Mehdi Rzin

Matteo Meneghini

Fabiana Rampazzo

Veronica Gao Zhan

Carlo De Santi

Riad Kabouche

Malek Zegaoui

F Medjdoub

Gaudenzio Meneghesso

Enrico Zanoni

V Gao Zhan

Linearity and robustness evaluation of 150-nm AlN/GaN HEMTs

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

GaN based High Electron Mobility Transistors (HEMTs) hold tremendous interest for millimeter wave applications especially with the emergence of the fifth generation (5G) communications technology, this is thanks to their high electron mobility and breakdown voltage [START_REF] Yuk | Future Directions for GaN in 5G and Satellite Communications[END_REF]. For high power amplifiers (HPA) operating beyond 30 GHz, innovative device downscaling with ultra-short gate length (LG < 200 nm) and ultra-thin barrier layer are needed whilst maintaining high power added efficiency (PAE) combined with high output power in millimeter wave range. Recently, AlN/GaN HEMTs with sub-10nm AlN barrier layer demonstrated promising performance with PAE exceeding 40 % at 40 GHz [START_REF]High PAE high reliability AlN/GaN double heterostructure[END_REF]. AlN/GaN HEMTs with a maximum output current density of 2.3 A/mm and a peak extrinsic transconductance of 480 mS/mm [START_REF]AlN/GaN Insulated-Gate HEMTs with 2.3 A/mm output current and 480 mS/mm transconductance[END_REF] have also been reported.

The transconductance of GaN based HEMTs exhibits significant changes with the gate bias, and this is considered as a critical issue for the linearity performance of the device [4] [5] [START_REF]Improving the Linearity of GaN HEMTs by Optimizing Epitaxial Structure[END_REF]. This limits large signal operation at high frequencies as well as power linearity, since the gain of a HEMT is critically dependent on the transconductance. The devices used in multichannel transmissions tend to produce intermodulation distortion, which leads to the degradation of the system signal-to-noise ratio. The third-order intermodulation distortion (IM3) is considered as the most important parameter for the linearity performance of the devices. In order to reduce IM3, the transconductance needs to remain constant during a wide operating range of the gate bias [START_REF]A δ-Doped InGaP/InGaAs pHEMT With Different Doping Profiles for Device-Linearity Improvement[END_REF].

In this paper, we evaluate for the first time the linearity and robustness of 150-nm AlN/GaN HEMTs on SiC substrate with C-doped GaN buffer and 4-nm AlN barrier. In addition to their outstanding RF performance in terms of high output power combined with high power added efficiency at 40 GHz, AlN/GaN HEMTs show good robustness after 24 hours stress test.

Samples details

The AlN/GaN heterostructures were grown by metal organic chemical vapor deposition (MOCVD) on 4 inch SiC substrates [START_REF]Comparison of C-Doped AlN/GaN HEMTs and AlN/GaN/AlGaN Double Heterostructure for mmW Applications[END_REF]. The HEMT structure consists of transition layers to GaN, a 1 μm-thick Cdoped GaN buffer layer followed by an undoped GaN channel, a 4.0 nm ultrathin AlN barrier layer and a 10-nm-thick in situ Si3N4 cap layer. The in-situ SiN layer is used both as early passivation as well as to prevent strain relaxation. Ohmic contacts were formed directly on the top of the AlN barrier by etching the in-situ SiN cap layer. A Ti/Al/Ni/Au metal stack was used for ohmic contacts followed by Vapor Deposition (PECVD). Room-temperature Hall measurements showed high electron sheet concentrations of 1.8×10 13 and 1.6×10 13 cm -2 with an electron mobility about 1100 cm 2 V -1 s -1 .

DC and double pulse characterization

Fig - 1 shows transconductance (gm) versus VGS characteristics of a representative AlN/GaN HEMT. The extrinsic transconductance peaks at 385 mS/mm at VDS = 10 V and VGS = -1.3 V. The presented devices are indeed still limited by the access resistances. Typical contact resistances are as high as 0.5 ohm.mm severely reducing the extrinsic transconductance. Intrinsic transconductance is in the range of 600 mS/mm for this device design. This high value results from the short gate-to-channel distance [START_REF]High-Performance Low-Leakage-Current AlN/GaN HEMTs Grown on Silicon Substrate[END_REF]. Interestingly, the transconductance of AlN/GaN HEMTs exhibits small changes with the gate bias for VDS > 4 V, unlike for AlGaN/GaN HEMTs. This is owing to the ultra-thin AlN barrier with high Al content, and to the less negative threshold voltage [START_REF]Low interface state density AIN/GaN MISFETs[END_REF] [START_REF]High-mobility window for two-dimensional electron gases at ultrathin AlN/GaN heterojunctions[END_REF]. Fig - 2 shows the double pulse gm versus VGS characteristics at VDS = 8 V at quiescent bias (QB) conditions (0 V, 0 V), (-5 V, 0 V), (-5 V, 5 V) and (-5 V, 10 V) of a representative AlN/GaN HEMT. The pulse width was set to 1 µs with a duty cycle of 1 %. The current collapse rate corresponding to the gate-lag is 16.5 %. Moreover, the drain lag rate between (-5 V, 0 V) and (-5 V, 10 V) corresponds to 13.3 %. A positive shift of threshold voltage, that is correlated with current collapse, is identified at QB (-5 V, 0 V) that increases by increasing drain source QB. Moreover, the transconductance versus VGS remains high at positive VGS under the different QB conditions

Semi on-state step stress

Before starting short or long term reliability tests at the bias targeted for the final application for these devices (in this case high power amplifiers (HPA) in class AB operation), it is mandatory to evaluate the limits and the safe operating area (SOA) of the process. DC step stress test is submitted at semi onstate (VGS > VTH) condition. For this bias point, the gate is biased at VGStress = -1.5 V and the drain voltage is increased from 0 V to device failure by a step of 5 V during 2 min each step [START_REF]Reliability of GaN High-Electron-Mobility Transistors: State of the Art and Perspectives[END_REF]. In this case, the drain voltage compliance was limited to 100 V. During the step stress test, drain and gate currents are monitored, electroluminescence is used after each step in on-state (peak of EL bell versus VGS) and offstate conditions, to check the distribution of the electroluminescence emission on the gate fingers during stress, if there are hot spots induced by the stress showing the creation of leakage paths or for the onset of the device failure. Fig - 3 shows gm-VGS characteristics, at VDS = 0.1 and 10 V, after each step of semi on-state step stress at VGStress = -1.5 V. As illustrated in Fig - 4, the transconductance peak as well as the threshold voltage remain constant until VDStress = 25 V. Moreover, gmpeak decreases almost linearly for VDStress > 25 V to drop by 26% at VDStress = 100 V. Furthermore, the threshold voltage shifts positively for VDStress > 25 V to shift by 46 mV at VDStress = 100 V. Moreover, an abrupt decrease of forward and reverse gate voltage, in addition to an

DC 24h robustness stress

This part focuses on the robustness of AlN/GaN HEMTs, by using 24 hours DC stress test, to assess the root cause for degradation mechanisms under stress limiting the electrical performance by considering semi on-state condition with VDStress ≤ 25 V. In this part, we discuss the robustness of AlN/GaN HEMTs at the bias conditions: (VGStress, VDStress) = (-1.2 V, 10 V), (-1.5 V, 15 V). (-2 V, 20 V) and (-2.5 V, 25 V). Different electrical parameters are recorded after interrupting the stress at 1h, 2h, 4h, 8h, 16h and 24h. Fig - 7 illustrates ID-VDS characteristics of a representative AlN/GaN HEMT with VGS from -3 V to 0 V with a pronounced kink effect. Circular dots are ID-VDS bias-points adopted for 24-hours tests. The bias conditions at VDS = 20 V with ID = 150 mA/mm and VDS = 25 V with ID = 100 mA/mm were chosen according to the bias targeted for the final application for these devices (i.e. high power amplifiers in AB class operation).

(VGStress; VDStress) = (-1.2 V, 10 V)

Fig - 8 shows the variation of gate current during Fig - 7. ID vs VDS output characteristics of representative AlN/GaN HEMT, with VGS from -3 V to +1 V in 0.5 V steps. Circular dots are ID-VDS bias-points adopted for 24-hours tests 24h stress test at different steps: 1h, 2h, 4h, 8h, 16h and 24h, carried out at (VGStress , VDStress) = (-1.2 V, 10 V).

Although the gate current is slightly noisy during the stress, it remains stable at a steady level around 1 µA/mm. Moreover, electroluminescence (EL) emission intensity is uniformly distributed along the gate fingers during the stress, highlighting the good distribution of carriers in the channel, as shown in Fig - 9. Moreover, no hot spots are identified during stress. The microscopy image of the stressed device after 24h stress shows no damage in the device structure. The gate current, during the stress, becomes more noisy starting from 10h stress, however it remains stable with no apparent abrupt increase with |IG| ≤ 1 µA/mm. EL intensity after 1h and 24h stress at (VGStress , VDStress) = (-1.5 V, 15 V) is higher on the edge of gate fingers, as shown in Fig - 12. Furthermore, gmpeak values after the different steps of the 24h stress are lower by less than 5 % with their initial values before stress as illustrated in Fig - 13.

Thus, no abrupt leakage currents increase or device degradation are identified after 24h stress at (VGStress , VDStress) = (-1.5 V, 15 V). The gate current, during the stress, becomes more noisy starting from 12h stress, with an abrupt increase from 4 µA/mm to 100 µA/mm. This result suggests that the gate current degradation, during the stress, is field driven. EL intensity is uniform along both gate fingers with some regions where EL intensity is higher after 1h stress, as shown in Fig - 15, with EL intensity increasing after 24h stress. The gate current at the early hour of stress is at higher value of 100 µA/mm, that keeps increasing during the first 12 hours of stress and stabilizes at about 3 mA/mm at the last 12 hours of the stress.

The gate current increase during 24h stress is field dependent and associated with EL hot spots along the gate fingers as shown in Fig - 18.

The monitoring of gmpeak during 24h stress shows a constant decrease of about 10 % during all the stress, as shown in Fig - 19. The uniform degradation of gmpeak is not correlated with gate current increase during stress. The latest is related more likely to Schottky contact degradation and not to the AlN/GaN heterostructure.

Discussion and conclusions

In this work, we have demonstrated the potential of AlN/GaN HEMTs in terms of linearity and robustness under high electric field up to 25 V. The high performance and reliability are attributed to the high epilayer and processing quality as well as the use of the in-situ SiN passivation enabling in particular an enhancement of the surface robustness.

The degradation of gate current during stress is field dependent related more likely to degradation of Schottky contact or gate/barrier interface.

In spite of adopting only SiN passivation without air bridges or field plates, these devices exhibit state-of-the-art power-added-efficiency of about 50% at 40 GHz, that has been achieved for an output power density in excess of 3 W/mm using gate width of 2x50 µm demonstrating the potential of AlN/GaN heterostructure for millimetre wave applications.

Fig - 1 .

 1 Fig -1. gm versus VGS characteristics of a representative AlN/GaN HEMT a rapid thermal annealing at 875°C. T-shaped Ni/Au Schottky contacts with 150-nm footprint and a top gate length of 400 nm were defined by e-beam lithography. The in-situ SiN underneath the gate pattern was fully removed using a low-power SF6 plasma etching. Device isolation was achieved by nitrogen implantation. A 200-nm thick Ex-situ SiN film was deposited by Plasma-Enhanced ChemicalVapor Deposition (PECVD). Room-temperature Hall measurements showed high electron sheet concentrations of 1.8×10 13 and 1.6×10 13 cm -2 with an electron mobility about 1100 cm 2 V -1 s -1 .

Fig - 2 .

 2 Fig -2. Double pulse gm-VGS characteristics at different QB conditions of a representative AlN/GaN HEMT.

V

  Fig -3. gm-VGS characteristics after each of semi onstate step stress at VGStress = -1.5 V

Fig - 10 Fig - 8 .Fig - 9 .Fig - 10 .

 108910 Fig -8. Variation of gate current during 24h stepstress experiment carried out at (VGStress , VDStress) = (-1.2 V, 10 V) with the electrical parameters monitored at 1h, 2h, 4h, 8h, 16h and 24h of AlN/GaN HEMT

Fig - 11 .Fig - 12 .

 1112 Fig -11. Variation of gate current during 24h stepstress experiment carried out at (VGStress , VDStress) = (-1.5 V, 15 V) with the electrical parameters monitored at 1h, 2h, 4h, 8h, 16h and 24h of AlN/GaN HEMT

1. 3 .

 3 Fig -14shows gate current variation during 24h stress test at different steps: 1h, 2h, 4h, 8h, 16h and 24h, at (VGStress , VDStress) = (-2 V, 20 V).

Fig - 14 .

 14 Fig -14. Variation of gate current during 24h stepstress experiment carried out at (VGStress , VDStress) = (-2 V, 20 V) with the electrical parameters monitored at 1h, 2h, 4h, 8h, 16h and 24h of AlN/GaN HEMT 1h 24h

  Fig -17shows the variation of gate current during 24h stress test at different steps: 1h, 2h, 4h, 8h, 16h and 24h, carried out at (VGStress , VDStress) = (-2.5 V, 25 V) of AlN/GaN HEMT.

Acknowledgements

The authors would like to thank the European Defense Agency (EDA) for the support to this work in the context of the project entitled "European Gallium Nitride Industry supply Chain (EUGaNIC)" funded by Germany, France, Italy and Sweden and coordinated by UMS GmbH in the frame of the Project no B-1447-IAP1-GP of the European Defense Agency.


T D Dao

Q Serra

S Berger

E Florentin
email: eric.florentin@insa-cvl.fr

Error estimation of Polynomial Chaos approximations in transient structural dynamics

Keywords: A posteriori error estimate, Polynomial Chaos Expansion, Structural dynamics

Usually, within stochastic framework, a testing dataset is used to evaluate the approximation error between a surrogate model (e.g. a Polynomial Chaos expansion) and the exact model. We propose here another method to estimate the quality of an approximated solution of a stochastic process, within the context of structural dynamics. We demonstrate that the approximation error is governed by an equation based on the residue of the approximate solution. This problem can be solved numerically using an approximated solution, here a coarse Monte Carlo simulation. The developed estimate is compared to a reference solution on a simple case. The study of this comparison makes it possible to validate the efficiency of the proposed method. This validation has been observed using different sets of simulations. To illustrate the applicability of the proposed approach to a more challenging problem, we also present a problem with a large number of random parameters. This illustration shows the interest of the method compared to classical estimates.

Introduction

In the last few years, stochastic methods have caught the attention of engineers and researchers in many scientific domains. Indeed, these methods allow the prediction within a confidence level of the possible values of the response of a stochastic system of interest, when the input parameters may vary or when some randomness may occur. Introducing variabilities has several applications such as risk management, optimisation and robust design, the robust optimal design being sometimes different from the optimal deterministic design [START_REF] Tootkaboni | Topology optimization of continuum structures under uncertainty -A Polynomial Chaos approach[END_REF]. It is an interesting tool to estimate the robustness of a numerical method [START_REF] Touzeau | Accuracy and robustness analysis of a geometrical finite element model updating approach for material parameters identification in transient dynamic[END_REF]. The biggest drawback of stochastic problems is the huge numerical cost needed to scan the stochastic space. A review of several stochastic methods has been proposed in [START_REF] Xiu | Fast numerical methods for stochastic computations: a review[END_REF][START_REF] Stefanou | The stochastic finite element method: Past, present and future[END_REF].

Monte Carlo (MC) simulation [START_REF] Fishman | Monte Carlo: concepts, algorithms, and applications[END_REF] is the most direct approach and is often used as a reference. It consists in the evaluation of the quantity of interest (QoI) from the stochastic system on a large sample of realizations. The advantage is that it does not require any complex computer code dedicated to stochastic problems, but its drawback is the slow convergence. Thus, it implies a huge computational cost. Variance reduction techniques, for example like Latin Hypercube sampling (LHS) or quasi Monte Carlo (qMC) simulations have been developed to get a better convergence [START_REF] Caflisch | Monte Carlo and quasi-Monte Carlo methods[END_REF].

Several methodologies based on a spectral expansion have been developed to tackle the cost associated with sampling methods. Perturbation methods are based on a Taylor series expansion around the mean value of parameters.

They are limited to the case of small variations of the random field around its mean value, and they cannot be applied to non-smooth functions. The Polynomial Chaos (PC) expansion is a spectral method, consisting in searching an approximation of the stochastic response in the space spanned by a finite basis of orthonormal polynomials. It lies on the Homogeneous Chaos theory started by Wiener [START_REF] Wiener | The Homogeneous Chaos[END_REF] in the case of Gaussian random variables. It has been extended later to non-Gaussian random fields [START_REF] Askey | Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials[END_REF][START_REF] Xiu | The Wiener Askey Polynomial Chaos for Stochastic Differential Equations[END_REF] and called generalized Polynomial Chaos expansion (gPC). Different interesting applications have been presented in [START_REF] Ghienne | Stochastic model reduction for robust dynamical characterization of structures with random parameters[END_REF][START_REF] Weiss | Uncertainty quantification for a 1d thermo-hyperelastic coupled problem using polynomial chaos projection and p-fems[END_REF][START_REF] Hosseini | Solution of stochastic nonlinear time fractional pdes using polynomial chaos expansion combined with an exponential integrator[END_REF][START_REF] Wang | Uncertainty propagation analysis of t/r modules[END_REF]. To determine the coefficients of the expansion, two strategies are classically adopted. An intrusive strategy is obtained when the PC expansion is used together with a Galerkin approach, resulting in a very large numerical system where the unknowns are the amplitudes of the coefficient of the expansion. However, it results in a large computational cost and in the need to write a dedicated program. The second kind of methods is non-intrusive, it only uses as inputs the outputs of a deterministic program. Two approaches are classically used to evaluate the expansion coefficients. The first one among them is called Non-Intrusive Spectral Projection [START_REF] Crestaux | Polynomial chaos expansion for sensitivity analysis[END_REF] and consists in the numerical evaluation of the projection (scalar product in stochastic space) of the QoI on the space spanned by the polynomial basis. The second one relies on linear regression to compute the best (in a least square sense) coefficients of the PC expansion.

Compared to the static case, in transient dynamics the time can simply be seen as an additional parameter. Using the same basis in each time step is called a time-frozen PC expansion and results in large errors in the evaluation of the long-term response [START_REF] Wan | Long-term behavior of polynomial chaos in stochastic flow simulations[END_REF]. Several strategies to add physical information to the variation of the spectral coefficients have been proposed in [START_REF] Gerritsma | Time-dependent generalized polynomial chaos[END_REF][START_REF] Ozen | A dynamical polynomial chaos approach for long-time evolution of SPDEs[END_REF]. To control the error made by the PC expansion, which is the distance between the response and its projection in the space spanned by the chosen polynomials, error estimates are needed. Several works investigated the a priori convergence behavior of gPC [START_REF] Wan | Long-term behavior of polynomial chaos in stochastic flow simulations[END_REF][START_REF] Todor | Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients[END_REF]. Recently, a number of authors have developed different strategies to build adaptive gPC, with an error control based on the variance decrease rate [START_REF] Wan | An adaptive multi-element generalized polynomial chaos method for stochastic differential equations[END_REF], the residual decrease rate [START_REF] Chouvion | Development of error criteria for adaptive multi-element polynomial chaos approaches[END_REF][START_REF] Mac | Residual-based a posteriori error estimation for stochastic magnetostatic problems[END_REF], or scalar values related to the residue of the least square problem solved in regression such as the R 2 coefficient or the Leave One Out LOO error [START_REF] Blatman | Adaptive sparse polynomial chaos expansion based on least angle regression[END_REF][START_REF] Sudret | Polynomial chaos expansions and stochastic finite element methods[END_REF][START_REF] Mai | Hierarchical adaptive polynomial chaos expansions[END_REF]. However, even if relevant, such indicators only provide a qualitative information on the quality of the meta-model, and thus need to fix some tolerance value arbitrarily. Another possibility, often used in statistical learning, consists in measuring the empirical error (Mean Square Error MSE) on a testing dataset. However, the small size of this dataset leads to a non-robust estimate of the MSE.

As far as the authors know, no a posteriori error estimate has been proposed yet to quantify, even approximately, the distance between the response and the PC metamodel in the transient dynamic framework. It is a problem especially because the Polynomial Chaos suffers from the curse of dimensionality, with a computational cost becoming unaffordable for expensive full-scale industrial finite element problems with numerous random parameters. In such a case, only low-order PC approximations are affordable, no a priori tolerance is known to set a value for LOO error, so quantitative a posteriori error estimates are of first importance to evaluate the degree of validity of the metamodel.

In this paper, we propose to use a coarse Monte Carlo simulation to solve a stochastic equation associated to the reference problem. The goal is to evaluate at a low numerical cost the error due to the PC expansion. The manuscript is organized as follows. Theories about MC simulation and PC expansion are recalled in sections 2, an example consisting in a two degrees of freedom (dof) problem is presented. The error estimation strategy is presented in section 3. Additional numerical examples are provided in section 4 before concluding in section 5.

Usual algorithms for solving linear transient dynamic problems with parametric uncertainties

Reference problem: mechanical system with parametric uncertainties

In a general way, assuming that the problem has been discretized in a finite number (n) of discrete unknowns, the dynamic equilibrium of a linear mechanical system can be written for each time t ∈ [0, t f ] as:

         M ẍ(t) + C ẋ(t) + Kx(t) = F(t) x(t = 0) = x 0 ẋ(t = 0) = ẋ0 (1)
where M, C and K are respectively the (n×n) mass matrix, damping matrix and stiffness matrix of the discrete system, x is the n-vector of displacements which are unknown, x 0 and ẋ0 are respectively the initial displacement and initial velocity, F is the vector of loads and t f the final time. The dots above previous expressions are used in place of time derivatives to alleviate the expression.

In the case of parametric uncertainties, some of the values in the matrices and in the vector of external loads can be random and independent parameters, so that their values observation θ. Consequently, the dynamical response x is a random process. In the following, we denote as r the number of random independent parameters. To these parameters are associated the vector of uniform dimensionless parameters ξ ∈ [-1; 1] r . Thus, each the variables appearing in

k u k s c m u m s x 2 (t)
x 1 (t) Eq. ( 1) can be eventually expressed as a function of the dimensionless parameters ξ. To alleviate the expressions, the random vector realization ξ(θ) is written ξ.

z(t)
As an example, we will consider the following stochastic problem with two degrees of freedom, illustrated in Fig. 1, where the quantity of interest is the random process vector:

x(t) = [x 1 , x 2 ] T (2)
The mass, damping and stiffness matrices are given by:

M = m s 0 0 m u , C = c -c -c c , K = k s (ξ) -k s (ξ) -k s (ξ) k s (ξ) + k u (ξ) (3)
The load vector consists in a prescribed base displacement z(t), shown in Fig. 2:

F(t) = [0, k u z(t)] T (4)
The initial conditions are taken equal to zero. Values of the parameters are provided in Tab.

Notations

This section aims at defining the notations appearing in the following parts of the paper. The following notations are used for the mean E[•] and the variance V[•] of any random vector:

µ • = E[•] ; σ • = V[•] (5)
These quantities depend on time. We also need to define time-averaged values of a function defined over [0, t f ]. We use as time-average function the operator defined for any function of time by:

• = 1 t f t f 0 •dt (6)

Polynomial Chaos approximation

In order to reduce the CPU cost when solving stochastic problems, the Polynomial Chaos approximation is often used to solve the stochastic problem associated to Eq. [START_REF] Tootkaboni | Topology optimization of continuum structures under uncertainty -A Polynomial Chaos approach[END_REF]. A polynomial chaos approximation of a random process consists in the projection of the exact random process in the space spanned by a family of polynomials. Thus, for any realizations of the vector of input parameters, the QoI of the metamodel is computed through a linear combination of polynomials [START_REF] Kewlani | A polynomial chaos approach to the analysis of vehicle dynamics under uncertainty[END_REF][START_REF] Nechak | A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems[END_REF][START_REF] Blatman | An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis[END_REF]. Thus the exact realizations of the random variable x are approximated for the same parameters by the realizations of the approximate random variable x PC such as:

x(ξ, t) ≈ x PC (ξ, t) = n PC j=1 X j (t)φ j (ξ) (7)
where φ j are the chaos polynomials and X j are the n PC chaos coefficients. Although theoretically any infinite family makes it possible to represent exactly the random process, numerically a finite family is needed, so that there is a relationship between the choice of the polynomial family and the convergence of the expansion Eq. ( 7). An optimal choice of the polynomial family depending on the distribution of the random input parameter ξ is recalled in Tab. 2 for continuous distributions [START_REF] Askey | Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials[END_REF].

Random variable ξ Wiener-Askey PC φ(ξ) Support Gaussian Hermite (-∞, +∞) Uniform Legendre [a,b] Gamma Legendre [0, ∞) Beta Jacobi [a,b]
Table 2: Correspondence between the distribution of the input parameter and the optimal choice of the polynomial basis Several methods exist to estimate the chaos coefficients X j . Non-intrusive methods are often preferred because they do not require a dedicated computational code, but only repeated access to a deterministic code. Among nonintrusive versions can be found the Non-Intrusive Spectral Projection and the regression method. We refer the interested reader to [START_REF] Crestaux | Polynomial chaos expansion for sensitivity analysis[END_REF][START_REF] Sudret | Polynomial chaos expansions and stochastic finite element methods[END_REF][START_REF] Blatman | Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach[END_REF] for details. Because it is often used in recent literature, we only present the regression approach here. The regression approach consists in searching an optimal set of chaos coefficients X j in the least square sense:

X 0 (t), X 1 (t), ..., X n PC (t) = Argmin X 0 (t),X 1 (t),...,X n PC (t)          1 Q Q q=1         x(ξ q , t) - n PC j=1 X j (t)φ j (ξ q )         2          (8)
where

{ξ 1 , ξ 2 , ..., ξ Q } are the Q realizations of the input parameters.
This minimisation is a classical least square minimisation. Introducing the so-called information matrix Φ which is the matrix of the polynomials evaluated at the sampling points:

Φ = (Φ q j ) q∈{1..Q}, j∈{1..N} with Φ q j = φ j (ξ q ), (9)
finding the solution results in computing the pseudo inverse of Φ denoted Φ + :

Φ + = Φ T Φ -1 Φ T (10)
Finally, the number of coefficients is determined by the number of random input parameters r and the maximal order p fixed for each polynomial. Thus, a full tensor product of monovariate polynomials leads to n PC = (p + 1) r polynomials.

Sparse truncation techniques [START_REF] Blatman | Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach[END_REF] are often used to reduce computational burden. We will only consider the classical truncation scheme here, which conduces to a reduced number of polynomials:

n PC = (p + r)! p!r! . ( 11
)
With the regression method, a minimum of Q = n PC simulations are necessary, which may be performed in practice at the Q = (p + 1) r Gauss collocation points or at points chosen with a Latin Hypercube Samples (LHS) method [START_REF] Berveiller | Stochastic finite element: a non intrusive approach by regression[END_REF]. In the latter case, Q = kn PC simulations (with k a small integer usually equal to 2, 3 or 4) are used.

In this work, we do not aim at reducing the cost of the surrogate model, but we aim at predicting the estimation error within a reasonable cost.

Error due to the PC approximation

Using x PC as an approximation of x following Eq. ( 7) leads to an error defined as:

e = x PC -x (12)
A measure of the error, called "generalization error", can be defined:

ε 2 = E[e T e] (13)
This global measure of the Mean Square Error corresponds to the sum of the local contribution ε 2 i of each component of error e i :

ε 2 = n i=1 ε 2 i = n i=1 µ e 2 i ( 14
)
Several error indicators have been used in the literature to estimate this error with a reasonable cost. Among them, we choose two estimates. The first is the empirical error on a training dataset, because it is intrinsic to the PC approximation and is naturally available directly with the results. Therefore it is largely used in the literature, in particular in work based on adaptive schemes [START_REF] Cheng | Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression[END_REF]. The second one is the The Leave One Out error which is a particular case of the k-fold cross validation techniques. It is also very practical to implement. Let us recall that we are interested into a cheap error estimates that can be easily implemented into industrial softwares.

The empirical error on training dataset. The Empirical Error (EE) is a measure of the residue of the minimization Eq. ( 8). The idea is to approximate the empirical error using the different Q sampling points defined in the PC approximation. The empirical error ε emp is then simply computed at a very low cost using:

ε 2 emp = 1 Q Q q=1 x(ξ q ) -x PC (ξ q ) 2 = 1 Q Q q=1 e(ξ q ) T e(ξ q ) ( 15
)
It can be remarked here that the regression approach of the PC naturally minimizes the empirical error. This estimate can also be seen as a coarse Monte Carlo approximation of ε using the sampling points.

Local error estimates ε emp,i can be used to have some information about the local quality of the i th component of

x. We have:

ε 2 emp = n i=1 ε 2 emp,i with ε 2 emp,i = 1 Q Q q=1 e 2 i (ξ q ) ( 16
)
The Leave One Out error. The Leave One Out (LOO) error [START_REF] Allen | The prediction sum of squares as a criterion for selecting predictor variables[END_REF][START_REF] Molinaro | Prediction error estimation: a comparison of resampling methods[END_REF] is a measure of the error derived from crossvalidation techniques. Considering first a set of Q deterministic samples, it is possible to take Q -1 samples arbitrarily -that is a set {1, ..., Q}\q -and to derive the associated PC expansion: x (\q) PC .

The error on the sample ξ q between the new PC approximation and its actual value is thus given by:

∆(ξ q ) = x (\q) PC (ξ q ) -x(ξ q ) ( 17
)
The total leave one out error is then computed by taking the root mean square value of the errors ∆(ξ q ) according to:

ε 2 LOO = 1 Q Q q=1 ∆(ξ q ) T ∆(ξ q ) ( 18
)
The LOO error estimate can be obtained by post-processing one single run of Polynomial Chaos as [START_REF] Blatman | Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis[END_REF]:

ε 2 LOO = n i=1 ε 2 LOO,i with ε 2 LOO,i = 1 Q Q q=1 1 h 2 q e 2 i (ξ q ) ( 19
)
where h q are the diagonal terms of the matrix :H = Id -ΦΦ + and where Id is the identity matrix. We can remark that the empirical and LOO errors only differ by the h q ponderation.

The interest of ε LOO or ε emp is that their calculation is based only on the values of the QoI in the training dataset, so the numerical cost is very low. However, they only provide qualitative information on the level of error, as it will be illustrated in the next section. An arbitrary tolerance values has to be fixed to evaluate if the PC expansion approximates the random variable correctly.

The empirical error on a testing dataset. A third measure is very classical in statistical learning. It is another empirical measure, based on a coarse Monte-Carlo method. One can remark that the empirical error based on the training dataset (values used to compute the surrogate) is minimal, because the surrogate coefficients are computed through a leastsquare minimization of this empirical error. So, one can think that these points are not appropriate to evaluate the error. The idea is to measure an empirical error on arbitrary random points in the stochastic space, different from the training points. The cost is much larger, because the exact response of the QoI is needed on the testing dataset. In typical applications the following recipe is often applied: 70% of the sampling points are affected to training, and 30% are affected to tests. This way of measuring the error is more costly, but leads to error levels much more realistic than with the empirical error on training datasets. However, the inherent question is the choice of the sampling points, and its consequence on the variability of the error estimate.

Reference solution : Monte Carlo estimation of mean response

Monte-Carlo method is a sampling based technique, resulting from the law of large numbers. If a large number of realizations θ are observed, the statistical moments converge toward the exact moments. Thus, it is often called 'brute force' because it consists simply in the evaluation of the deterministic problem for a large number of realizations.

If all the parameters ξ are independent, and for a large number of realizations θ, the central limit theorem provides a bound of the error between the first actual moment and the statistical mean computed with n MC realizations ξ

i , i ∈ {1, 2, . . . n MC }.
Using the n MC realizations of x 1 , the first component of x, one can compute its empirical mean µ emp,n MC x 1 and its empirical standard deviation σ emp,n MC x 1 as follows:

µ emp,n MC x 1 = 1 n MC n MC i=1 x 1 (ξ i ) (20)
and:

(σ emp,n MC x 1 ) 2 = 1 n MC -1 n MC i=1 (x 1 (ξ i ) -µ emp,n MC x 1 ) 2 (21)
The difference e

1 = |µ x 1 -µ emp,n MC x 1
| represents the error on the mean of x 1 due to the Monte Carlo estimation and satisfies:

| µ x 1 -µ emp,n MC x 1 | ≤ C MC σ x 1 √ n MC (22)
where C MC is a constant depending on the level of confidence desired, and following Student's law -typically the value C MC = 1.96 corresponding to a 95% confidence interval is chosen associated to a normal distribution when , providing the Monte Carlo error estimate 95% :

x 1 ,95% = 1.96

σ emp,n MC x 1 √ n MC (23)
The convergence rate (here in √ n MC ) obtained by sampling randomly the space of random parameters ξ can be improved by using quasi-random sequences such as a Sobol's sequence, a Halton's sequence, or by using a Latin Hypercube discretization of the random space [START_REF] Blatman | Quasi random numbers in stochastic finite element analysis[END_REF][START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF].

In the following, because the actual moments µ x 1 ,µ x 2 are not known, they are computed approximately with a Monte Carlo method using n MC = 10 6 . Thus, in the following, we consider the following equality as true for each quantity •:

µ • = µ emp,10 6 • (24)
On Fig. 3 we compare the time-averaged mean value of the square of the first degree of freedom x 1 , that is In blue, a bar corresponding to the approximation µ x 2 1 ± x 2 1 ,95% is plotted for each value of n MC .

µ emp,n MC
On Fig. 3 it can be observed that even for a small number n MC , the error estimate x 2 1 ,95% allows to predict accurately the variability in the Monte Carlo estimate.

Example of the Monte Carlo estimation of mean response for PC expansion

To illustrate the previous section, results of simulation are provided in time domain. We compare here very fine Monte Carlo MC result (n MC = 10 6 ) to Polynomial Chaos (PC) for several orders of polynomials p. The QoI is the displacement of the car x 1 (ξ, t). Convergence of the PC approach can be observed in Fig. 5, which shows the value of the time-averaged generalization error µ e 2 (reference computed by Monte-Carlo method with 10 6 samples) for different polynomial orders. Thus, PC makes it possible to predict accurately the response of the stochastic random variables with a computational cost several order lower than classical Monte Carlo techniques.

Finally, classical error indicators are compared to exact error ε 2 in Fig. 6 for p=2.

It can be observed, that neither the qualitative or quantitative agree with the generalization error. In the following, the objective is to show if another method can predict more accurately the error, especially for low polynomial orders, at a cost reasonably higher than LOO.

Estimation of the a posteriori error through the resolution of a coarse Monte Carlo problem

We have seen in the previous section that classical error measures are not perfectly adequate because they only provide a qualitative value of the generalization error. In this part, we aim at developing an algorithm allowing to predict the numerical value of the generalization error, with a low numerical cost.

Strategy

Introducing error e defined in Eq. ( 12) into equation Eq. ( 1) the problem can be written:

M(ξ)ë(ξ, t) + C(ξ)ė(ξ, t) + K(ξ)e(ξ, t) = R(ξ, t) ( 25
)
where R is the residue associated to the PC approximated solution of the dynamic equilibrium:

R(ξ, t) = M(ξ) ẍPC (ξ, t) + C(ξ) ẋPC (ξ, t) + K(ξ)x PC (ξ, t) -F(ξ, t) (26)
Thus, the generalization error e is solution of a linear differential equation in time of second order. The idea presented in this paper consists in using a coarse Monte Carlo scheme to solve equation Eq. ( 25). Validation of the approach is carried out by comparison with a reference value obtained with a reference solution.

Coarse Monte Carlo

We recall that the reference is obtained by setting n MC = 10 6 , however the associated cost is prohibitive in most practical applications. We propose here to evaluate the mean square error using a coarse Monte Carlo simulation i.e.

n MC << 10 6 . Then, an estimate of the mean square error is defined as:

ε 2 MC = µ e 2 MC ,n MC (27)
where e MC is a solution of Eq. ( 25) on the coarse MC sampling. Its time-averaged value is denoted ε 2 MC . Then, the time-averaged efficiency of the estimate ε 2 MC is quantified by the scalar η 2 defined by

η 2 = ε 2 MC ε 2 (28)
We will study local error associated to dof x 1 (resp. x 2 ), we denote η 1 (resp. η 2 ) its efficiency. Similarly η 12 corresponds to the efficiency of the global error estimate.

This quantity can be defined either for local error (i.e. e i ) as well as for global error (i.e. e) and depends on the number of samples considered and on the choice of the samples.

From Eq. ( 23), it can be seen that error of Monte Carlo decreases when n MC is large or when the variance of the random variable is low. Fig. 7 presents the time-averaged variance of the response and the time-averaged variance of the error, for each of the dofs and for the global response, computed with a 10 6 random Monte Carlo scheme. It is clear that if the Polynomial Chaos order is high enough, the error is low, and then the variance of the error is much lower than the variance of the response. In such a case, a coarse Monte Carlo approach may be efficient on the error problem Eq. ( 25), while it is not converged if applied on the initial problem Eq.( 1). This is the key point of the present paper.

Finally, an additional parameter subjected to investigation is the choice of points, that is the design of experiment leading to ε 2 MC . Three possibilities are tested here: a random choice of sampling (Rand), a Latin Hypercube with random smoothing (LHS), or a deterministic choice based on a uniform sampling (US) in every stochastic direction.We choose these three DoE because of their simplicity and very common use in the field of mechanical engineering. Lots of refined samplings could lead to finer results. We decided here to choose the simplest ones, with the idea to illustrate the methodology with generic sampling schemes. Different variants can be derived from this choice, for example the Sobol, and Halton sampling [START_REF] Szepietowska | Sensitivity analysis based on non-intrusive regression-based polynomial chaos expansion for surgical mesh modelling[END_REF] or the sequential sampling schemes [37] Variances of response and of error

V [x 2 1 ] V [e 2 1 ] V [x 2 2 ] V [e 2 2 ] V [x 2 1 + x 2 2 ] V [e 2 1 + e 2 2 ]
Figure 7: Variances of the response x 2 and of the error e 2 .

Results

The Polynomial Chaos surrogate is generated by using regression approach. We choose as deterministic samples the roots of the polynomials of order p+1. Error estimates are presented for several polynomial orders in the following.

Fig. 8 and Fig. 9 present the efficiency η 2 for three choices of sampling when the number of realizations increases.

The meta-model approximation has been carried out with a polynomial order p = 2 and p = 4 respectively. US being a deterministic sampling, it leads to only one result. On the contrary, the estimates based on LHS or random sampling (Rand) depend on the sampling, so different tries with the same size do not lead to the same value of the predicted estimate. It can be seen first that the three sampling strategies converge to the exact value η 2 = 1 for the two cases p = 2 and p = 4. Even if consists in a reassuring result, we recall that the aim is to develop a cheap error estimate, using thus a small number of sampling points. For small values of n MC , due to the small numbers of realizations of the resolution of the error equation Eq. ( 25), the error estimate ε 2 MC is subject to large variations. It can be observed that a LHS sampling decreases this variability and should be tried first instead of a pure random sampling. Moreover, Uniform Sampling outperforms the two other sampling strategies in our example: because the sampling strategy is deterministic, only one computation leads to a result, which belongs to the range of possible values obtained with a random sampling strategy. Even if this choice may not be the best sampling which could be achieved with a random sampling, the accuracy of the estimate is quite correct. However, this sampling suffers from the curse of dimensionality, so that this approach may not be applicable when a large number of stochastic parameters is considered. In this case, a LHS sampling is preferable.

Next, in order to investigate the behaviour of the method when a small number of samplings is used, we complete Fig. 8 and Fig. 9 by plotting in Fig. 10 to Fig. 13 the efficiency of the method using an uniform sampling (US) of 3 to 10 points for each stochastic parameter. The convergence of η 2 exhibits a smooth behaviour. Classically, in the literature, an error estimated is considered correct if its efficiency is between 0.5 and 2. Here, even the smallest sampling satisfies this condition.

Finally, Fig. 14 shows that the local and global mean square errors, obtained with 16 and 81 samples (uniforming sampling of 4 or 9 points for each stochastic variable), are well predicted. A trading between precision and cost has to be found, as a larger number of points leads to a better superposition, with a higher numerical cost. Here, the use of 81 points is presented for illustration, but this choice is not necessary since the predicted error is already correct with only 16 samples. Furthermore, in comparison with classical error estimates such as the empirical error and LOO error , it can be observed on Fig. 6 that the method presented here, with only 16 additional computations, enables to predict much more accurately the estimation error. This result is true for local and global errors as well.

Application to a larger benchmark problem

The next section proposes an illustration of the method to a large scale problem in this section. A truss structure is constituted of 9 different beams. Each beam has its own material properties, considered as constant on the length of the beam. Among their properties, two are considered as stochastic: the Young's modulus and the density. Filnally, it leads to 18 stochastic parameters. The truss is discretized using Euler-Bernoulli beams with 3 degrees of freedom per node (two translations and one rotation). The structure is subjected in time domain to a half-sine load on one node, starting at 10ms. Time integration is carried out using a classical Newmark scheme. We consider here only the error between the stochastic surrogate and the stochastic problem, so we do not take into account the error due to the time discretization and to the finite element discretization. An illustration of the truss structure is given in Fig. 15. To evaluate the error, considering that the exact stochastic response is not reachable with the Monte Carlo method due to the computational cost, we use the developed strategy in this paper. The exact error is not computable at an affordable cost so there is no way to know a priori about the minimal size of the DoE (Design of experiment). Consequently, the error is computed with the present approach for a growing number of LHS-samples, until convergence of the time-averaged predicted mean square error on the horizontal displacement. We chose to stop iterations when the relative tolerance between two steps is lower than 10%. The DoE is computed using a LHS design, the uniform sampling in a 18-dimension space being too expensive.

Parameter

Fig. 16 presents the evolution of EE, LOO and present approach approximated errors for the horizontal displacement with time. It can be observed that the EE and LOO estimates do not predict the peak of error at proximity of the load, predicted with the method developed here. Figure 16: Error estimates applied on the truss structure, for a second order polynomial chaos surrogate.

Conclusion

The Polynomial Chaos Expansion is a very powerful tool to study the response of models subjected to parametric uncertainties. Several error values can be found in the literature, but it has been shown that even if their values provide a qualitative answer about the good fit between the exact random unknown and its PC approximation, they do not provide accurate quantitative values. When applied to structural dynamics, it has been demonstrated that the error is solution of an equation similar to the equation of the dynamics of the system, but where the source term is equal to the residue obtained by using the surrogate instead of the exact random variable in the equation of the dynamics. It has to be noted here that the presented approach should be also applicable to other kinds of metamodels. It has been illustrated numerically that solving the error equation with a Monte Carlo approach, even coarsely, provides a good information on the error level. We recommend the use of quasi Monte Carlo samplings to reduce the variance of the predictions. Active learning strategy based on sequential sampling could be a promising solution to be studied. In a forthcoming paper, a variant of this method will be studied on non linear problems. The idea is to solve the error problem using a dedicated Polynomial Chaos approximation.

Figure 1 : 2 -Figure 2 :

 122 Figure 1: 2-dofs system considered in the study

x 2 1 to its converged value µ x 2 1 2 1 2 1of the mean µ x 2 1 .

 11221 for different (small) values of the number of samples n MC . Different values of µ emp,n MC x are obtained when different samples are used. Consequently, even when the number of samples is fixed the empirical mean may vary. In Fig. 3, for a given number of realization n MC , 100 different sets of n MC realizations are computed and lead to different approximations µ emp,n MC x Each value is presented by a + on Fig. 3. The value of µ x 2 1 is 0.07505.

Figure 4 :

 4 Figure 4: Comparison between MC and PC of mean of QoI and 95% confidence interval of the two degrees of freedom.

Fig. 4

 4 Fig. 4 presents the mean of the QoI (thick line) as well as the 95% confidence interval of the QoI as a function of the time.The mean of the PC surrogate is exactly obtained with the first term of the expansion. It can be observed that both orders lead to a good superposition of both the mean of the QoI and the confidence interval. Furthermore we observe the well known result stating that in time simulations, time-frozen PCE lose accuracy as time increases [15] . Indeed, superposition is observed until 4s with p=2 while superposition occurs until 6s with p=4. In Tab. 3 the amount of deterministic simulation used for both MC and PC are given. For PC, it corresponds to the roots of the chosen Legendre polynomials. MC p=1 p=2 p=3 p=4 Number of computations 10 6 4 9 16 25

Figure 5 :

 5 Figure 5: Convergence of Polynomial Chaos.

Figure 6 :

 6 Figure 6: Evolutions over time of different error estimates.

Figure 8 :

 8 Figure 8: Local and global error estimates for several sampling strategies, with a 2nd order surrogate.

Figure 9 :

 9 Figure 9: Local and global error estimates for several sampling strategies, with a 4th order surrogate.

Figure 10 :

 10 Figure 10: η 2 12 as a function of n MC , with US method and p = 2.

Figure 11 :

 11 Figure 11: η 2 12 as a function of n MC , with US method and p = 3.

Figure 12 :

 12 Figure 12: η 2 12 as a function of n MC , with US method and p = 4.

Figure 13 :

 13 Figure 13: η 2 12 as a function of n MC , with US method and p = 5.

Figure 14 :

 14 Figure 14: Exact and Estimated mean square errors for US sampling with a polynomial order of the surrogate p = 2.

Figure 15 :

 15 Figure 15: Truss structure The material parameters are provided in Tab. 4. The finite element model results in 267 dofs. The stochastic space is a 18-dimension space. The quantity of interest is the horizontal displacement of the node loaded by the force. The metamodel is built with a second order PC expansion. It results in 190 stochastic coefficients to solve, and thus to 380 deterministic computations. The testing dataset consists in 20 samples computed with LHS.

Table 1 :

 1 1. Values of input parameters

	Parameter Probability law	Interval
	k s	Uniform	[360 . . . 440] N/m
	k u	Uniform	[1800 . . . 2200] N/m
	m s	Determinist	20kg
	m u	Determinist	40kg
	c	Determinist	4Ns/m
	z max	Determinist	0.2m
	x 0	Determinist	[0,0]
	ẋ0	Determinist	[0,0]

Table 3 :

 3 Number of deterministic samples.

Table 4 :

 4 Values of input parameters of the truss structure.


Yannick Dufil

Virginie Gadenne

Pascal Carrière

Jean-Michel Nunzi

Lionel Patrone

Growth and organization of (3-Trimethoxysilylpropyl) diethylenetriamine within reactive amino-terminated self-assembled monolayer on silica

Keywords:

Alkane chains are the most commonly used molecules for monolayer fabrication. Long chains are used for their strong van der Waals interactions inducing good layer organization. Amine function-terminated alkyl chains are of great interest and are widely used for further surface functionalization. Since it is mandatory that such layers be organized to provide amine moieties at the surface, the present study deals with exploring amine-terminated SAM formation as an alternative to the usual aminopropylalkylsilane SAM. Additionally, using a long NH2terminated alkyl chain allows the formation of hydrogen bonding thanks to the two NH moieties born along the chain. Furthermore, such hydrogen bonding makes possible to shorten the molecule length while preserving a well-organized monolayer. For this purpose we performed a complete study of the grafting of (3-Trimethoxysilylpropyl) diethylenetriamine (DETAS) on native silicon oxide using various solvents, relative humidity and temperature values. Grafting kinetics was monitored by ellipsometry and goniometry, and SAM structure and organization using AFM and ATR-FTIR spectroscopy. Hydrogen bonding was evidenced within the SAM growth process and in the final complete SAM. We believe such study enables a better control of good quality DETAS SAM in order to improve their efficiency in further surface functionalization applications.

Introduction

Monolayer self-assembling is a convenient, simple and versatile technic to build ordered bi-dimensional nanoscale structures. It has been widely used in order to tailor interfacial properties, such as wettability or hydrophobicity, as well as the work-function of most materials. Self-assembling molecules possess three parts which can be modified to meet the required criteria: the head, the body, and the tail. By proper modification of the body of the molecule one can act on the ability of the molecule to form ordered monolayers. Particularly, organosilane-based self-assembled monolayers (SAMs) have attracted much attention for several years [START_REF] Schreiber | Structure and growth of self-assembling monolayers[END_REF][START_REF] Ulman | An Introduction to Ultrathin Organic Films[END_REF][START_REF] Wang | Silicon surface modification with a mixed silanes layer to immobilize proteins for biosensor with imaging ellipsometry[END_REF][START_REF] Oh | Characteristics of DNA microarrays fabricated on various aminosilane layers[END_REF][START_REF] Cheng | Site-selective self-assembly of MPA-bridged CuHCF multilayers on APTMS-supported gold colloid electrodes[END_REF][START_REF] Keating | Kinetics and thermodynamics of Au colloid monolayer self-assembly: undergraduate experiments in surface and nanomaterials chemistry[END_REF][START_REF] Zhang | Synthesis, characterization and evaluation of uniformly sized core-shell imprinted microspheres for the separation trans-resveratrol from giant knotweed[END_REF][START_REF] Ledung | A novel method for preparation of disulfides on silicon[END_REF][START_REF] Chen | Fullerene self-assembly onto (MeO)3Si(CH2) 3NH2modified oxide surfaces[END_REF][START_REF] Kanwal | Three layered organic memory devices from C60 molecules and insulating polymers[END_REF][START_REF] Manjumdar | Fullerene-based bistable devices and associated negative differential resistance effect[END_REF][START_REF] Chauhan | Self-assembly of the 3-aminopropyltrimethoxysilane multilayers on Si and hysteretic current-voltage characteristics[END_REF][START_REF] Fiorilli | Vapor-phase self-assembled monolayers of aminosilane on plasma-activated silicon substrates[END_REF][START_REF] Gupta | Selfassembled and electrochemically deposited mono/multilayers for molecular electronics applications[END_REF][START_REF] Wang | Morphology and amine accessibility of (3-aminopropyl) triethoxysilane films on glass surfaces[END_REF][START_REF] Allen | Macro-, micro-and nanoinvestigations on 3-aminopropyltrimethoxysilane self-assembly-monolayers[END_REF][START_REF] Morrill | Imaging 3-aminopropyltriethoxysilane self-assembled monolayers on nanostructured titania and tin (IV) oxide nanowires using colloidal silver nanoparticles[END_REF][START_REF] Pavlovic | Surface functionalization of silicon oxide at room temperature and atmospheric pressure[END_REF]. Those SAMs are known as being very stable and robust due to chemical bonding with the surface and siloxane cross-linking. Since they self-assemble on silicon dioxide they offer the possibility to build molecular devices compatible with siliconbased technology. Moreover, lots of commercial organosilane molecules with different terminal moieties make them very attractive to functionalize oxide surfaces. Among these molecules, 3aminopropyltrimethoxysilane (APTMS) is widely used to immobilize various entities via chemical bonding, including proteins [START_REF] Wang | Silicon surface modification with a mixed silanes layer to immobilize proteins for biosensor with imaging ellipsometry[END_REF], DNA [START_REF] Oh | Characteristics of DNA microarrays fabricated on various aminosilane layers[END_REF], metal nanoparticules [START_REF] Cheng | Site-selective self-assembly of MPA-bridged CuHCF multilayers on APTMS-supported gold colloid electrodes[END_REF][START_REF] Keating | Kinetics and thermodynamics of Au colloid monolayer self-assembly: undergraduate experiments in surface and nanomaterials chemistry[END_REF] or other specific molecules [START_REF] Zhang | Synthesis, characterization and evaluation of uniformly sized core-shell imprinted microspheres for the separation trans-resveratrol from giant knotweed[END_REF][START_REF] Ledung | A novel method for preparation of disulfides on silicon[END_REF][START_REF] Chen | Fullerene self-assembly onto (MeO)3Si(CH2) 3NH2modified oxide surfaces[END_REF] such as fullerenes [START_REF] Chen | Fullerene self-assembly onto (MeO)3Si(CH2) 3NH2modified oxide surfaces[END_REF]. Most research groups prepared APTMS SAMs from a liquid phase using various solvents (water, ethanol, toluene, and so on) [START_REF] Wang | Silicon surface modification with a mixed silanes layer to immobilize proteins for biosensor with imaging ellipsometry[END_REF][START_REF] Cheng | Site-selective self-assembly of MPA-bridged CuHCF multilayers on APTMS-supported gold colloid electrodes[END_REF][START_REF] Chen | Fullerene self-assembly onto (MeO)3Si(CH2) 3NH2modified oxide surfaces[END_REF][START_REF] Chauhan | Self-assembly of the 3-aminopropyltrimethoxysilane multilayers on Si and hysteretic current-voltage characteristics[END_REF], and various grafting times [START_REF] Chauhan | Self-assembly of the 3-aminopropyltrimethoxysilane multilayers on Si and hysteretic current-voltage characteristics[END_REF][START_REF] Fiorilli | Vapor-phase self-assembled monolayers of aminosilane on plasma-activated silicon substrates[END_REF][START_REF] Gupta | Selfassembled and electrochemically deposited mono/multilayers for molecular electronics applications[END_REF][START_REF] Wang | Morphology and amine accessibility of (3-aminopropyl) triethoxysilane films on glass surfaces[END_REF][START_REF] Allen | Macro-, micro-and nanoinvestigations on 3-aminopropyltrimethoxysilane self-assembly-monolayers[END_REF]. However, APTMS very often leads to disordered multilayers [START_REF] Morrill | Imaging 3-aminopropyltriethoxysilane self-assembled monolayers on nanostructured titania and tin (IV) oxide nanowires using colloidal silver nanoparticles[END_REF]. Therefore, in this work we have been interested in an alternative amino-organosilane that could enable preparation of more ordered amine-terminated SAMs on a silicon surface. For this purpose, we have chosen to study the SAM formation and organization of (3-Trimethoxysilylpropyl) diethylene-triamine organosilane (DETAS). SAMs from such molecule have been studied for different applications such as immobilization of other organic species on top [START_REF] Gao | Immobilization of pyrene via diethylenetriamine on quartz plate surface for recognition of dicarboxylic acids[END_REF], controlling the diffusion of evaporated metals on the SAM [20], coating of silica nanoparticle for water treatment without organic solvent [21], patterning by using DETAS selfassembled monolayer to promote cell growth at specific sites of the surface [22], or specific robust adsorption of PEDOT nanofilms [23]. For such possible applications, it is important that DETAS SAMs should be well organized. Regarding SAM formation, such organosilane is bearing a longer alkyl chain that enables a higher flexibility to promote molecular interaction, and including also two NH moieties capable of introducing structuring hydrogen bonding between neighboring molecules in the SAM. 1), acetone, methanol, ethanol, iso-propanol, butanol, toluene, dichloromethane, trichloromethane, hydrogen peroxide 30% (H2O2), and sulfuric acid (H2SO4) are of synthesis grade and absolute ethanol of analytical grade, and all products were used with no further purifications. The water used in all experiments was deionized and purified by a Millipore system and its resistance was ~18 MΩ.

Experimental details

Chemicals

DETAS [(3-Trimethoxysilylpropyl

) diethylene-triamine] ([MeO]3-Si-C3[NH- C2]2-NH2) (Fig.

Sample cleaning and SAM preparation

Si substrates of ~1 cm by 1.5 cm were degreased in three consecutive ultrasonic baths of 10 min each subsequently with acetone, dichloromethane, and iso-propanol. The samples were then transferred into a mixture of 3 volumes of H2O2 and 7 volumes of H2SO4 heated at 110 °C for 30 min to remove the last traces of organic pollutant on the surface. After that, the samples were thoroughly rinsed with deionized water and stored into a water beaker upon usage for no more than few minutes. The samples were then dried under a nitrogen flow and quickly plunged into a DETAS solution. The temperature was controlled with an oil bath on a thermostated plate by a Ministat from Huber, inside a glove box (818-GB from Plas-Lab) filled with grade 4.5 nitrogen. Atmospheric relative humidity was monitored (using a TFA Dostmann 30.5013). Relative humidity was controlled with desiccants and a heating system. After SAM preparation, samples were rinsed with clean solvent used for grafting, and treated with ultrasonic bath of the same solvent for 5 min. After that, samples were dried under nitrogen flow and characterized right away.

SAM characterization

Ellipsometric measurements were carried out on a Sentech SE400 at 70° angle with a 632.8 nm laser. Parameters used were 3.8750 and 0.0018 for the real and imaginary indices of the silicon substrate, 1.46 and 0 for the native oxide layer, and 1.45 and 0 for the indices of DETAS with an expected thickness of 14 Å [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF]. Thickness of the native silicon oxide layer was recorded on control samples that underwent the same cleaning process as the grafted samples. At least five different points of the sample surface were measured, possibly on different samples, in order to give a mean value with error bars amplitude between the minimum and the maximum values measured. Goniometric measurements were conducted with a Kruss drop shape DSA 10 MK2 analyzer, with deionized water as liquid. The sample was cleaned under nitrogen flow after measuring. AFM measurements were performed in Tapping mode with 40 N/m force constant silicon tips, using a Bruker Multimode 8 equipped with a Nanoscope V electronics. Fourier transformed infrared spectra were recorded on a Thermofisher IS50 spectrometer in single reflection ATR mode at an incidence angle of 45° on diamond. 96 scans were accumulated in a spectral range within 4000-400 cm -1 with a resolution of 4 cm -1 . Background spectrum of the bare silicon substrate was recorded in the same conditions and subtracted.

Results and discussion

Growth kinetics of SAMs of DETAS

In order to identify the time-scale necessary to reach a complete single monolayer coverage of DETAS on silica, the SAM growth kinetics was studied with a concentration of 5.10 -2 mol•l -1 in different solvents under 33 ± 5% relative humidity. High temperature (31.5 ± 1.5 °C) grafting was conducted in alcohol solvents and lower temperature (11.0 ± 0.5 °C) in toluene. Grafting durations chosen for kinetics study were 1 s, 5 s, 10 s, 30 s, 1 min, 5 min, 10 min, 1 h, 3 h and 24 h. Time evolution of the thickness is plotted in Fig. 2 for the different growth conditions.

All experimental data could be well fitted using a simple Langmuir type growth equation:

Th = A exp(-t/τ ) + Thmax (1)
Where Th stands for the thickness, A coefficient arises from a delay in the grafting process, is the time constant and Thmax is the thickness reached at the end of the grafting. Values of the fitting parameters are displayed in Table 1. As can be seen, the thickness jumps from 0 to about 8 Å at t = 0 s, upon dipping in the solution. This maybe an indication that DETAS molecules form a film at the surface of the solution that is deposited on the surface while dipping, thus forming a pseudo Langmuir-Blodgett film on the sample surface.

As can be seen in Table 1, the nature of the solvent and the grafting temperature seem to play very little role in the growth kinetics since the growth time constants and delay coefficients are comparable within confidence limits.

Growth time constant values are in the range of 7000-10000 s (~1.9-2.8 h) and the final SAM thickness is obtained after ~30000 s, i.e., ~8 h. The growth kinetics study of DETAS SAM was performed by Demirel et al. [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF] at a concentration of ~4.10 -3 M in ethanol. 1 Authors found that DETAS SAM reached a maximum thickness of ~15 Å after ~12 h grafting. Given that in our experiments the concentration is about ten times higher, we should expect a growth process ten times quicker. But as shown in a previous study on alkylsilane growth [START_REF] Desbief | Impact of chain length, temperature, and humidity on the growth of long alkyltrichlorosilane selfassembled monolayers[END_REF], the temperature and the humidity both play a crucial role in the SAM growth kinetics. No information is given in the article by Demirel et al. [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF] on the relative humidity conditions. However, as they worked at ambient temperature, our temperature around 31.5 °C was higher, which was shown to slower the SAM growth with favoring a more disordered phase that hinders molecule diffusion.

The first steps of the growth are visible in the semi-logarithm time scale presented in the insets of Fig. 2. As can be seen, the thickness remains at about 8 Å during ~2000 s in the same manner as in the growth analysis by Demirel et al. [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF]. It may indicate a first step of growth were the molecules are lying bent on the surface before a possible interplay of both van der Waals bonds between methylene groups and hydrogen bonds between the amino NH and NH2 moieties. This is further supported by the quicker increase that we observe for the water contact angle than for the thickness, as in Ref. [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF] (Fig. 3). Indeed, as can been seen, compared to the first order Langmuir type fitting function of the thickness time evolution, water contact angle values are positioned at lower time within the first 200 min of SAM growth. This can be explained by molecules lying flatter on the surface at the early stage of growth (t < 200 min), which means the water droplet probes a mixture made of methylene CH2 groups which contribute to increase the water contact angle, with both more hydrophilic terminal amino NH2 moieties and uncovered oxide surface. On the contrary, at the end of the growth, water contact angles are well measured on NH2 groups, with an end value of ~60° compatible with the values found in the literature for NH2 terminated SAMs [START_REF] Kurth | Thin films of (3-aminopropyl)triethoxysilane on aluminum oxide and gold substrates[END_REF] and particularly with the value of 65° reported for the final DETAS SAM [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF]. Fig. 2 shows complete SAMs were obtained after at least 8 h whatever the solvent, temperature and humidity. Therefore, in the following, to warrant a complete SAM coverage we fixed the grafting duration to 24 h.

Effect of the relative humidity

As the importance of dipping time and concentration is quite wellknown, we first investigated the effect of relative humidity (RH) on the formation of self-assembled monolayers of DETAS on native silicon oxide. For this purpose, among possible solvents for DETAS we chose to use toluene rather than alcohols because its non-polar nature makes it more sensitive to the water content in solution. The effect of relative humidity on SAM formation was tested from 25% RH up to 80% RH with a DETAS solution in toluene at 1.10 -2 mol•l -1 at room temperature (22 ± 2 °C) and the dipping time was set to 24 h. The SAM thickness, measured by ellipsometry, was plotted versus relative humidity in Fig. 4.

Contact angle measurements were also recorded but are irrelevant with using toluene as a solvent as we have noticed that dipping a substrate into toluene is enough to considerably modify its contact angle with water (from 10° to 30° depending of the surface), unless it is cleaned in ultrasonic bath followed by hot acid bath. As can be seen in the graph, a SAM with about the desired thickness of 14 Å [START_REF] Demirel | Formation and organization of amino terminated self-assembled layers on Si(001) surface[END_REF] is achieved provided that relative humidity is kept under ~40%. When relative humidity gets higher, the thickness increases by around two orders of magnitude which can be explained by multilayer formation. The sudden thickness increase above 40% RH indicates a change in the behavior of DETAS molecule in solution at this threshold value. Our hypothesis is that DETAS behavior in non-polar solvent is analogous to the behavior of surfactants. As relative humidity increases, water dissolves in toluene until it reaches equilibrium allowing water to regroup around DETAS molecules forming micelle-like structures. Then these reverse micelles can react to form oligomers that are transferred to the surface before chemisorption. A visible layer was observable on each interface with the solution (solutionair, solutionsubstrate, and solution-beaker) which further confirms the surfactant-like behaviour of these structures. A foam was also observable in the solution for higher concentration of DETAS which is consistent with an oligomerized micelle-like behavior. AFM analysis was conducted on samples at high relative humidity at different grafting times, revealing cauliflower-like structures growing on the substrate at the beginning of the grafting and a superposition of many layers of oligomerized reverse micelles at the end of the grafting (Fig. 5). Therefore, in the following of the study we worked at low humidity to avoid such reverse micelle superimposition in the final SAM.

Low humidity Self-assembled monolayers

Regarding low relative humidity (< 40%) grafting, smooth (0.28 nm average roughness) monolayers about 14 Å thick were obtained as confirmed by ellipsometry together with AFM profiling by denting the self-assembled monolayer with a soft fabric. AFM image of DETAS self-assembled monolayer grafted with low relative humidity in toluene and ethanol (Fig. 6) exhibits similar smooth layers with very few defects. Therefore, using alcohol as solvent will be preferred in the following since such solvent in which DETAS is better solubilized are less sensitive to the relative humidity than toluene. Some white spots of about 1-30 nm remain on the layers, which can be attributed to some oligomers that have formed either on the surface or in solution. Lowering further the relative humidity may improve the monolayer quality.

Siloxane monolayer organization is usually driven by van der Waals interaction between long alkyl chains. This process is well known and leads to remarkable self-assembled monolayers provided the alkyl chain is long enough [START_REF] Ulman | An Introduction to Ultrathin Organic Films[END_REF][START_REF] Desbief | Impact of chain length, temperature, and humidity on the growth of long alkyltrichlorosilane selfassembled monolayers[END_REF]. DETAS however does possess two secondary amine groups inside the alkyl chain alongside with a primary amine group at the head of the molecule. Those amine groups are able to form hydrogen bonds inside the layer to stabilize more efficiently the monolayer. Hydrogen bonds are stronger than van der Waals interaction by several orders of magnitude, allowing better organization of the monolayer than pure alkane chains with comparable short length. The ATR-FTIR spectra of a self-assembled monolayer of DETAS are shown in Fig. 7 in different spectral regions.

In Fig. 7a, one can see the two doublet peaks positioned at mean values of ~3712 and ~3612 cm -1 that are assigned to hydrolyzed methoxysilane groups that have not crosslinked in the usual condensation process [27]. We can note that at the base of these two doublet peaks, there is a broad band which seems to indicate that some of those silanol groups are hydrogen bonded. The band at 3406 cm -1 and the large one within 3300-2800 cm -1 that could be decomposed into three Lorentzian bands at 3264, 3142 and 3017 cm -1 are assigned to the N-H stretch of primary and secondary amines [28][START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies: Tables and Charts[END_REF][START_REF] Pang | Amino-functionalized alkylsilane SAMassisted patterning of poly(3-hexylthiophene) nanofilm robustly adhered to SiO2 substrate[END_REF]. The band at 3264 cm -1 can be a consequence of the shift to lower frequency of the 3406 cm -1 band for the secondary amines that experience hydrogen bonding in the SAM. The other two bands of the primary amine stretch at 3142 and 3017 cm -1 are forming a broad band and indicate that there is hydrogen bonding within the self-assembled monolayer. Fig. 7b exhibits two bands at 1568 and 1494 cm -1 that can be attributed to amine group deformation. Usually such modes appear at higher frequencies but again such a shift towards lower frequencies may be due to hydrogen bonding [START_REF] Okabayashi | Diffuse reflectance infrared Fourier transform spectral study of the interaction of 3-aminopropyltriethoxysilane on silica gel. Behavior of amino groups on the surface[END_REF]. The extent of such shift depends on the intensity of the hydrogen bonding. For example, Chiang et al. [START_REF] Ishida | The structure of aminofunctional silane coupling agents: 1. γ-Aminopropyltriethoxysilane and its analogues[END_REF] assigned the band at 1561 cm -1 for aminopropyltrimethoxisilane SAM deposited on silica gel to NH2 deformation modes with strong hydrogen bonding. In Fig. 7c, the peak at 701 cm -1 is assigned to N-H swaging on both primary and secondary amines [START_REF]Spectroscopic Properties of Amines -Chemistry LibreTexts[END_REF]. Furthermore, formation of siloxane bonds can be inferred from the band located at 1019 cm -1 [START_REF] Tian | Infrared characterization of interfacial Si-O bond formation on silanized flat SiO2/Si surfaces[END_REF][START_REF] Santhosh Kumar | Bulk superhydrophobic materials: a facile and efficient approach to access superhydrophobicity by silane and urethane chemistries[END_REF].

Carbon chain length and concentration of alcohols on grafting

Effects of concentration on the self-assembled monolayer was also tested for DETAS concentrations from 10 -3 to 5.10 -1 mol•l -1 in three alcohol solvents that were shown to be less sensitive to relative humidity than toluene, certainly because of their high polarity. No micellar behavior should appear due to the high solubility of water and DETAS molecules in alcohols. Relative humidity was around 41%, temperature 33 °C and grafting time was 24 h for all samples. The thickness, measured by ellipsometry, plotted versus DETAS concentration is given in Fig. 8 for each of the three solvents used.

As can be seen in the graph, low concentration below 10 -2 mol•l -1 leads to incomplete self-assembled monolayers whereas higher concentration around 5.10 -1 mol•l -1 leads to SAMs with the expected thickness, but presenting high deviation in the thickness measurement, which indicates that a lot of defects or adsorbates are present at the SAM surface. A minimum concentration of 2.5.10 -2 mol•l -1 seems to be needed to achieve a good quality compact layer. Concentration can be further increased up to 5.10 -2 mol•l -1 without damaging the monolayer quality. It is also noticeable that the longer the alcohol chain, the better the quality of the monolayer, suggesting that the alkane chain is playing a role in the organization of the self-assembled monolayer. Similarly, the addition of long alkane chain solvent like hexadecane can improve organization and quality of octadecyltrichlorosilane SAMs [START_REF] Desbief | Impact of chain length, temperature, and humidity on the growth of long alkyltrichlorosilane selfassembled monolayers[END_REF].

Conclusion

Amino-terminated self-assembled monolayers are currently widely used in both industrial and research-oriented applications. We demonstrated that relative humidity plays a key role in the formation of DETAS SAMs with nonpolar solvent like toluene and leads to multilayers through micellar deposition when it reaches 40% and above. Observation suggests that for obtaining a good DETAS SAM quality, a minimal DETAS concentration of 2.5 10 -2 M together with a long chain of alcohol solvent are needed. Kinetics show an immediate molecule adsorption within the very first moment of the SAM formation process which is uncommon for organosilane grafting. Thickness remains almost constant for more than 30 min before increasing with the molecule organization through hydrogen bonding between primary (NH2) and secondary (NH) amine moieties, which was evidenced by infrared spectroscopy. Further steps are in progress concerning the functionalization to immobilize species on top of DETAS SAMS.

Fig. 1 .

 1 Fig. 1. Schematic representation of (3-Trimethoxysilylpropyl) diethylene-triamine (named "DETAS"), approximately 14 Å in length.

Fig. 2 .

 2 Fig. 2. Grafting kinetics curves of DETAS self-assembled monolayer at 5.10 -2 M and under 33 ± 5% RH in: (a) toluene at 11.0 ± 0.5 °C, (b) methanol at 31.5 ± 1.5 °C, (c) ethanol at 31.5 ± 1.5 °C. Inset shows the curve with logarithm time scale to highlight the early stage of the growth.

Fig. 3 .

 3 Fig. 3. Top: Grafting kinetics of DETAS self-assembled monolayer at 10 -2 M, 30 °C, under 80% RH in toluene for the thickness (filled black squares) and water contact angle (filled red circles). Fitting curves using a first order Langmuir type law was obtained from thickness data. Bottom: scheme of the proposed grafting process with: (a) molecules are adsorbed towards the surface, (b) molecules are lying flat on the surface, (c) molecules are beginning to stand up at the surface, (d) final organized compact SAM. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .

 4 Fig. 4. Effect of relative humidity on the thickness of self-assembled monolayers of DETAS prepared on silica from toluene solution (1.10 -2 M, 22 ± 2 °C).

Fig. 5 .

 5 Fig. 5. AFM images (10 × 10 µm 2 ) showing: (a) cauliflower structures growing in the first stage of grafting, and (b) reverse micelle deposition in the later stage of grafting in toluene. Grafting conditions were: 10 -2 M of DETAS in toluene, ambient temperature, high humidity > 40%.

Fig. 6 .

 6 Fig. 6. AFM image of DETAS self-assembled monolayer: (a) 5.10 -2 M in ethanol, 26 °C and 20% RH; (b) 10 -2 M in toluene, 24 °C and 23% RH.

Fig. 7 .

 7 Fig. 7. ATR-FTIR spectra of DETAS self-assembled monolayer on silicon substrate (prepared from a solution 5.10 -2 M in EtOH, 26 °C and 20% RH). Lorentzian fits have been performed (dashed green and red curves).

Fig. 8 .

 8 Fig. 8. Influence of both concentration and alcohol carbon chain length on the thickness of DETAS SAM prepared at 33 °C and 41% RH. Grafting duration is 24 h.

Table 1

 1 Fitting values of the growth kinetics of SAM of DETAS in different solvents and temperature.

	Solvent	Temperature (°C)	A	τ (10 3 s)	Thmax (Å)
	Toluene	11.0 ± 1.0	-5.9 ± 0.3	10.6 ± 3.6	14
	MeOH	31.5 ± 1.0	-5.8 ± 0.3	8.6 ± 3.9	14
	EtOH	31.5 ± 1.0	-6.4 ± 0.3	7.4 ± 1.2	14

The concentration used in reference

is 1% in volume ratio. With a density of 1.03 for DETAS it corresponds to a mass concentration of ~10 -2 g/ml which gives a molar concentration of ~4.10 -3 M with a molar weight of 265 g.

Acknowledgements

FFCR-FCRF "New collaboration programme" between France and Canada is acknowledged for financial support through the "SAMOS" project, and particularly for funding the PhD of Y. Dufil. Funding from PHC Barrande 2018 (40672RL), Campus France, Ministères de l'Europe et des Affaires étrangères (MEAE) et de l'Enseignement supérieur, de la Recherche et de l'Innovation (MESRI), as well as financial support from Ministry of Education, Youth and Sports, within the projects LTC17058 and 8J18FR011, and by COST Action CA15107 MultiComp are also acknowledged. Y.D., V.G., and L.P. thank P. Fitl, J. Lančok, D. Tomeček and J. Vlček. J.-M.N. thanks CNRS and Aix-Marseille Université for funding his invitation at IM2NP. Equipment was mainly funded by the "Objectif 2" EEC program (FEDER), the "Conseil Général du Var" Council, the PACA Regional Council, Toulon Provence Méditerranée and ISEN-Toulon which are acknowledged.

Authorship contribution statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.