Moulay Saïd

El Youssoufi

Christian Saix

Frédéric Jamin

Thermo-mechanical Behaviour of a Soil. Yield Surface Evolution

This paper presents a study of temperature influence on the yield surface for saturated and unsaturated soils. It lies within the framework of characterization and modelling of soil behaviour on thermo-hydro-mechanical loading. In this study, clayey silty sand samples are subjected to mechanical consolidation tests at different temperatures and fixed suctions. These experimental tests make it possible to determine the yield surface evolution according to temperature ranging between 20 • C and 60 • C. For the studied soil, the experimental study shows a thermo-extensible nature of the yield surface. An explanation of this result is proposed while gone by microscopic considerations in touch with capillary meniscuses evolution according to temperature and suction.

Introduction

This paper presents a contribution to the experimental study and the modelling of temperature influence on the yield surface (border surface between the elastic and plastic domains) of soils. Since the sixties, many studies have been interested in the temperature influence on the mechanical behaviour of saturated soils. In the clayey media case, these researches are generally based on problems involved by storages of radioactive waste in deep geological formation, but applications can concern all porous media subjected to various thermal gradients (geothermic structures). In the case of soils with lower depth, we can quote buried high-voltage cables, different waste storage and notably domestic. For these depths, the hypothesis of a completely saturated soil isn't always acceptable and it's necessary to take into account the effect of suction. In the isothermal unsaturated case, several authors have studied the suction influence on the mechanical behaviour of soils. For example, Alonso et al. [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF] proposed a general constitutive model that takes into account the effect of suction in the mechanical behaviour of soils based on the Modified Cam-Clay model.

In the non-isothermal saturated case, several studies put clearly in evidence the temperature influence on the yield surface. Some authors used the Modified Cam-Clay model that appears to be a good estimate of the behaviour to take into account the temperature effect on the yield surface [START_REF] Cui | A thermomechanical model for saturated clay[END_REF][START_REF] Graham | Modified Cam-Clay modeling of temperature effects in clays[END_REF][START_REF] Hueckel | Thermoplasticity of saturated clays: Experimental constitutive study[END_REF][START_REF] Laloui | Thermo-plasticity of clays: A simple constitutive approach[END_REF][START_REF] Tanaka | Stress-strain behaviour of reconstituted illitic clay at different temperatures[END_REF].

In the non-isothermal unsaturated case, Saix et al. [START_REF] Saix | Eléments de couplage thermomécanique dans la consolidation de sols non saturés[END_REF] have studied experimentally the effect of a thermal hardening in elastic limits in vertical total stress for a given suction. Jamin et al. [START_REF] Jamin | Temperature influence on the yield surface for unsaturated soils[END_REF] were interested more particularly in the determination, for various temperature, of a yield surface f (p, q, s) = 0, where p is the net mean stress, q the deviatoric stress and s the suction. The first results of the experimental study show that this surface can be approximate by the Modified Cam-Clay model in the case of clayey silty sand and presents a thermo-extensible nature according to temperature for no null suction. This study has been pursued by adopting news loading paths in the aim of specifying the Alonso et al. [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF] model with temperature. This evolution is studied through the temperature influence on the preconsolidation stress, the slope of the critical state line and the stiffness parameters.

In a first part, a bibliographical synthesis leads to emphasize the most essential results of the literature relating to the temperature influence on the model parameters in the saturated and unsaturated cases. Then, we present the studied soil, the experimental device and experimental process. Lastly, the experimental results are presented and are followed by a physical interpretation.

Yield surface model

Hydro-mechanical modelling

In the isothermal unsaturated case, Alonso et al. [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF] proposed a constitutive model to describe the unsaturated soils behaviour. This model takes into account the Modified Cam-Clay yield surface developed in Cambridge University by Roscoe and Burland [START_REF] Roscoe | On the generalized stress-strain behaviour of wet clay[END_REF]. This yield surface depends on the net mean stress p, the deviatoric stress q and the suction s defined in axisymetric triaxial conditions by:

⎧ ⎨ ⎩ p = 1 3 (σ z + 2σ x ) -p * g q = σ z -σ x s = p * g -p * e ( 1
)
where σ z = principal vertical stress, σ x = principal radial stress, p * g = pressure of the gas phase and p * e = pressure of the liquid phase. The equation of this yield surface in the stress space (p, q, s) is given by:

q 2 -M 2 (p + p s (s)) (p 0 (s) -p) = 0 (2)
where M = slope of the critical state lines not dependant on suction, p s (s) = parameter introducing the evolution of apparent cohesion with suction and p 0 (s) = isotropic preconsolidation stress function of suction. In the case of a mechanical loading with constant suction, the relation between the isotropic preconsolidation stresses p 0 (s) for unsaturated conditions at a suction s and p 0 (0) for saturated conditions (s = 0) is written [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF]:

p 0 (s) p c = p 0 (0) p c λ(0)-κ λ(s)-κ (3)
where p c = reference stress, κ = elastic stiffness parameter for changes in net mean stress, λ(s) and λ(0) = stiffness parameters for changes in net mean stress for virgin states on the soil with constant suction s for the former and in saturated condition (s = 0) for the latter, with:

λ(s) = λ(0) (1 -r) e -βs + r ( 4
)
where r = constant equal to the ratio λ(s → ∞)/λ(0) and β = parameter controlling the rate of increase of soil stiffness with suction. Equation (3), named Loading Collapse, plays an important role in the model. It shows an increase of the preconsolidation stress according to the suction, and also the phenomenon of collapse observed with the decrease of the suction (wetting path). In the case of an increase in suction with constant stress, Alonso et al. [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF] give as assumption that the limit of the elastic domain is defined by:

s = s 0 (5)
with s 0 = hardening parameter of the suction increase yield curve. This equation is called Suction Increase.

Thermo-mechanical modelling

In the non-isothermal saturated case, several authors have studied the temperature influence on the mechanical behaviour of clayey soils. In the oedometric tests case, we generally note a decrease on the preconsolidation stress when the temperature increases [START_REF] Eriksson | Temperature effects on consolidation properties of sulphide clays[END_REF][START_REF] Tidförs | Temperature effect on preconsolidation pressure[END_REF]. In the axisymetric triaxial tests case, Hueckel and Baldi [START_REF] Hueckel | Thermoplasticity of saturated clays: Experimental constitutive study[END_REF], Tanaka et al. [START_REF] Tanaka | Stress-strain behaviour of reconstituted illitic clay at different temperatures[END_REF], Cui et al. [START_REF] Cui | A thermomechanical model for saturated clay[END_REF] and Graham et al. [START_REF] Graham | Modified Cam-Clay modeling of temperature effects in clays[END_REF] have shown that the ellipse of the Modified Cam-Clay model contracts with an increase of temperature with the result that the slope of the critical state line M remains constant so independent of temperature. Besides, other authors note a decrease of the critical state line M with an increase of temperature.

Thermo-hydro-mechanical modelling

Romero [START_REF] Romero | Characterisation and thermo-hydro-mechanical behavior of unsaturated Boom clay: An experimental study[END_REF] has studied coupled effect between the suction and the temperature on the mechanical behaviour of Boom clay in drained condition. The results show, on the one hand, an increase of the yield surface with an increase of suction at constant temperature and, on the other hand, a decrease of this surface with an increase of temperature at constant suction.

However, Jamin et al. [START_REF] Jamin | Temperature influence on the yield surface for unsaturated soils[END_REF] obtained different results on clayey silty sand. Indeed, they noted that, in unsaturated conditions, the yield surface shows a thermo-extensible nature with an increase of temperature. Consequently, in the non-isothermal unsaturated case, the temperature is introduced into parameters in relation (2) of the Modified Cam-Clay yield surface which is defined within a general framework by:

q 2 -[M (T )] 2 (p + p s (s; T )) (p 0 (s; T ) -p) = 0 (6)
where M (T ) = slope of the critical state line depending on the temperature, p 0 (s; T ) = preconsolidation stress depending on suction and temperature and p s (s; T ) = parameter introducing the evolution of apparent cohesion with suction and temperature. This parameter defined by Gens [START_REF] Gens | Constitutive laws[END_REF] is given by:

p s (s; T ) = k (1 -f (T )) s (7)
with f (T ) = function taking into account the variation of apparent cohesion due to the temperature and k = constant.

Temperature influence on stiffness parameters

In a general way, the tests carried on various soils do not show any temperature influence on the values of stiffness parameters. In the saturated case, Campanella and Mitchell [START_REF] Campanella | Influence of temperature variations on soil behavior[END_REF] and Eriksson [START_REF] Eriksson | Temperature effects on consolidation properties of sulphide clays[END_REF] undertook mechanical consolidation tests at various temperatures. Tidförs and Sällfors [START_REF] Tidförs | Temperature effect on preconsolidation pressure[END_REF] carried out mechanical consolidation tests while varying the temperature during the loading on Bäckebol clay. In the unsaturated case, Saix et al. [START_REF] Saix | Eléments de couplage thermomécanique dans la consolidation de sols non saturés[END_REF] studied the temperature influence on stiffness parameters evolution. All results show an independence of these stiffness parameters with respect to the temperature. Despax [START_REF] Despax | Influence de la température sur les propriétés mécaniques des argiles saturées[END_REF] also supposes an independence of the stiffness parameters with respect to the temperature in spite of the slight incidence observed. More recently, Graham et al. [START_REF] Graham | Modified Cam-Clay modeling of temperature effects in clays[END_REF] take a slight increasing of elastic stiffness with an increase of temperature.

All the results presented in the literature do not allow proposing general conclusions. The variety of the studied soils, the experimental conditions, preparation and loading make necessary to lay out complementary tests to confirm or cancel the evoked tendencies.

Studied soil, experimental device and procedure

Studied soil

The soil used in this study is clayey silty sand, which particle size does not exceed 2 mm. A mineralogical analysis has shown that this soil is mainly consti-tuted of calcite (50%) and quartz (40%). The clayey fraction represents only 10% and is essentially constituted of smectite, chlorite and phyllite [START_REF] Devillers | Transfert d'eau dans les sols faibles teneurs en eau. 16me Congrs Mondial de Science du Sol[END_REF]. This material, with a liquid limit w L = 25% and a plasticity index I P = 10.5%, is similar to a clay of low plasticity according to the LCPC classification (Laboratoire Central des Ponts et Chaussées, France) and USCS (Unified Soil Classification System, U.S.A.). The reconstituted soil sample have, after the preparation, a dry density ρ d = 1500 kg/m 3 (void ratio e = 0.77) and a water content w = 18% (degree of saturation S r = 63.5%). The interest of this soil type is to make it possible to obtain an unsaturated state for low values of suction.

Experimental device

The experimental characterization and the modelling of the soil behaviour under thermo-hydro-mechanical loading use essentially a thermal triaxial apparatus (Fig. 1). This experimental device allows reaching the volume soil sample changes of very slight slenderness (70 mm in diameter and 15 mm in height) during different thermo-hydro-mechanical loading paths. The fixed experimental variables are σ x , (σ z -σ x ), p * g , p * e and T . The measured experimental variables are the height variation ∆h, the radius variation ∆r and the water volume variation ∆V e going in or out of the sample. These measures, allow reaching the void ratio e and the water content variation ∆w.

Dh

Experimental procedure

Experimental procedure (Fig. 2) consists in realizing thermo-mechanical consolidation tests with a constant suction s at different temperatures T . These tests follow loading paths [START_REF] Jamin | Contribution l'étude du transport de matire et de la rhéologie dans les sols non saturés différentes températures[END_REF] defined by a constant ratio η = q/p between deviatoric stress q and net mean stress p. The values of η used in this study are included between 0 (isotropic loading) and 3/2, the temperatures of different trials are 20, 30, 40, 50 and 60 • C, and the constant suctions are equal to 0 kPa (saturated conditions) and 5 kPa (unsaturated conditions). Whatever the loading path η and the temperature T , the mechanical loading consists in imposing increments of vertical total stress σ zp * g which takes the values between 10 and 300 kPa. From the consolidation curve obtained, we determine the value of preconsolidation vertical total stress σ zp * g η,L for the different ratios η and various temperatures T .

We deduce the preconsolidation stress couples (p η ; q η ) L using the following relation:

(p η ; q η ) L = 3 3 + 2η σ z -p * g η,L ; 3η 3 + 2η σ z -p * g η,L (8)

Results and discussions

Following the bibliographical synthesis presented in the section 1, tests were carried out on clayey silty sand, in saturated and unsaturated conditions. The latters allow to specify the temperature influence on the stiffness parameters at constant suction. Then we present the preconsolidation stress evolution according to temperature and propose an estimation of the yield surface in accord with the Alonso et al. model [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF]. Lastly, we propose a physical interpretation of the experimental results.

Stiffness parameters evolution

The evolution of the elastic stiffness κ and plastic stiffness λ, in the saturated case (Fig. 3), shows an increase with the temperature for the plastic stiffness.

In the unsaturated case, all measured stiffness in the tests carried out, show a slight incidence of temperature.

Temperature influence on the preconsolidation stress

In the saturated case, the results show the independance of the preconsolidation stress p 0 (0; T ) with an increase of temperature. In the unsaturated case, the results (Fig. 4) show, at constant suction s = 5 kPa, an increase in the preconsolidation stress p 0 (5; T ) with the temperature T .

Yield surface evolution

From the obtained tests values and using the relation (8), we deduce the preconsolidation stress couples (p η ; q η ) L . In the stress space (p, q), the thermoextensible nature of the yield surface is clearly in evidence (Fig. 5).

From the relation (6), the values taken into account in the modelling of the yield surface are given in table 1. On the one hand, we note an increase in the preconsolidation net stress leads to the temperature influence on the preconsolidation vertical total stress (Fig. 4). On the other hand, it's turned out that the slope of the critical state line does not undergo notable variation whatever the test temperature. These results don't ensure to estimate the temperature influence on the apparent cohesion which has been supposed here constant whatever the temperature value.

Results interpretation

In the unsaturated case, at ambient temperature, the undertaken experimental studies [START_REF] Alonso | A constitutive model for partially saturated soils[END_REF][START_REF] Cui | Yielding and plastic behaviour of an unsaturated compacted silt[END_REF] show an increase of the preconsolidation net mean stress with an increase in suction. In the unsaturated clayey silty sand case for constant suction, our results also show an increase of the preconsolidation net mean stress with temperature [START_REF] Jamin | Contribution l'étude du transport de matire et de la rhéologie dans les sols non saturés différentes températures[END_REF]. According to the Jurin's law:

s = 2Γ R with Γ = Γ (T ) and dΓ dT < 0 (9)
where Γ = surface air-water tension and R = meniscus radius, theses tendencies can have a physical interpretation. Indeed, at constant temperature, an increase in suction causes a decrease of the meniscus radius. In addition, at constant suction, an increase in temperature also causes a decrease in the meniscus radius; the surface air-water tension decreases according to temperature. Consequently, an increase in suction at constant temperature or an increase in temperature at constant suction leads to a decrease in the meniscus radius. This reduction of meniscus radius results in an increase of the cohesive forces between grains by capillarity, which causes to limit their rearrangements. Indeed, the arrangements are the main factors of soils behaviour irreversibility. This limitation corresponds to an increase of the preconsolidation net mean stress, and so to an extension of the yield surface.

Conclusion

Temperature influence (between 20 • C and 60 • C) on the yield surface of clayey silty sand was studied. In the saturated case, we have noted the independance of the preconsolidation stress with an increase of temperature. In the unsaturated case, at constant suction (s = 5 kPa), the experimental results show the thermo-extensible nature of the yield surface for the studied soil. These results appear to be in opposition with those proposed by Romero [START_REF] Romero | Characterisation and thermo-hydro-mechanical behavior of unsaturated Boom clay: An experimental study[END_REF] who observed, for Boom clay, a thermo-shrinkage nature of the yield surface whatever the suction value. This opposition lets to think that the clayey fraction of the studied soil is a determining factor in the yield surface evolution. Indeed, Romero [START_REF] Romero | Characterisation and thermo-hydro-mechanical behavior of unsaturated Boom clay: An experimental study[END_REF] studied a clay (Boom clay) whereas the clayey fraction of clayey silty sand, studied here, is only 10%. Similar tests on a material, whose clayey fraction would be intermediate, should bear new informations on a possible transition between the thermo-shrinkage and thermo-extensible natures of the yield surface.

Fig. 1 .

 1 Fig. 1. Thermal triaxial apparatus

Fig. 2 .

 2 Fig. 2. Consolidation curve example. Yield surface in (p, q) plan

Fig. 3 .

 3 Fig. 3. Temperature influence on the stiffness parameters

Fig. 4 .Fig. 5 .

 45 Fig. 4. Temperature influence on the preconsolidation stress p 0 (s; T )

Table 1 .

 1 Parameters of Modified Cam-Clay model at constant suction s = 5 kPa Temperature 30 • C 40 • C 50 • C 60 • C

	M	1.25 1.25 1.25 1.25
	p 0 (5; T )	34.5	39	43.5	51


Mathieu Jeannin

Névine Rochat

Kuntheak Kheng

Gilles Nogues

Cathodoluminescence spectroscopy of plasmonic patch antennas: towards lower order and higher energies

I. INTRODUCTION

Advances on classical and quantum solid state light emitters and optoelectronic devices bring great interests in controlling light emission properties of nano-sized emitters. As an example, several approaches are employed in the field of semiconducting quantum dots, from microcavities 1 and other photonic structures [START_REF] Claudon | A highly efficient single-photon source based on a quantum dot in a photonic nanowire[END_REF][START_REF] Lodahl | Interfacing single photons and single quantum dots with photonic nanostructures[END_REF][START_REF] Gschrey | Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing threedimensional in situ electron-beam lithography[END_REF] to plasmonic antennas [START_REF] Curto | Unidirectional emission of a quantum dot coupled to a nanoantenna[END_REF][START_REF] Pfeiffer | Enhancing the optical excitation efficiency of a single self-assembled quantum dot with a plasmonic nanoantenna[END_REF][START_REF] Curto | Multipolar radiation of quantum emitters with nanowire optical antennas[END_REF][START_REF] Jeannin | Deterministic radiative coupling between plasmonic nanoantennas and semiconducting nanowire quantum dots[END_REF] . In the latter case, a great effort has been made to develop new antenna geometries that could increase the coupling strength between a single emitter and the antenna. Additionally, researchers are investigating novel possibilities of controlling the radiation pattern of the coupled structure.

One of the emerging strategies is to confine the plasmon field inside an insulating layer comprised between a metallic nanoparticle and a continuous metallic film. Such structures were initially investigated using colloidal plasmonic particles [START_REF] Mock | Distancedependent plasmon resonant coupling between a gold nanoparticle and gold film[END_REF][START_REF] Mubeen | Plasmonic properties of gold nanoparticles separated from a gold mirror by an ultrathin oxide[END_REF] . Due to the strong electric field enhancement possible inside the insulating layer, these systems were soon proposed as promising plasmonic cavities. Recent demonstrations include a 1900-fold increase in emission intensity for colloidal quantum dots [START_REF] Hoang | Ultrafast room-temperature single photon emission from quantum dots coupled to plasmonic nanocavities[END_REF] , and reaching up to the strong coupling regime for single molecules placed between colloidal Au spheres and an Au mirror [START_REF] Chikkaraddy | Single-molecule strong coupling at room temperature in plasmonic nanocavities[END_REF] . All of these systems rely on the coupling between the dipolar localized surface plasmon mode supported by the nanoparticle and the metallic mirror. Another key properties of the particle-on-a-mirror geometry is that the coupling between the plasmon mode supported by the particle and the film allows to dramatically change the emission diagram of the coupled system. This was especially investigated using Au disk antennas on a Au mirror. Because of geometrical and electric field profile similarities to their radio-frequency counterparts, these laterally confined insulator-metal-insulator-metal (IMIM) nanostructures were then designated as plasmonic patch antennas [13][START_REF] Filter | Circular optical nanoantennas: an analytical theory[END_REF][START_REF] Bigourdan | Design of highly efficient metallodielectric patch antennas for single-photon emission[END_REF] . They have been successfully coupled to colloidal quantum dots [START_REF] Belacel | Controlling spontaneous emission with plasmonic optical patch antennas[END_REF] and studied by cathodoluminescence [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . Contrary to the nanoparticle-on-a-mirror experiments, circular patch antennas reported up to now have large diameters. They can be thought as circular cavities for surface plasmon polariton modes. Their large size and subsequent large number of supported mode is the key to the control of their radiation pattern.

In this article, we use cathodoluminescence (CL) spectroscopy and imaging on Au and Al circular plasmonic patch antennas in the intermediate regime between large antenna and small particle. We characterize circular antennas supporting single mode operation as well as highly multipolar modes. The CL signal collected from plasmonic antennas is closely related to the electromagnetic local density of states (LDOS). In [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF][START_REF] García De Abajo | Optical excitations in electron microscopy[END_REF][START_REF] Losquin | Link between cathodoluminescence and electron energy loss spectroscopy and the radiative and full electromagnetic local density of states[END_REF] , it is shown that the signal measured in CL corresponds to the integration over the electron beam path of the partial, radiative LDOS projected along the electron beam path. It allows to precisely characterize the antennas, to determine their spectral properties and to map the radiative LDOS on a nanometer scale for each antenna resonance. The paper is organized as follows:

we first present the patch antennas fabrication and the CL system. We then introduce the fundamental properties of Au patch antennas and a semianalytical model proposed to explain their mode structure [START_REF] Filter | Circular optical nanoantennas: an analytical theory[END_REF][START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . We then study antennas of smaller dimensions, down to ∼170 nm in diameter, allowing for the study of the fundamental antenna mode. Finally, using Al patch antennas we demonstrate that changing the antenna material makes it possible to shift their operation range down to a wavelength of 450 nm. Indeed, Al is seen as a promising candidate for blue and U.V. plasmonics [START_REF] Knight | Aluminum plasmonic nanoantennas[END_REF][START_REF] Gérard | Aluminium plasmonics[END_REF][START_REF] Martin | Fabrication of aluminium nanostructures for plasmonics[END_REF][START_REF] Sobhani | Pronounced linewidth narrowing of an aluminum nanoparticle plasmon resonance by interaction with an aluminum metallic film[END_REF] in spite of the current low quality of Al films [START_REF] Mcpeak | Plasmonic films can easily be better: Rules and recipes[END_REF] . It has the additional advantage to be compatible with current CMOS fabrication technologies, easing a possible large-scale implementation of Al-based plasmonic devices. The patch antennas are fabricated on Si substrates. A first, optically thick layer of metal (Au or Al, 100 nm) is evaporated on the substrate using e-beam evaporation. In order to get the best optical properties, we aimed at lowering as much as possible the surface roughness and granular structure of the films. Au evaporation is performed under ∼ 10 -6 mbar at an evaporation rate of ∼ 2 Å/s. Al has a tendency to oxidise a lot more easily than Au, and therefore smooth Al films are hard to obtain with the vacuums levels available in conventional e-beam evaporation machines. A possible way to improve the quality of the film is to use a higher evaporation rate, in our case 2 nm/s 24 . The spacing oxide layer is then deposited using atomic layer deposition, and its thickness is controlled using an interferometric measurement after deposition.

II. SAMPLE FABRICATION AND EXPERIMENTAL DETAILS

All the top layer plasmonic structures are fabricated by electron beam lithography. An electron-sensitive, positive resist (Poly(metyl methacrylate), or PMMA) is spin-coated on the sample and soft-baked on a hot plate at 180 • C for 5 min. The disks are patterned using an electron beam exposing the resist upon electron impact. Each disk is separated from its neighbours by a distance greater than 2 µm to avoid any coupling between two adjacent structures. The resist is then developed, and 35 nm of metal (Au or Al) is deposited using electron gun evaporation. A final lift-off process is performed in 1-methyl-2-pyrrolidinone (NMP) heated at 80 • C to remove the remaining resist.

Three IMIM systems are under investigation. In each case, the Al 2 O 3 oxide layer thickness is 57±2 nm. First, Au patch antennas were fabricated on an Au mirror, as shown in Fig. 1(a), with diameters ranging from 280 to 900 nm. Second, Au patch antennas on an Al mirror were fabricated with diameters ranging from 120 to 930 nm (Fig. 1(b)). Finally, Al antennas on an Al mirror were fabricated, with diameters also ranging from 120 to 930 nm (Fig. 1(c)). As can be seen in the scanning electron microscope (SEM) images in Fig. 1, the Au deposition results in a very smooth Au film and antenna surface. Magnified SEM images revealed an Au grain size of around 10 nm. The Al film appears a lot rougher, as can be inferred from Fig. 1(c). We noticed that Al grains tend to coalesce, so the smoothness of the top surface degrades upon increasing the deposited thickness. Hence, the top surface of the Al mirror (100 nm thick) shows Al grains sizes of around 40 nm, and is a lot rougher than the Al antenna surface (35 nm thick) which has Al grains of only 20 nm in size. Note that in the case of bimetal antennas the roughness of the Al film also slightly degrades the quality of the Au top disk, as seen in Fig. 1(b).

Our cathodoluminescence setup consists in FEI (quanta 200) SEM fitted with a drilled home made asymmetric parabolic mirror allowing for electrons to pass through, and providing a very large numerical aperture (NA) ranging from 0.5 (detector side) to 0.9 (parabola side). We use an acceleration voltage of 30 keV and a beam current around 7 nA. The hole allowing for the electrons to pass is 500 µm in diameter, preventing light collection in the vertical direction with an angular divergence of a few degrees. The CL signal collected by the mirror is focused on the entrance slit of a spectrometer (Horiba Jobin-Yvon IHR550) by a spherical mirror, preventing chromatic aberrations. The light is then dispersed by a 150gr/mm grating blazed at 550 nm and sent either to a charged-coupled device (CCD) camera (Andor Newton) or energy-filtered through the exit slit of the spectrometer and sent to an avalanche photodiode (APD) to obtain energy-selective CL images.

Complementary experiments were carried out using a Attolight commercial Rosa CL setup allowing for hyperspectral imaging. The electron beam acceleration voltage is 10 keV, with a beam current of 25 nA. The light is collected using a Cassegrain-type objective (NA 0.72) embedded inside the electronic column of the scanning electron microscope. The collected light is focused on the entrance slit of a spectrometer (Horiba Jobin-Yvon IHR320), dispersed using a 150gr/mm grating blazed at 500 nm, and detected using a CCD camera (Andor Newton).

III. CATHODOLUMINESCENCE OF AU PATCH ANTENNAS

Let us first describe the spectral properties of the Au patch antennas. To obtain the antennas CL spectrum, we raster scan the electron beam over the antenna surface, using a square scanning area exactly circumscribing the disk. The beam scanning time is set to be much smaller than the integration time (∼1 min) so that every point on the antenna surface can be considered as equally excited. A background spectrum is acquired by scanning the exact same area on a region where no antenna is present, recording the CL response of the bare substrate. This substrate spectrum is subtracted from the antenna's CL response. We note that the background luminescence from the substrate has an intensity of ∼ 4000 s -1 , while the CL signal originating from localized surface plasmon modes supported by the antenna is ∼ 20 s -1 in intensity. The background subtraction procedure does not suppress entirely the contribution of the substrate luminescence to the collected CL signal. A typical antenna spectrum is presented in Fig. 2(a) (590 nm diameter). It presents a broad peak indicated in red dashes. This signal comes from residual luminescence due to Au interband transitions for wavelengths below 600 nm [START_REF] Beversluis | Continuum generation from single gold nanostructures through near-field mediated intraband transitions[END_REF] as well as from the diffraction by the patch edge of the surface plasmon polaritons excited outside the antenna region at larger wavelength.

Sharper peaks are visible at 650 and 750 nm, which we attribute to localized surface plasmon resonances (LSPR) of the antenna. CL images of the LDOS corresponding to each LSPR are obtained by slowly scanning the electron beam over the antenna surface, collecting the light emitted in a 50 nm spectral bandwidth around the resonance. This filtered CL emission is then detected by an APD and correlated to the electron beam position. The spectral integration bandwidth is selected to maximise the collected intensity, matching the resonance bandwidth but ensuring that we collect only light emitted from a single resonance.

In the rest of the article, the LDOS images are normalized in intensity. The typical emission probability is of the order of 10 -6 photons per incident electrons. More information on the LDOS image acquisition is given in the Appendix section. Note that the CL images do not reveal the antenna mode structure, but rather map the probability of exciting a given antenna mode at each electron beam position. The LDOS images corresponding to the two LSPR peaks observed in Fig. 2(a) are shown in the corresponding iv and v insets in Fig. 2

(b).

We subsequently repeat this procedure for several antenna diameters, and obtain the antenna spectrum with its LSPR peaks and their corresponding LDOS patterns. In Fig. 2(c), we gather all the antenna spectra in the form of a waterfall plot. The experimentally measured antenna diameters are indicated, and fabrication imperfections are responsible for the non-even spacing between the different antennas sizes. It is also represented by the color-coded intensity spectral map in Fig. 2(c) which represents the CL spectra as a function of wavelength and increasing diameter. Note that the vertical scale has a fixed width for each spectrum. This representation will be used again in Fig. 6.

The LDOS imaging allows us to track a given LSPR with respect to the antenna diameter change. It reveals that the successive LSPRs of the patch red-shift with increasing antenna diameter. The red-shifting resonances branches are marked with red solid lines serving as guides for the eye. All the resonances belonging to a given branch show the same spatial structure, as represented in Fig. 2(b) for the first six branches. Each pattern is composed of one or more concentric rings. In addition, considering successive LDOS patterns (from i to vi) we observe an alternation of bright and dark antenna center, in agreement with previous reports [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . Furthermore, we do not observe any azimuthal dependence of the LDOS. This will be discussed in the modelling section.

IV. SEMI-ANALYTICAL MODEL

The optical properties of plasmonic patch antennas can be described by the semianalytical model of ref. [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] which is adapted from the fully analytical work of ref. [START_REF] Filter | Circular optical nanoantennas: an analytical theory[END_REF] . The analytical model considers an arbitrary stack of circularly symmetric layers of fixed radius R, different thickness t j and permittivity ε j , where j is the layer number, embedded in an homogeneous medium of permittivity ε d . A sketch of the geometry is shown in Fig. 3(a).

The stack is considered as a circular cavity supporting Bessel-type surface plasmon polariton (SPP) modes. The dispersion relation of the different SPP modes and their vertical mode profile are numerically calculated from the equivalent infinite (R → ∞) multi-layer system [START_REF] Davis | Surface plasmon modes in multi-layer thin-films[END_REF] .

In the case of our insulator-metal-insulator-metal (IMIM) geometry, only two distinct propagating SPP modes are present, referred to as symmetric and antisymmetric due to the symmetry of the E z component of the electric field. The vertical profiles of E z are sketched in Fig. 3 The electric field is decomposed into eigenmodes of the circular geometry in which the z, ρ and ϕ dependences can be separated as: where a(z) is the vertical mode profile sketched in red in Fig. 3(b), J m is the Bessel function of the first kind of azimuthal order m, ρ is the radial vector and k n,m is the surface plasmon polariton wave vector supported by the infinite multilayer structure following the dispersion relation of Fig. 3(c). The eigenfunctions are further determined by the resonance condition:

E n,m z (ρ, z) = a(z)J m (k n,m ρ) exp (imϕ) (1)
Re [k n,m ] 2R + ϕ m = 2x n (J m ) (2)
where Re [k n,m ] denotes the real part of the SPP wave vector. The phase factor ϕ m accounts for the phase shift acquired upon reflection at the patch edge ρ = R. x n is the n -th zero of the Bessel function J m . While ϕ m can be numerically calculated for each azimuthal number m according to ref. [START_REF] Filter | Circular optical nanoantennas: an analytical theory[END_REF] , we used the empirical formulas and method reported in ref. [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] instead. The main approximation is that ϕ m is considered the same for all antenna modes (n, m). Equation 2 gives the resonant frequencies ω n,m of the disk after inversion of the SPP dispersion relation. The LDOS Γ (ω, ρ) probed during the CL experiment can thus be expressed as:

Γ (ω, ρ) ∝ n,m |E n,m z (ρ)| 2 L (ω, ω n,m , γ n,m ) (3)
where the energy of the mode profile

E n,m z (ρ) is normalized in space through 2πρ |E n,m z (ρ)| 2 dρdz = 1, and L (ω, ω n,m , γ n,m
) is a normalized Lorentzian function centred at frequency ω n,m with a full width at half maximum γ n,m so that L (ω, ω n,m , γ n,m ) dω = 1. Following ref. [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] , the width of the Lorentzian function is given by:

γ n,m /ω = (2Im [k n,m ]) 2 + γ 0 (1 + ω b norm ) -1 (4)
where γ 0 and b are adjustable parameters used to fit the resonance bandwidth of the antennas, Im [k n,m ] is the imaginary part of the SPP wave vector, and ω norm = ω/(1800 THz).

The frequency broadening of the antennas resonances describes the loss-induced frequency broadening of the plasmonic patches. Note that as the LDOS is related to the square modulus of the electric field, one can only detect the radial dependence of the plasmon mode, and not its azimuthal dependence.

Using the previous relations we compute the different resonances contributing to the LDOS for each radius of the the fabricated antennas. The fitting parameter ϕ m was obtained in [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] by matching the resonance of the fundamental antenna mode to the results of numerical simulations. However, due to the large disk size, this mode had a very low energy and could not be experimentally observed. Here instead, we match the position of all the antenna modes with the experimentally determined resonances of branches i, ii and iii in Fig. 2(c).

In this case, we find that a correction on the SPP dispersion relation is necessary to reproduce the experimental mode dispersion. The semi-analytical model thus seem to underestimates the mode wave vector by a factor 1.5 to 2. This might be due to the fact that the model considers the disks as cavities for propagating SPPs, whereas for small disks the large lateral confinement may result in larger SPP effective wave vectors, as already described in the case of plasmon waveguides 28,29 . Fitting the experimental data, we find γ 0 = 0.05, b = 1.7, and

ϕ m = 0.5.
Comparing the results of the simulations with the experimentally determined spectra and LDOS reveals further information on the mode structure supported by the patch antennas. As noted in [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] , it is important to realize that the patch antenna resonances are composed of superposition of several Bessel modes with increasing quantum numbers. This superposition arises from the overlap between the frequency broadened Bessel modes due to ohmic and radiation losses [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . However, in our case, the small size of our patch antenna allows us to image LDOS corresponding to single mode resonances. Doing so, we arrive at a one-to-one correspondence between the LDOS model and the experimental CL images, which was not possible in [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . Note that the relative amplitude between the resonances is not reproduced.

This is because the model only describes the LDOS supported by the particle, while CL is sensitive to the radiative component of the LDOS. The radiative efficiency and radiation pattern of each resonance is thus left out in the model and is responsible for this discrepancy.

Finally, the broad continuum of CL signal below 600 nm is not reproduced by the model, because it is simply due to luminescence from the Au layers and thus not related to the plasmonic mode structure of the antennas.

V. TOWARDS LOWER ORDERS

Figure 4 shows that the antenna can sustain resonances corresponding to a single plasmon mode. Reaching low order mode operation, or even single mode operation is important to efficiently couple the patch antennas to localized emitters placed inside the spacing layer, such as in [START_REF] Belacel | Controlling spontaneous emission with plasmonic optical patch antennas[END_REF] . Large patch antennas support a high number of competing mode, which gives them their interesting directional beaming properties, as shown in depth in [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] . However, a large radiative enhancement requires a strong coupling between the emitter and a single radiative mode of the antenna. This is why the strongest reported Purcell factors in the patch geometry involve small metallic particles as in refs. [START_REF] Hoang | Ultrafast room-temperature single photon emission from quantum dots coupled to plasmonic nanocavities[END_REF][START_REF] Chikkaraddy | Single-molecule strong coupling at room temperature in plasmonic nanocavities[END_REF] . Hence, a trade-off has to be found between directionality and radiative rate enhancement when coupling nano-emitters to such plasmonic patch antennas.

The lowest order radiative mode that can be sustained by the patch antenna is the fundamental (n = 1, m = 1) dipole-like mode. To reveal this fundamental mode, we fabricated smaller Au antennas on an Al mirror. The use of the Al mirror strongly reduces the luminescence contribution from the substrate, as will be evidenced in the next section. The CL spectrum and hyperspectral map of the CL signal as a function of the wavelength and antenna reduced radius are presented in Fig. 5 for a 170 nm diameter antenna.

This antenna clearly exhibits a dipole-like LDOS spatial pattern, characteristic of the fundamental (n = 1, m = 1) antenna mode, as can be seen in Fig. 5(b). However, the spectrum presented in Fig. 5(a) exhibit a very broad unresolved double peak resonance centred at 845 nm. Using the hyperspectral mode of the setup, we image the LDOS supported by the antenna on the high and low energy side of this resonance. Fig. 5(c) shows the LDOS using the integration bandwidths coloured in Fig. 5(a). It reveals that due to fabrication imperfections, the antenna in not perfectly circular, resulting on an apparent splitting of the fundamental (n = 1, m = 1) mode and breaking of the circular symmetry. We note the the CL signal seems stronger on the upper half of the antenna for both the horizontal (H) and vertical (V) modes in Fig. 5(c). We attribute this effect to the progressive SEM induced contamination of the antenna surface as we scan the beam over it (total acquisition time is ∼30 min)

VI. TOWARDS HIGHER ENERGIES

Having completely characterize the mode structure of small circular Au plasmonic patch antennas, we fabricated Al patch antennas on an Al mirror to extend their operation wavelength to higher energies. vide optical resonances for wavelengths from 550 to 750 nm, while the Al patch antennas in Fig. 6(c) exhibit resonances spanning wavelengths from 450 nm to 650 nm. We note however that there is a trade-off between the vertical confinement of the mode in the oxide layer and the high energy operation of the patch structures. Because most of the energy of the symmetric vertical mode profile is confined inside the oxide layer, increasing the strength of the mode requires a smaller oxide thickness. This leads in return to a stronger confinement of the electric field and hence a stronger refractive index sensed by the mode, which results in a red-shift of the resonant wavelengths for a given patch diameter. Furthermore, the plasmonic response of Al particles is limited to the blue and green parts of the visible electromagnetic spectrum, since a broad interband transition is present in Al at 800 nm [START_REF] Gérard | Aluminium plasmonics[END_REF] .

LDOS imaging experiment on each resonance confirm that the change in material allows to tune the resonant wavelengths of the antennas but does not affect their supported mode structure, which is only characteristic of the antenna geometry. We note that due to the roughness of the Al film, LDOS imaging was more complicated in the case of Al antennas.

The presence of such large Al grains produces a lot of parasitic signal arising from surface plasmon modes strongly localized on the grains. Parasitic CL signal also originates from the diffraction by the surface roughness of the continuum of surface plasmon polaritons excited by the electron beam.

VII. CONCLUSION

In conclusion, we have used cathodoluminescence spectroscopy and imaging to investigate the spectral and spatial plasmonic properties of circular plasmonic patch antennas made of Au and Al. In a first step, we have characterized Au patch antennas on an Au mirror.

Their characteristic spectrum shows several resonances above 600 nm. Energy-resolved CL imaging allowed us to image the radiative LDOS corresponding to the resonances. We have linked the CL properties of these antennas to a semi-analytical model presented in ref. [START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF] .

Contrary to what is reported in this reference, we arrive here at a one-to-one correspondence between the measured and simulated LDOS at the expense of correcting the surface plasmon To perform LDOS imaging, the CL signal in filtered in energy using the exit slit of the spectrometer and sent on an APD. The APD output is connected to an electronic pulse generator. We ensure that the amplitude and time of the pulse are set so that we operate in a photon counting mode. Typically, the pulse duration is one third of the pause time of the electron beam on each position. The resulting image is thus composed of pixels of discrete intensity values. A digitalisation step is perform to convert these discrete intensity values into a number of counting events. To account for the spatial extension of the electron beam, a spatial Gaussian filter of 10 nm width is applied on the signal.

We can compare the results obtained from the two CL setups using Fig. 7, where the CL

Figure 1 .

 1 Figure 1. Sketch of the three patch antennas configurations: (a) Au patch and Au mirror, (b) Au patch and Al mirror, and (c) Al patch and Al mirror. In each case, the Al 2 O 3 spacing layer thickness is 57 nm. Below each sketch, we show an SEM image of a typical antenna. Notice the difference in surface roughness from (a) to (c) due to the poor quality of the thick Al film.

Figure 2 .

 2 Figure 2. (a) Typical CL spectrum of a Au patch plasmonic antenna of 590 nm diameter. The background contribution is shown in red dashed line. The LSPRs of the antenna are indicated, with the Roman number corresponding to the LDOS images in panel (b) and branches in panel (d). (b) Corresponding energy-filtered CL images revealing the LDOS pattern associated with the first six resonances branches in (c). The yellow circles represent the antenna physical size. (c) Waterfall plot of the CL spectra obtained for different antenna diameters. The red lines follow the position of the resonances (marked in black) and serve as guides for the eye. They are numbered from the lowest order (smallest antenna) to the largest order in small Roman numbers. (d) Corresponding intensity color-coded spectral map of the antennas CL signal as a function of wavelength and antenna diameter. The resonance branches are evidenced in green lines as guides for the eye, with

  (b) in red (symmetric mode) and blue (antisymmetric mode). The symmetric mode has most of its energy confined inside the spacing layer, while the antisymmetric mode is rather confined at the top air/metal interface. The respective dispersion relations of the two modes are presented in Fig3(c), where the solid lines represent the real part of the dispersion relation, and the dashed lines its imaginary part. The antisymmetric mode shows low losses but is almost index-matched with the free-space dispersion relation in air, and is thus poorly confined at the metal surface. The symmetric mode significantly deviates from the free-space propagation in alumina, indicating a strong confinement. It also suffers from greater losses than the antisymmetric one. Because it presents a very strong electric field inside the alumina layer and matches the polarisation of the exciting field created by the electrons, this mode plays a dominant role in the CL experiment. Hence, as in refs.[START_REF] Mohtashami | Nanoscale excitation mapping of plasmonic patch antennas[END_REF][START_REF] Kuttge | Ultrasmall mode volume plasmonic nanodisk resonators[END_REF] , we restrict the subsequent analysis to the symmetric vertical mode profile.

Figure 3 .

 3 Figure 3. (a) Sketch of the geometry considered in the analytical model developed in 17 . (b) Sketch of the experimental geometry, with the two vertical mode profiles sustained by the patch antenna: symmetric (red) and antisymmetric (blue). Note that in our case, only the patch is of finite radius R. (c) Dispersion relations of the symmetric (red) and antisymmetric (blue) mode profiles. The real part of the wave vector k n,m is represented in solid lines, and its imaginary part in dashed lines.The black dashed lines are the photonic dispersion relations in air and alumina.

Figure 4 (

 4 Figure 4(a) shows the comparison between the measured spectrum (thick red line) and the predicted resonances (thin lines) for a 270 nm diameter antenna (smallest antenna in Fig. 2(c)). The main contribution to the LDOS originates from the three Bessel modes indicated in thin solid lines, with their radial and azimuthal number (n, m) given in the legend. The simulated spatial LDOS map is shown in Fig. 4(b), while the top insets in panel (a) show the measured LDOS using a spectral bandwidth represented by the shaded area in the spectrum. We can see that the resonance at 750 nm is composed of the (n = 1, m = 2)

Figure 4 .

 4 Figure 4. (a) Comparison between the measured antenna CL spectrum (thick red line) and the simulated resonances (thin lines) of a 270 nm diameter antenna. The background contribution is indicated in red dashes. The principal Bessel contributions to the simulated spectrum are shown in thin solid lines. The corresponding Bessel numbers (n, m) are indicated in the legend. (b) Simulated spatial LDOS map: as it has no azimuthal dependence, it is represented as the function of the wavelength and the normalized radial coordinate ρ/R which varies form -1 to +1 along the diameter of the antenna. Note the separation between the m = 0 mode producing a maximum of LDOS at the antenna center, and the m = 0 modes. (c) Measured CL images using the integration bandwidth indicated by the colored area in (a). The yellow circles indicate the antenna physical size. The Roman numbers refer to the mode branches numbers in Figs. 2(b)-2(c).

Figure 5 .

 5 Figure 5. CL spectrum (a) and hyperspectral CL map (b) of a 170 nm diameter antenna. The hyperspectral map in (b) is obtained by summing the spectra of all the pixels whose distance from the antenna center is between ρ and ρ + dρ and normalized to the corresponding surface element. It is symmetric by construction with respect to ρ = 0. It clearly reveals the fundamental antenna mode n = m = 1. (c) LDOS CL image for the spectral integration bandwidths H and V indicated in (a). The degeneracy of the mode is lifted due to the imperfect shape of the disk and lead to the breaking of the circular symmetry. As a consequence we observe a horizontal (H) and vertical (V) splitting of the corresponding LDOS images.

Figure 6 .

 6 Figure 6. CL intensity spectral maps of Au patch antennas on a Au mirror (a), Au patch antennas on an Al mirror (b) and Al patch antennas on an Al mirror (c) representing the CL spectrum of each antenna as a function of wavelength and diameter. Note that in panel (c), the first two spectra show stronger resonances, and have been divided by 2 to increase the overall contrast. Panel (a) also corresponds to Fig. 2(c).

Figure 6

 6 Fig 6(a) and (b) show that the presence of the Al mirror strongly reduces the luminescence contribution from the substrate, resulting in a low background signal for wavelengths below 550 nm in panel (b). Further comparison between panels (b) and (c) show the dispersion relation of the successive antenna resonances, including the dipole-like, fundamental mode

  wave vector dispersion relation by an empirical factor. The increasing complexity of the mode structure supported by the antennas comes from the broadening of the modes due to ohmic and radiation losses. Hence, a single antenna resonance can be composed of several overlapping antenna modes. We then further reduced the size of the circular antennas and used hyperspectral CL imaging to reveal the fundamental mode of Au patch antennas on an Al mirror. Finally, using Al patch antennas on an Al mirror, we demonstrated the possibility to fabricate Al patch antennas providing optical resonances down to 450 nm.

Figure 7 .

 7 Figure 7. (a) CL spectra for a 435 nm diameter Au antenna on an Al mirror, as presented in Fig 6(b). The red line corresponds to the 10 keV excitation, while the blue line corresponds to the 30 keV excitation. (b) Hyperspectral LDOS imaging with the 10 keV beam as a function of wavelength and reduced antenna radius. (c) LDOS images obtained using the 30 keV setup by integrating the CL signal in the shaded spectral regions. They correspond to the second and third mode branches in Fig. 2(b)-(c), starting from the smallest antenna. The yellow circles indicate the antenna physical size.

  properties of a 435 nm diameter Au antenna on an Al mirror are shown. Panel (a) represents the two CL spectra and panel (b) the hyperspectral LDOS map obtained with the 10 keV beam. Panel (c) shows the LDOS images obtained with the 30 keV beam and integrated over the spectral width shaded in under the spectrum. Comparing the LDOS obtained using hyperspectral imaging in Fig. 7(b) and Fig. 7(c) show the same mode structure, i.e. the bright center and double ring structure of the resonance at 575 nm and the double ring structure of the resonance at 700 nm.

R. Esteban, T. V. Teperik, and J.-J. Greffet, "Optical patch antennas for single photon emission using surface plasmon resonances," Phys. Rev. Lett. 104 (2010).

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10484-10503 (2000).

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures," Phys. Rev. B 63, 125417 (2001).

ACKNOWLEDGMENTS

We acknowledge the help of Institut Néel technical support teams Nanofab (clean room) and optical engineering (CL, Fabrice Donatini). The hyperspectral CL experiments were performed on the Nanocharacterization platform (PFNC) at MINATEC.

FUNDING This work was supported by the French National Research Agency, Labex LANEF du Programme d'Investissements d'Avenir ANR-10-LABX-51-01 through the LANEF PhD Grant and the LANEF equipment project TREE.