The base of the Y is fixed on the movable part (top) of the XYZ stage (see also Fig.(2.22) at page[START_REF] Castelvecchi | Physicists doubt bold report of metallic hydrogen[END_REF] ). The DAC is screwed to a cylindrical holder made of aluminum (mass of DAC with holder 0.6 Kg), which is in turn fixed on the top of the Y piezo shaker. Two vertical springs are hooked up to the holder in order to statically compensate the weight force of the DAC. The voltage for controlling the position of the Y piezo shaker is provided by the internal LIA (lock-in amplifier) function generator, which generates a sinusoidal voltage of amplitude V LIA , frequency f and phase φ LIA

Table 2 . 1 :

 21 Summary of the solutions containing gold BPs with PEG as surfactant. The reported dimensions come from TEM images. Sample 1 has a very low size and shape dispersion, while sample 2 is more polydisperse and contains also gold spheres.

single metal nanoparticle under high pressure 93

	2.2 A
	.54)

  and Γ chem ≈ 13.6%Γ. In this case one can write

	ΔΓ Γ	= 0.024	Δγ rad γ rad	+ 0.136	ΔΓ chem Γ chem	+ 0.06	Δγ e γ e	+ 0.56	Δγ ph γ ph	+ 0.22	Δγ s γ s	, (2.63)

  Q-factors obtained probing different NDs at different frequencies ω pr . This property has been experimentally verified for a couple of NDs, repeating the P&P experiment with very different ω pump and ω pr . The results, reported in Fig.(3.17), show little fluctuation in f i and Q i attributed to their incertitudes. The phases φ i are simply shifted of π when the probe wavelength passes in the other SPR flank. On the other hand, A i exhibit strong variations, which make unsuitable a quantitative comparisons between mode amplitude of different NDs. The results of the study performed on circular NDs from S1 and S2 are summarized in Fig.(3.18). The frequencies from S2 have been multiplied by 19/40 in order to compensate the difference in h with S1 (see Section (3.1.4) ). Overall, three different modes (mode1, mode2, mode3 ) have been detected. For aspect ratio η in the range[2.4-3], each NDs exhibit mode1 and mode2 with similar frequencies and phases. In order to distinguished between them, they were classified in accordance to the criterion A 1 > A 2 and Q 1 < Q 2 6

Optical and electronic properties of metal nano-objects

Ruby fluorescence characterization in air should be accomplished in every experiment, since every ball show a slightly different fluorescence spectrum.

The piezo stage is not essential in this monitoring phase, but is useful in maximizing the fluorescence signal from the ruby ball and essential in localizing and studing single nano-objects.

The use of a pulsed laser source is not relevant for the purposes of linear SMS technique. On the other hand it becomes fundamental to further implement nonlinear methods within the same setup, like the pump&probe technique or the second-harmonic generation.

"BP4" in the two upper graphs and through the complete pressure cycle of "BP5" in the lower ones.

actually, there is no need of restricting the application of the Poynting vector to completly closed surfaces in the acoustic case.

in complex notation it becomes P = -1 2 v * •T.

heat diffusion within nanoparticle can be neglected, as it is much faster than the surrounding matrix

as long as the experimental probe wavelength is kept in the same SPR flank, as illustrated in Fig.(3.17).

although at η = 3 this distinction becomes vague.

= n

P T M (λ, P ).

4Me:1Et properties under high pressure

semi-analytical model

Bibliography

Appendices

Appendix A

Gaussian beam diffraction limit

When a lens is illuminated by a Gaussian beam, the diffraction-limited diameter of the image spot has the form [128] spot = Kλ N.A. , (A. [START_REF] Wyckoff | Crystal Structures[END_REF] where K is a constant dependent on the pupil illumination and on the definition of diameter, λ is the wavelength and N.A. the numerical aperture. The intensity profile of the spot is strongly dependent on the radiation profile at the entrance pupil of the lens. If the pupil illumination has a Gaussian profile, the image spot has a Gaussian profile:

where W 0 is the total beam spot power that coincides with the incident one (W 0 = W i ), FWHM is the full width at half maximum, and (r-r 0 ) the radial distance from the optical axis. On the other hand, if the pupil is brightened with a plane wave top uniform illumination, obtained for example by a strongly truncated Gaussian profile, the spot results in an Airy disc:

where W 0 is a fraction of the incident power (W 0 = W i (1w L ), with w L being the entrance pupil power loss ratio of Eq. (A.7) below (w L = 0 when no truncation occurs), bessel j1 [...] stands for the Bessel function of first kind and first order, A and B are numerical constant (A = 0.83159179; B = 3.23266). When the pupil illumination is between these two extremes, a hybrid intensity profile occurs.

Gaussian beam diffraction limit

In the case of the Airy disc, the diameter of the spot can be defined considering the first dark ring:

However, when the pupil illumination is not uniform, the image spot intensity never falls to zero making it necessary to define the spot size differently. Commonly it is considered as the diameter at the half intensity point, the so called full width at half maximum (FWHM ), or the radius at the 1/e 2 intensity point, the beam waist w. When the image spot has an ideal Airy disc intensity profile, the diffraction limit of Eq. (A.4) can be rewritten as

An alternative characterization of the Airy disc intensity profile is made approximating the central lobe with a Gaussian profile with same central amplitude or with free amplitude. The fitting width results

Gaussian (max-fixed)

, (max-free)

In the second case, where the amplitude is treated as a free parameter, the width is smaller because the maximum amplitude is overestimated. This introduces a systematic error if one tries to fit an experimental Airy profile with a Gaussian function.

The constant K of Eq. (A.1) is a function of the truncation ratio

where w G is the waist of the incident Gaussian beam and D the limiting aperture diameter of the lens. If T ≥ 2 (high truncation) the image spot intensity profile approaches the Airy pattern, whilst when T ≤ 0.

Appendix B Diamond properties under high pressure

The refractive index of the diamond n d is a function of the wavelength λ as well as the pressure P . In standard condition, the dependence of n d on λ was studied by Edwards et al. [START_REF] Edwards | Infrared refractive index of diamond[END_REF]. Previous data from the UV to the near IR region were reviewed and collected, and a phenomenological formula to estimate n d in the far IR region was obtained

where Λ 1 = 5.2, λ 1 = 72 nm, Λ 2 = 0.226, λ 2 = 312 nm, Λ 3 = 0.0076, λ 3 = 2785 nm. This expression is valid in the whole range 400 -4000 nm.

The diamond refractive index is also pressure dependent. This aspect was studied in the range 0-40 GPa for several wavelength (400-900 nm) by Eremets et al., [START_REF] Eremets | Refractive index of diamond under pressure[END_REF] resulting to be a quasi-linear function of the pressure. Realizing a linear approximation of these data in the range 0-20 GPa allows to estimate the pressure dependence

The slope coefficient p(λ) is always negative and is reported in

where λ is the light wavelength expressed in µm. For the the ethanol, in the range 500-1600 nm and at 20 The imaginary part κ Et remains always within the range 10 -6 -10 -8 in the region 600-1100 nm, and hence the absorption is negligible. The refractive index of the PTM mixture 4Me:1Et can be estimated combining Eq. (C.1) and (C.