Tomas Ménard

Emmanuel Bernuau

Emmanuel Moulay

Patrick Coirault

Continuous-discrete

come

Introduction

The study of nonlinear observers has been an active research area for several decades [START_REF] Besançon | Nonlinear observers and applications[END_REF]. Indeed, they are powerful tools with a large number of possible applications since they allow to reconstruct all the states of a system from the model and the measured outputs only. This can be useful when all the states cannot be measured, for example when sensors do not exist which is the case in robotics to reconstruct stiness [START_REF] Ménard | A stiness estimator for agonisticantagonistic variable-stiness-actuator devices[END_REF] or in process control for some specic variables [START_REF] Gehan | Dissolved oxygen level output feedback control based on discrete-time measurements during a pseudomonas putida mt-2 fermentation[END_REF]. Another reason can be that the sensors are too expensive in some cases such as an industry production for example. Another use can be to check that the available sensors are working correctly by using such observers for fault detection and isolation [START_REF] Zhang | Adaptive observerbased fast fault estimation[END_REF].

Many approaches have then been developed to design such observers. One of the rst ideas has been to transform nonlinear systems into a linear ones so that observers can be constructed [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]. Transformations Email addresses: e-mail: tomas.menard@unicaen.fr (Tomas Ménard), e-mail: emmanuel.bernuau@agroparistech.fr (Emmanuel Bernuau), emmanuel.moulay@univ-poitiers.fr (Emmanuel Moulay), patrick.coirault@univ-poitiers.fr (Patrick Coirault).

into more general forms have also been considered such as in [START_REF] Plestan | Linearization by generalized inputoutput injection[END_REF] for example. General classes of systems such that observers can be designed have also been considered. For Single-Input Single-Output (SISO) uniformly observable systems, a high gain observer can always be constructed [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF], though the change of coordinates can be dicult to nd. General classes of Multi-Input Multi-Output (MIMO) uniformly observable systems have also been considered in [START_REF] Farza | Observer design for a class of MIMO nonlinear systems[END_REF]. Some other approaches rely on the specic structure of the system, such as backstepping [START_REF] Fossen | Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping[END_REF] or adaptive observers [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF].

Homogeneity has been considered more recently for observer design [1113], though it has been introduced early in [START_REF] Zubov | Systems of ordinary dierential equations with generalized-homogeneous right-hand sides[END_REF] and [START_REF] Rothschild | Hypoelliptic dierential operators and nilpotent groups[END_REF]. Homogeneity is an interesting property since it is well suited to approximation or domination design [START_REF] Qian | Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems[END_REF][START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]. Other properties can also be obtained from homogeneity of systems, such as nite-time or xed-time stability for example [START_REF] Polyakov | Nonlinear feedback design for xed-time stabilization of linear control systems[END_REF][START_REF] Polyakov | Finite-time and xed-time stabilization: Implicit Lyapunov function approach[END_REF]. Several structures of observers have been developed. One way is to use an inductive approach such as backstepping [START_REF] Li | A dual-observer design for global output feedback stabilization of nonlinear systems with loworder and high-order nonlinearities[END_REF]. One can also rely on the fact that the considered homogeneous observer is close to an observer whose convergence is already established [START_REF] Ménard | Fixed-time observer with simple gains for uncertain systems[END_REF]. Another possible approach is to consider an Implicit Lyapunov Function (ILF) [START_REF] Lopez-Ramirez | Finite-time and xed-time observer design: Implicit Lyapunov function approach[END_REF].

In the aforementioned works, the outputs are supposed to be available continuously, but this is rarely the case Preprint submitted to Automatica in practice. Indeed, most observers are implemented digitally on computers or micro controllers, the output is then usually sampled and the sampling times may be aperiodic [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. This may be the case when there are packet dropouts in Networked Control Systems (NCS) [START_REF] Wang | Networked control systems[END_REF] or in order to lower the load of a network when it is used by many agents at the same time [START_REF] Menard | Leader-following consensus for multi-agent systems with nonlinear dynamics subject to additive bounded disturbances and asynchronously sampled outputs[END_REF]. One way is to use a fully discrete-time approach [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF], but one loses the physical meaning of the states and only semiglobal stability might result. Another way is to use the emulation method [START_REF] Nesic | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], which means that a continuous observer is rst synthesized and discretized afterward. A third way is to adapt the continuous observer to obtain a so called continuous-discrete time observer which uses the discrete-time measurements and provide a continuous estimation of the states. Several designs have been proposed following this way. A rst idea is to use observers whose state is reset whenever a new measurement is received. A high gain approach has been used in [START_REF] Nadri | Design of a continuous-discrete observer for state ane systems[END_REF] for a class of linear systems up to input injections and for a class of uniformly observer systems whose nonlinear part is input dependant in [START_REF] Andrieu | Continuous discrete observer with updated sampling period[END_REF]. LMIs have been used to guarantee the stability in [START_REF] Dinh | Continuousdiscrete time observer design for Lipschitz systems with sampled measurements[END_REF] for a class of nonlinear systems using reachability analysis, in [START_REF] Mazenc | Continuous-discrete observers for time-varying nonlinear systems: A tutorial on recent results[END_REF] using an hybrid approach, for a class of systems whose dynamics is assumed to belong to a polytopic set of matrices in [START_REF] Etienne | Observer synthesis under time-varying sampling for Lipschitz nonlinear systems[END_REF], for a class of linear systems with noise analysis in [START_REF] Ferrante | State estimation of linear systems in the presence of sporadic measurements[END_REF] and for the same class of linear systems but with a less conservative approach in [START_REF] Sferlazza | Time-varying sampled-data observer with asynchronous measurements[END_REF]. Another idea is to use continuous observers based on an output error predictor which is reset when a new measurement is received. This approach has been rst used in [START_REF] Karafyllis | From continuous-time design to sampled-data design of observers[END_REF] and then in [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF] using a high-gain approach and in [START_REF] Ferrante | L 2state estimation with guaranteed convergence speed in the presence of sporadic measurements[END_REF] by using LMIs. Both continuous and discontinuous observers have been combined in [START_REF] Ferrante | Observer design for linear aperiodic sampled-data systems: A hybrid systems approach[END_REF].

It has been shown recently that sampled homogeneous systems enjoy good properties. Indeed, depending on the degree of homogeneity either global practical stability or semi-global asymptotic stability can be obtained [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF]. Furthermore, it has been shown in [START_REF] Josse | Robustness of sampled-data homogeneous systems[END_REF] that robustness with respect to exogenous disturbance in the dynamics is also obtained. Such results have also been obtained for time-delay systems which can be seen as a more general framework than sampled systems, such as in [START_REF] Emov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF][START_REF] Zimenko | A note on delay robustness for homogeneous systems with negative degree[END_REF].

In this paper, we are particularly interested in the adaptation of homogeneous observers of negative degree (the precise meaning of homogeneous observer will be detailed in the following) in order to cope with the sampling of the output. Several features of the proposed design are worth being emphasized: i) the proposed observer error is globally uniformly ultimately bounded. ii) a Lyapunov function is explicitly constructed for the continuous-discrete time observer.

iii) the vector eld representing the original system may be non-homogeneous, which is not the case of [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF]. iv) simulations show that the performances of the proposed scheme are better than the one proposed in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF].

The paper is organized as follows. The class of systems and the dierent assumptions are stated in section 2. Section 3 is devoted to the presentation of the proposed continuous-discrete time observer. Some simulations are provided in section 4 in order to show the performances of the proposed scheme. Finally, section 5 ends the paper.

2 Class of considered systems

Let us consider the following system

ẋ(t) = f (x(t), u(t)) y(t) = h(x(t)) (1)
with x ∈ R n , u ∈ R, y ∈ R. The function f : R n × R → R n is assumed to be continuous and h : R n → R is assumed to be C 1 and homogeneous of degree d h > 0 with respect to some weights (α 1 , . . . , α n ) ∈ R n + . One assumes that a continuous observer, meaning that it has access to the output continuously for all t ≥ 0, exists for system [START_REF] Besançon | Nonlinear observers and applications[END_REF], which is in the following form

ẋ(t) = f (x(t), u(t)) -G(ŷ(t) -y(t)) ŷ(t) = h(x(t)) (2)
where the function G : R → R n is assumed to be homogeneous of degree α < 0 with respect to the weight d h .

One assumes that the states of observer (2) converge to the states of system [START_REF] Besançon | Nonlinear observers and applications[END_REF]. More precisely, one assumes the following.

Assumption 1 There exists a class C 1 homogeneous Lyapunov function U : R n → R of degree d U > max{-α, d h , α 1 , . . . , α n } with respect to the weights (α 1 , . . . , α n ), whose derivative along the trajectories of the following error system

ė(t) = f (x(t), u(t)) -f (x(t), u(t)) -G(ŷ(t) -y(t)) (3)
is negative denite, where e = x -x, i.e. there exists ε > 0 such that

U | (3) (e(t)) ≤ -ε (U (e(t))) d U +α d U . ( 4
)
Remark 1 Since the function U is homogeneous of degree d U with respect to the weights (α 1 , . . . , α n ), several useful properties can be derived using Lemma 4.2 in [START_REF] Bhat | Geometric homogeneity with applications to nite-time stability[END_REF]. Firstly, the function ∂U/∂e j is homogeneous of degree d U -α j and continuous, hence there exists

κ j ≥ 0 such that ∂U ∂e j (e) ≤ κ j n l=1 |e l | d U -α j α l . (5)
Secondly, using the fact that U is positive denite gives the existence of a constant > 0 such that

n l=1 |e l | d U +α α l ≤ (U (e)) d U +α d U . (6)
Remark 2 It can be noted that the vector eld f is not assumed to be homogeneous. Nevertheless, in general, some homogeneous properties will be required to obtain a homogeneous Lyapunov function U for the error system.

For example, such Lyapunov functions can be found for systems where f is composed of an homogeneous drift term f 1 (x), which is homogeneous of degree α < 0 with respect to the weights (α 1 , . . . , α n ), and a term f 2 (x, u) which will be dominated in the stability analysis, such as in [START_REF] Li | A dual-observer design for global output feedback stabilization of nonlinear systems with loworder and high-order nonlinearities[END_REF] for example.

Remark 3 While the form of observer (2) seems restrictive, it should be noted that it is typically obtained in other coordinates than those of the system and an estimate of the state can be obtained by inversing the change of coordinates.

Main results

The aim of this work is to construct a continuous-discrete time observer for system [START_REF] Besançon | Nonlinear observers and applications[END_REF], that is an observer that provides a continuous estimate of the state of system (1) when the output y is only available at some sampling times (t k ) k∈N . The sampling times are assumed to verify:

0 = t 0 < t 1 < . . . < t k < . . . (7) 0 < T min < t k+1 -t k < T max , ∀k ∈ N (8)
The proposed observer is given by

ż(t) = -∂h ∂x (x(t))G(ẑ(t)) ẋ(t) = f (x(t), u(t)) -G(ẑ(t)) (9)
when t ∈ [t k , t k+1 ), and

ẑ(t k ) = ŷ(t k ) -y(t k ) x(t k ) = x(t - k ) (10)
Let us now write the error dynamics for observer [START_REF] Fossen | Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping[END_REF].

One denotes e 0 (t) = ẑ(t) -(ŷ(t) -y(t)), e = x -x = (e 1 , . . . , e n ), then, for t ∈ [t k , t k+1 ), one has

ė0 (t) = ∂h ∂x (x(t))f (x(t), u(t))) - ∂h ∂x (x(t))f (x(t), u(t)) ė(t) = (f (x(t), u(t)) -f (x(t), u(t)) -G(ŷ(t) -y(t))) -(G(ẑ(t)) -G(ŷ(t) -y(t))) (11)
The whole error state is denoted E = (e 0 , . . . , e n ) ∈ R n+1 and the considered candidate Lyapunov function V (E, t) is dene for a given set of sampling instant verifying (7)(8) as follows. Let E ∈ R n+1 and t ≥ 0, then there exists a unique k ∈ N such that t ∈ [t k , t k+1 ) and V (E, t) is dened as

V (E, t) = U (e) + exp(δ(t k+1 -t))|e 0 | d U d h , (12)
with δ > 0. Note that the function V is a valid candidate Lyapunov function, indeed, it is positive denite since exp(δ(t k+1 -t)) ∈ [1, exp(δT max )] for all t ∈ [t k , t k+1 ).

Before stating the main result, some additional assumptions have to be made in order to obtain the stability of the continuous-discrete time observer. In particular, it is required that some Hölder properties hold:

Assumption 2 (i) For all x 1 = (x 1 1 , . . . , x 1 n ), x 2 = (x 2 1 , . . . , x 2 n ) ∈ R n and u ∈ R, there exists ν 1 ≥ 0 such that ∂h ∂x (x 1 )f (x 1 , u) - ∂h ∂x (x 2 )f (x 2 , u) ≤ ν 1 n i=1 |x 1 i -x 2 i | d h +α α i . (13)
(ii) For all z 1 , z 2 ∈ R, i = 1, . . . , n, there exists ν i 2 such that

|G i (z 1 ) -G i (z 2 )| ≤ ν i 2 |z 1 -z 2 | α i +α d h . ( 14
)
Theorem 1 (Ultimate Boundedness) Consider system [START_REF] Besançon | Nonlinear observers and applications[END_REF] and observer (9) [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF] subject to Assumption 2, then for any T max ≥ T min > 0 the error system [START_REF] Qian | Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm[END_REF] is globally stable and there exist K ≥ 0, independent of T max , such that the set

             E =        e 0 e 1 . . . e n        ∈ R n+1 : U (e) + |e 0 | d U d h ≤ K(T max ) d U -α              (15)
is globally attractive.

Remark 4 Theorem 1 states that the proposed observer cannot explode in nite-time even if the upper bound on the sampling periods period increases. This is not the case for most of the existing observers since if the sampling periods increases too much, the observer usually becomes unstable. This of course does not mean that the observation error stay small since it increases as T max increases. Nevertheless, this is a very interesting feature since it allows to keep satisfactory performances even when approaching the limit where other observers become unstable as it can be seen in the example section. This can also be useful to allow a very fast recovery of the estimation for example when the sampling instants are irregular and the sampling periods become large at some moments.

Proof. The proof is divided into three steps. In Step 1, one shows that the Lyapunov function candidate V is decreasing when new measurements are available, that is for t = t k with k ∈ N. In Step 2, one is interested in the behavior of the error dynamics when the dynamics is continuous, that is when t ∈ [t k , t k+1 ) with k ∈ N. An over-valuation of the derivative of V is then obtained. In Step 3, the results of Steps 1 and 2 are combined to show that the observer error is globally uniformly ultimately bounded.

Step 1.

Step 1 is rather direct, indeed, since the dynamics of e are continuous (e(t - k ) = e(t k )) and since e 0 (t k ) = 0, one has

V (E(t - k ), t - k ) = U (e(t - k )) + exp(δ(t k -t k ))|e 0 (t - k )| d U d h = U (e(t k )) + |e 0 (t - k )| d U d h ≥0 ≥ U (e(t k )) + exp(δ(t k+1 -t k )) |e 0 (t k )| d U d h =0 ≥ V (E(t k ), t k ) (16)
Step 2. The derivative of the function V along the trajectories of system [START_REF] Qian | Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm[END_REF], for t ∈ [t k , t k+1 ), is given by

V (E(t), t) |(11) = (17)
U (e(t)

) |(3) -δ exp(δ(t k+1 -t))|e 0 (t)| d U d h + n j=1 ∂U ∂e j (e(t)) (G j (ẑ(t)) -G j (ŷ(t) -y(t))) + d U d h exp(δ(t k+1 -t)) e 0 (t) d U -d h d h × ∂h ∂x (x(t))f (x(t), u(t)) - ∂h ∂x (x(t))f (x(t), u(t))
One now states two facts whose proof can be found in the appendix.

Fact 1. For any γ 1 > 0, there exists

Γ 1 ≥ 0 such that n j=1 ∂U ∂e j (e) (G j (ẑ) -G j (ŷ -y)) (18)
≤ γ 1 (U (e))

d U +α d U + Γ 1 |e 0 | d U +α d h
holds for all E ∈ R n+1 .

Fact 2. For any γ 2 > 0, there exists Γ 2 ≥ 0 such that

d U d h exp(δ(t k+1 -t)) e 0 d U -d h d h (19)
× ∂h ∂x (x)f (x, u) - ∂h ∂x (x)f (x, u) ≤ γ 2 (U (e)) d U +α d U + Γ 2 (exp(δ(t k+1 -t))) d U +α d U -d h |e 0 | d U +α d h holds for all t ∈ [t k , t k+1 ) and E ∈ R n+1 .
Using inequality (4), with Facts 1 and 2 in [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF] gives

V (E(t), t) ≤ -ε (U (e(t))) d U +α d U (20)
-δ exp(δ(t k+1 -t))|e 0 | d U d h + γ 1 (U (e(t))) d U +α d U + Γ 1 |e 0 (t)| d U +α d h + γ 2 (U (e)) d U +α d U + Γ 2 (exp(δ(t k+1 -t))) d U +α d U -d h |e 0 (t)| d U +α d h ≤ -(ε -γ 1 -γ 2 ) (U (e(t))) d U +α d U (21)
-δ exp(δ(t k+1 -t))|e 0 (t)| d U d h + (Γ 1 + Γ 2 ) (exp(δ(t k+1 -t))) d U +α d U -d h × |e 0 (t)| d U +α d h
One now states another fact whose proof can be found in the appendix.

Fact 3 For δ = 1/T max , there exists K 1 ≥ 0 independent of T max such that the following inequality holds true

-δ exp(δ(t k+1 -t))|e 0 (t)| d U d h (22)
+ (Γ 1 + Γ 2 ) (exp(δ(t k+1 -t))) d U +α d U -d h |e 0 (t)| d U +α d h ≤ - ε 2 (exp(δ(t k+1 -t))) d U +α d U |e 0 (t)| d U +α d h + K 1 T d U +α -α max for all t ∈ [t k , t k+1 ).
Using Fact 3 in equation ( 21) and taking γ 1 , γ 2 such that

ε -γ 1 -γ 2 = ε/2 gives V (E(t), t) ≤ - ε 2 (U (e(t))) d U +α d U - ε 2 exp(δ(t k+1 -t)) d U +α d U |e 0 (t)| d U +α d h + K 1 T d U +α -α max ≤ - ε 2 (U (e(t))) d U +α d U - ε 2 exp(δ(t k+1 -t))|e 0 (t)| d U d h d U +α d U + K 1 T d U +α -α max ≤ - ε 2 U (e(t)) + exp(δ(t k+1 -t))|e 0 (t)| d U d h d U +α d U + K 1 T d U +α -α max ≤ -ν (V (E(t), t)) d U +α d U + Γ (23)
by applying Lemma 1-(iii) with p = d U +α d U and where

Γ = K 1 T d U +α -α max and ν = ε 2 .
Step 3. From the results of Steps 1 and 2, it is clear that V (E(t), t) is decreasing outside of the interval

0, Γ ν d U d U +α , then either V (E(t), t) enters the interval 0, Γ ν d U d U +α

Noting in addition that

Γ ν d U d U +α = K(T max ) d U -α and U (e) + |e 0 | d U d h ≤ V (E, t), ∀E ∈ R n+1 , ∀t ≥ 0 (24)
ends Step 3.

Example

One considers here the following system

             ẋ1 (t) = x 2 (t) -0.1 x 1 (t) 0.8 ẋ2 (t) = x 3 (t) + 0.1 x 1 (t) 0.6 + 0.1 x 2 (t) 0.75 ẋ3 (t) = -0.5 x 1 (t) 0.4 -0.5 x 2 (t) 0.5 -x 3 (t) 2/3 + u(t) y(t) = x 1 (t) (25
) with u(t) = 5 cos(t) + sin(2t) and where z α = sign(z)|z| α for all z ∈ R and α ≥ 0. The system is initialized as x(0) = [-10, -5, 2] T . The sampling times belong to the interval [T min , T max ] which will be dened It can be seen that system ( 25) is homogeneous of degree -0.2 with respect to the weights (1, 0.8, 0.6) and an homogeneous observer is given by

             ẋ1 (t) = x2 (t) -0.1 x1 (t) 0.8 -k 1 x 1 (t) -x1 (t) 0.8 ẋ2 (t) = x3 (t) + 0.1 x1 (t) 0.6 + 0.1 x2 (t) 0.75 -k 2 x 1 (t) -x1 (t) 0.6 ẋ3 (t) = -0.5 x1 (t) 0.4 -0.5 x2 (t) 0.5 -x3 (t) 2/3 + u(t) -k 3 x 1 (t) -x1 (t) 0.4
(26) with the gains chosen as k 1 = 3, k 2 = 3 and k 3 = 1. The adapted continuous-discrete time observer as proposed in this paper is then given by

                     ż(t) = -k 1 ẑ(t) 0.8 ẋ1 (t) = x2 (t) -0.1 x1 (t) 0.8 -k 1 ẑ(t) 0.8 ẋ2 (t) = x3 (t) + 0.1 x1 (t) 0.6 + 0.1 x2 (t) 0.75 -k 2 ẑ(t) 0.6 ẋ3 (t) = -0.5 x1 (t) 0.4 -0.5 x2 (t) 0.5 -x3 (t) 2/3 + u(t) -k 3 ẑ(t) 0.4 ẑ(t k ) = x 1 (t k ) -x1 (t k ) (27)
The stability of the continuous observer and a homogeneous Lyapunov function can be obtained by following the approaches developed in [START_REF] Li | A dual-observer design for global output feedback stabilization of nonlinear systems with loworder and high-order nonlinearities[END_REF] or [START_REF] Bernuau | Homogeneous continuous nite-time observer for the triple integrator[END_REF] but are rather involved and then not reported here. Assumption 2-(i) is veried, indeed one has

∂h ∂x (x)f (x, u) = x 2 -0.1 x 1 0.8 (28)
then

∂h ∂x (x 1 )f (x 1 , u) - ∂h ∂x (x 2 )f (x 2 , u) ≤ |x 1 2 -x 2 2 | + 0.1| x 1 1 0.8 -x 2 1 0.8 | ≤ |x 1
Assumption 2-(ii) is also veried, indeed one has

|G i (z 1 ) -G i (z 2 )| = k i | z 1 iα+1 -z 2 iα+1 | ≤ k i 2 -iα |z 1 -z 2 | iα+1
by applying Lemma 1-(i) since (iα + 1) ∈ (0, 1) for i = 1, 2, 3.

For the considered example, there is a convenient way to explain the ultimate boundedness of the proposed observer. Indeed, the predictor ẑ for the output error is nite-time stable and so it becomes zero after a nite time and the observer dynamics is then a copy of the system dynamics. This explains why the observer does not become unstable with a large T max contrarily to most existing approaches.

The observer proposed in this paper is compared with the ones proposed in [39, Proposition 15] and [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF] with the same gains k 1 , k 2 , k 3 . All the observers are initialized as x1 (0) = x2 (0) = x3 (0) = 0. The rst simulation is done with relatively small sampling times, that is T min = 0.05s and T max = 0.1s. As it can be seen on Figure 2 the three observers perform well. The quadratic errors are respectively equal to 0.77 for the proposed observer, 0.88 for the observer from [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] and 0.93 for the observer from [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF]. The second simulation is done with T min = T max = 1s.

In that case, it can be seen on Figure 3 that the proposed observer and the one from [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF] perform well while the one from [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] is not very accurate. The quadratic errors are respectively equal to 2.28 for the proposed observer, 661.25 for the observer from [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] and 2.88 for the observer from [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF].

The last simulation is done with T min = T max = 2.65s which corresponds to the maximum allowable sampling time for which the observer from [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF] is stable. It can be seen that the proposed observer is the only one that performs well. The quadratic errors are respectively equal to 128.9 for the proposed observer, 2.5 × 10 5 for the observer from [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] and 3.9 × 10 3 for the observer from [START_REF] Farza | Continuous-discrete time observers for a class of MIMO nonlinear systems[END_REF].

Conclusion

In this paper, we have proposed a new observer for homogeneous systems of negative degree whose output is available only at some discrete instants. It has been proved that the proposed observer error is globally uniformly ultimately bounded for any maximum sampling time. The proposed observer has been compared with a classical continuous-discrete time observer which has a maximum allowable sampling period and it has been shown in simulation that the proposed observer still performs well while the classical continuous-discrete time observer is at its limit. While the proposed observer perform well for small sampling times, it is only ultimately bounded, so future works will be concerned with the extension of the proposed observer so that it is exponentially stable for small maximum sampling periods and still practically stable for larger sampling periods.

A Technical lemmas

Lemma 1 [START_REF] Hardy | Inequalities[END_REF][START_REF] Li | A dual-observer design for global output feedback stabilization of nonlinear systems with loworder and high-order nonlinearities[END_REF] (i) Let p ∈ (0, 1) and x, y ∈ R, then the following inequality holds

| x p -y p | ≤ 2 1-p |x -y| p (A.1)
(ii) Let c, d, γ ≥ 0, then for all x, y ∈ R, one has

|x| c |y| d ≤ γ|x| c+d + d c + d c γ(c + d) c d |y| c+d (A.
2) (iii) Let p ∈ (0, 1) and x, y ∈ R, then the following inequality holds

(|x| + |y|) p ≤ |x| p + |y| p (A .3)
Lemma 2 Let α 1 , α 2 > 0 be such that α 1 > α 2 and let γ > 0 be any constant parameter, then for all x ≥ 0 the following inequality holds

x α1 + Γ ≥ γx α2 (A.4) with Γ = γ(x * ) α2 -(x * ) α1 ≥ 0 and x * = α2γ α1 1 α 1 -α 2 .
Furthermore, the inequality is optimal in the sense that the left hand-side and right hand-side of Inequality A.4 are equal for x = x * .

Proof. One can directly show that the function f (x) = γx α2 -x α1 has only one maximum point which is global. This maximum is reached at x * and is equal to Γ.

≤ n j=1 κ j n l=1 |e l | d U -α j α l |G j (ẑ) -G j (ŷ -y)| (B.2) ≤ n j=1 κ j n l=1 |e l | d U -α j α l ν j 2 |e 0 | α j +α d h (B.3) ≤ n j=1 n l=1 κ j ν j 2 |e l | 1 α l d U -αj |e 0 | 1 d h αj +α (B.4) ≤ n j=1 n l=1 κ j ν j 2 (B.5) × γ1 |e l | 1 α l d U +α + ρ j 1 (γ 1 ) |e 0 | 1 d h d U +α ≤ γ1   n j=1 κ j ν j 2   |e l | d U +α α l (B.6) +   n j=1 nκ j ν j 2 ρ j 1 (γ 1 )   |e 0 | d U +α d h
where γ1 can be chosen arbitrarily small, and

ρ i 1 (γ 1 ) = αj +α d U +α d U -αj γ1(dU +α) d U -α j α j +α . Using (6), one obtains n j=1 ∂U ∂e j (e) (G j (ẑ) -G j (ŷ -y)) (B.7) ≤ γ1   n j=1 κ j ν j 2   (U (e)) d U +α d U +   n j=1 nκ j ν j 2 ρ j 1 (γ 1 )   |e 0 | d U +α d h Fact 1 is then proved by γ 1 = γ1 n j=1 κ j ν j 2 and Γ 1 = n j=1 nκ j ν j 2 ρ j 1 (γ 1 ) . C Proof of Fact 2
Similarly to the proof of Fact 1, using Assumption 2-(i) and Lemma 1-(ii), one gets

d U d h exp(δ(t k+1 -t)) e 0 d U -d h d h × ∂h ∂x (x(t))f (x(t), u(t)) - ∂h ∂x (x(t))f (x(t), u(t)) ≤ d U d h exp(δ(t k+1 -t)) 1 d U -d h |e 0 | 1 d h d U -d h (C.1) × ν 1 n l=1 |e l | d h +α α l ≤ d U d h ν 1 n l=1 |e l | 1 α l d h +α (C.2)
× exp(δ(t k+1 -t)) with

Γ 3 = γ α 1 α 1 -α 2 3 α2 α1 α 2 α 1 -α 2 -α2 α1 α 1 α 1 -α 2
, that is

Γ3 Tmax = T α 1 α 1 -α 2 -1 max K α 1 α 1 -α 2 2 α 2 α 1 α 2 α 1 -α 2 - α 2 α 1 α 1 α 1 -α 2 (D.8) = T d U +α -α max K α 1 α 1 -α 2 2 α 2 α 1 α 2 α 1 -α 2 - α 2 α 1 α 1 α 1 -α 2

=K1

(D.9) Then Inequality [START_REF] Lopez-Ramirez | Finite-time and xed-time observer design: Implicit Lyapunov function approach[END_REF] is valid.

  and stay in there or it converges to a nite value a > Γ ν d U d U +α . It is direct to see that the second option is not possible because of inequality[START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF] and the results of Step 1.

Fig. 1 .

 1 Fig. 1. Measured output later depending on the simulations. The output is assumed to be measured with a Gaussian white noise with a standard deviation equal to 2, it is illustrated on Figure 1. It can be seen that system (25) is homogeneous of degree -0.2 with respect to the weights (1, 0.8, 0.6) and an homogeneous observer is given by

Fig. 2 .

 2 Fig. 2. States and estimations for Tmin = 0.05s and Tmax = 0.1s

Fig. 3 .

 3 Fig. 3. States and estimations for Tmin = Tmax = 1s

Fig. 4 .

 4 Fig. 4. States and estimations for Tmin = Tmax = 2.65s.

B Proof of Fact 1 Using Assumption 2 -

 12 (ii), Inequality[START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] and Lemma 1-(ii) with c = d U -α j and d = α j + α, x = |e l | 1 (G j (ẑ) -G j (ŷ -y)) (B.1)

  where γ2 ≥ 0 can be chosen as small as desired andρ 2 (γ 2 ) = d U -d h -d h . Fact 2 is then proved by taking γ 2 = γ2 d U d h ν 1 and Γ 2 = d U d h ν 1 ≤ exp(δ(t k+1 -t)) ≤ exp(δT max ) for all t ∈ [t k , t k+1). This is equivalent to -δ|e 0 (t)| = 1/T max . In order to prove Inequality (D.3) and nd K 1 , let us apply Lemma 2 withx = |e 0 (t)|, α 1 = d U d h , α 2 = d U +α d hand γ 3 = T max K 2 , which gives usx α1 + Γ 3 ≥ γ 3 x α2 (D.4) + Γ 3 ≥ T max K 2 |e 0 (t)|

									since 1 d U d h + (Γ 1 + Γ 2 )e	d U +α d U -d h |e 0 (t)|	d U +α d h
									≤ -	ε 2	e	d U +α d U	|e 0 (t)|	d U +α d h	d U +α -α max + K 1 T	(D.2)
									⇔ (Γ 1 + Γ 2 )e	d U +α d U -d h +	ε 2	e	d U +α d U	|e 0 (t)|	d U +α d h
									=K2
									≤ δ|e 0 (t)|	d U +α -α max d h + K 1 T d U	(D.3)
									1 d h	d U -d h	since δ ⇔ |e 0 (t)|	d U d h d U +α d h	(D.5)
	≤	d U d h + ρ l ν 1 2 (γ 1 n γ2 |e l | 1 α l d U +α l=1 d h	d U +α	(C.3)	⇔ ⇔ δ|e 0 (t)| 1 T max |e 0 (t)| d U d h + d U d h + Γ 3 T max T max Γ 3 ≥ K 2 |e 0 (t)| ≥ K 2 |e 0 (t)| d U +α d U +α d h d h	(D.6) (D.7)
	≤ γ2	d U d h	ν 1	l=1 n	|e l |	d U +α α l	(C.4)
		+			d U d h	ν 1	l=1 n	ρ l 2 (γ d U +α d h
	≤ γ2	d U d h	ν 1 (U (e))	d U +α d	(C.5)
		+			d U d h	ν 1	l=1 n	ρ l 2 (γ 2 ) exp(δ(t k+1 -t))	d U +α d U -d h |e 0 |	d U +α d h
									d h +α
							d U +α	d h +α γ2(dU +α)	d U n l=1 ρ l 2 (γ 2 ) .
	D Proof of Fact 3
	One can notice that Equation (22) is valid if the following
	inequality is valid
	-δ|e 0 (t)|	d U d h + (Γ 1 + Γ 2 ) (exp(δT max ))	d U +α d U -d h |e 0 (t)|	d U +α d h
	≤ -	ε 2	(exp(δT max ))	d U +α d U	|e 0 (t)|	d U +α d h	d U +α -α max + K 1 T	(D.1)

1 d U -d h |e 0 | 2 ) exp(δ(t k+1 -t)) 1 d U -d h |e 0 | 2 ) exp(δ(t k+1 -t)) d U +α d U -d h |e 0 |

-x 2 2 | + 0.1 × 2 -0.2 |x 1 1 -x 2 1 | 0.8by applying Lemma 1-(i) and the assumption holds since d h = 1, α = -0.2, α 1 = 1 and α 2 = 0.8.


domain: phys spi

text:
Guillaume Roland

Alain Portavoce

Maxime Bertoglio

Marion Descoins

Jacopo Remondina

Didier Dutartre

Frédéric Lorut

Magali Putero
email: magali.putero@univ-amu.fr

New insights in GeTe growth mechanisms

Keywords: GeTe, Phase Change Materials, crystallization, reactive diffusion, in situ X-Ray diffraction, in situ TEM, thin film

The compound GeTe, despite its simple stoichiometry, is rather unconventional and has been investigated both from a fundamental and technological perspective: it is of high interest for several technologies such as data-storage (phase change memories) and thermoelectricity. The understanding of GeTe growth is thus a key issue for technological applications and fundamental understanding. In this work, GeTe crystallization kinetics is compared to Ge/Te reactive diffusion kinetics using in situ X-ray diffraction measurements, as well as in situ transmission electron microscopy. GeTe crystallization from an amorphous solid solution exhibiting the stoichiometry of the compound GeTe is found to occur at the same temperature as for the reactive diffusion of an amorphous Ge layer on top of a polycrystalline Te layer. Furthermore, GeTe growth is of tridimensional type in the two cases, and can be modeled by the Johnson-Mehl-Avrami-Kolmogorov model. Using this model, the activation energies of nucleation and growth were determined for both crystallization and reactive diffusion. The results suggest that GeTe exhibits nucleation/reaction kinetics unusually low compared to atomic transport kinetics, contrasting with other germanides.

Introduction

Phase Change Random Access Memory (PCRAM) is one of the most promising technology for the next generation of non-volatile memories with Ferroelectric RAM and Magnetic RAM [START_REF] Lankhorst | Low-cost and nanoscale non-volatile memory concept for future silicon chips[END_REF]. PCRAM exploits the surprising phase change properties of Phase Change Materials (PCM) in order to achieve a memory effect. PCM can reversibly and quickly change between a high-resistivity amorphous (and in general low-reflectivity) state and a lowresistivity crystalline (and in general high-reflectivity) state through thermal annealing [START_REF] Raoux | Phase-change random access memory: A scalable technology[END_REF][START_REF] Wuttig | Phase-change materials for rewriteable data storage[END_REF][START_REF] Bruns | Nanosecond switching in GeTe phase change memory cells[END_REF][START_REF] Noé | Phase-change materials for nonvolatile memory devices: from technological challenges to materials science issues[END_REF].

This class of materials is mostly composed of chalcogenides glasses [START_REF] Raoux | Phase-change random access memory: A scalable technology[END_REF][START_REF] Noé | Phase-change materials for nonvolatile memory devices: from technological challenges to materials science issues[END_REF] , and has been used for nonvolatile optical memories such as CDs, DVDs and Blu-ray discs, using the large reflectivity difference between the two states [START_REF] Raoux | Phase-change random access memory: A scalable technology[END_REF][START_REF] Wuttig | Phase-change materials for rewriteable data storage[END_REF][START_REF] Noé | Phase-change materials for nonvolatile memory devices: from technological challenges to materials science issues[END_REF][START_REF] Yamada | Origin, secret, and application of the ideal phase-change material GeSbTe[END_REF]. Ge-Sb-Te based chalcogenide alloys or compounds (GST) are considered to be among the most promising PCM for PCRAM applications [START_REF] Raoux | Phase-change random access memory: A scalable technology[END_REF][START_REF] Noé | Phase-change materials for nonvolatile memory devices: from technological challenges to materials science issues[END_REF][START_REF] Yamada | Origin, secret, and application of the ideal phase-change material GeSbTe[END_REF]. However, their low crystallization temperatures ( 150-170°C for Ge2Sb2Te5 [START_REF] Noé | Phase-change materials for nonvolatile memory devices: from technological challenges to materials science issues[END_REF]) can be a limiting factor for industrial production and high reliability applications, such as automotive applications [START_REF] Privitera | Amorphous-to-crystal transition of nitrogen-and oxygen-doped Ge 2Sb2Te5 films studied by in situ resistance measurements[END_REF][START_REF] Gomiero | Crystallization Speed in Ge-Rich PCM Cells as a Function of Process and Programming Conditions[END_REF]. In contrast, the binary compound GeTe exhibits ultra-fast switching as well as a higher crystallization temperature than GST ( 180-230°C depending on oxidation [START_REF] Berthier | In situ observation of the impact of surface oxidation on the crystallization mechanism of GeTe phasechange thin films by scanning transmission electron microscopy[END_REF]). Consequently, GeTe is also a strong PCM contender for PCRAM applications [START_REF] Bruns | Nanosecond switching in GeTe phase change memory cells[END_REF], but is also considered as a good candidate for the development of GeTe-based RF switches [START_REF] Ghalem | Arrays of GeTe electrically activated RF switches[END_REF][START_REF] El-Hinnawy | A four-terminal, inline, chalcogenide phase-change RF switch using an independent resistive heater for thermal actuation[END_REF][START_REF] Singh | Chalcogenide Phase Change Material GeTe Based Inline RF SPST Series and Shunt Switches[END_REF]. Furthermore, numerous studies have also reported interesting GeTe properties for thermoelectric applications [START_REF] Suwardi | Effective enhancement of thermoelectric and mechanical properties of germanium telluride: Via rhenium-doping[END_REF][START_REF] Li | Electronic origin of the high thermoelectric performance of GeTe among the p-type group IV monotellurides[END_REF][START_REF] Perumal | High performance thermoelectric materials and devices based on GeTe[END_REF].

The control of the GeTe compound growth is of prime importance for all applications.

Two main techniques are commonly used in industrial processes for the solid-state production of thin film compounds: i) reactive diffusion between two films in contact [START_REF] Breil | Challenges of nickel silicidation in CMOS technologies[END_REF][START_REF] Pawlak | Investigation of Ni fully silicided gates for sub-45 nm CMOS technologies[END_REF][START_REF] Gambino | Silicides and ohmic contacts[END_REF] , which is driven by atomic transport and interfacial reaction, or ii) non-diffusive reaction [START_REF] Assaf | High Curie temperature Mn5Ge3 thin films produced by non-diffusive reaction[END_REF] that is also called "crystallization", since it consists in the formation of an ordered compound from a homogeneous solid solution in the amorphous state exhibiting the stoichiometry of the desired compound. GeTe crystallization has been mainly studied to date from homogeneous Ge(50%):Te(50%) amorphous (a-GeTe) thin films for PCRAM applications [START_REF] Kolb | Understanding the Crystallization Behavior of Surface-Oxidized GeTe Thin Films for Phase-Change Memory Application[END_REF][START_REF] Chen | Resolving Crystallization Kinetics of GeTe Phase-Change Nanoparticles by Ultrafast Calorimetry[END_REF]. The crystallization of a-GeTe thin films is characterized by homogeneous random nucleation of the GeTe compound, followed by its three-dimensional (3D) growth [START_REF] Berthier | In situ observation of the impact of surface oxidation on the crystallization mechanism of GeTe phasechange thin films by scanning transmission electron microscopy[END_REF][START_REF] Kolb | Understanding the Crystallization Behavior of Surface-Oxidized GeTe Thin Films for Phase-Change Memory Application[END_REF][START_REF] Libera | Time-resolved reflection and transmission studies of amorphous Ge-Te thin-film crystallization[END_REF]. Reported GeTe crystallization experiments were performed using differential scanning calorimetry (DSC) [START_REF] Matsushita | Measurements of activation energy in the initialization process of amorphous GeTe films[END_REF][START_REF] Sun | Crystallization kinetics of GeTe phase-change thin films grown by pulsed laser deposition[END_REF], in situ resistivity measurements [START_REF] Navarro | Electrical performances of SiO2-doped GeTe for phase-change memory applications[END_REF][START_REF] Fantini | N-doped GeTe as performance booster for embedded phase-change memories[END_REF][START_REF] Gallard | New insights into thermomechanical behavior of GeTe thin films during crystallization[END_REF] or in situ reflectivity measurements [START_REF] Libera | Time-resolved reflection and transmission studies of amorphous Ge-Te thin-film crystallization[END_REF][START_REF] Matsushita | Measurements of activation energy in the initialization process of amorphous GeTe films[END_REF][START_REF] Gourvest | Impact of Oxidation on Ge2Sb2Te5 and GeTe Phase-Change Properties[END_REF], allowing an activation energy linked to the crystallization process to be determined using the Kissinger analysis. This activation energy varies between 1.7 and 3.9 eV depending on the authors and the experimental technique [START_REF] Libera | Time-resolved reflection and transmission studies of amorphous Ge-Te thin-film crystallization[END_REF][START_REF] Matsushita | Measurements of activation energy in the initialization process of amorphous GeTe films[END_REF][START_REF] Sun | Crystallization kinetics of GeTe phase-change thin films grown by pulsed laser deposition[END_REF][START_REF] Navarro | Electrical performances of SiO2-doped GeTe for phase-change memory applications[END_REF][START_REF] Gourvest | Impact of Oxidation on Ge2Sb2Te5 and GeTe Phase-Change Properties[END_REF]. Reactive diffusion (RD) is commonly used to produce silicide and germanide ohmic contacts on the source, the drain and the gate of microelectronic transistors in the complementary-metal-oxide-semiconductor (CMOS) technology through the self-aligned-silicide (Salicide) process, based on the RD of a metallic film with the semiconductor [START_REF] Breil | Challenges of nickel silicidation in CMOS technologies[END_REF][START_REF] Pawlak | Investigation of Ni fully silicided gates for sub-45 nm CMOS technologies[END_REF][START_REF] Gambino | Silicides and ohmic contacts[END_REF]. RD depends on interfacial reaction and selfdiffusion [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF][START_REF] Gas | Formation of silicide thin films by solid state reaction[END_REF][START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF]. Thus, RD experiments can allow the kinetics of semiconductor/metal reaction and atomic transport to be assessed, which are of main interest for crystallization understanding and process simulation [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF]. Germanide RD on flat substrates generally leads to a two-dimensional (2D) growth [START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF][START_REF] Gaudet | Thin film reaction of transition metals with germanium[END_REF][START_REF] Perrin Toinin | First stages of Pd/Ge reaction: Mixing effects and dominant diffusing species[END_REF][START_REF] Perrin Toinin | Origin of the first-phase selection during thin film reactive diffusion: Experimental and theoretical insights into the Pd-Ge system[END_REF][START_REF] Schutter | Texture in thin film silicides and germanides: A review[END_REF], characterized by first an interfacial nucleation step leading to the extremely fast formation of an interfacial thin layer of few nanometers-thick (lateral growth), followed by the growth of this 2D layer in the direction perpendicular to the interface (normal growth). The growth of the layer can be simulated by linear-parabolic type of models [START_REF] Gas | Formation of silicide thin films by solid state reaction[END_REF], reproducing the compound linear growth versus time, limited by interfacial reaction kinetics, for small layer thicknesses, followed by a parabolic growth regime versus time, limited by atomic transport kinetics, once the growing layer has reached a critical thickness [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF][START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF]. However, only parabolic growth has been so far observed in the case of 2D germanide growth [START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF], meaning that germanium/metal reaction kinetics is generally several orders of magnitude faster than grain boundary self-diffusion kinetics in the compound.

In this work, GeTe crystallization kinetics in a 150 nm-thick a-GeTe thin film is compared to the GeTe growth kinetics observed during the RD of a 60 nm-thick Ge layer deposited on a 90 nm-thick Te layer using in situ X-ray diffraction (XRD) and in situ transmission electron microscopy (TEM). Surprisingly, GeTe follows a 3D growth exhibiting similar nucleation temperatures in the two cases. Activation energies of nucleation and growth are determined using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model [START_REF] Avrami | Kinetics of phase change. I: General theory[END_REF][START_REF] Avrami | Kinetics of phase change. II Transformation-time relations for random distribution of nuclei[END_REF][START_REF] Avrami | Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III[END_REF] to fit in situ XRD measurements acquired during isotherms.

Material and methods

The samples were elaborated in a commercial magnetron sputtering system exhibiting a base pressure of 10 -8 mbar. Te was sputtered from a 99.99% pure Te-target in RF mode, while Ge was sputtered from a 99.999% pure Ge-target in DC mode, using a 99.9999% pure Ar gas flow under a work pressure of 3.6  10 -3 mbar. Ge and Te fluxes were calibrated separately by measuring the thickness of sputtered films deposited at room temperature (RT) on the native oxide of Si(001) substrates by X-ray reflectivity. Two types of samples were elaborated. Sample #1 was meant to study GeTe crystallization. In this case, Ge and Te were co-deposited in the Ge(50 at.%):Te(50 at.%) stoichiometry at RT on the native SiO2 layer ( 2 nm-thick) of a Si(001) substrate. The SiO2 layer is kept as a diffusion barrier preventing Si atoms from the substrate to interact with the deposited layers. The 150 nm-thick GeTe layer was capped with 10 nm of pure Ge at the same temperature. The Ge layer was added aiming to limit the GeTe layer oxidation. The entire film was fully amorphous after deposition (Fig. 1). Sample #2 was destined for the study of solid state Ge/Te RD. To this aim, a 90 nm-thick Te layer was deposited on the same substrate as used for sample #1, before to be covered by a 70 nm-thick Ge layer. All the layers were deposited at RT, and the sputtering powers and deposition times were kept the same as for sample #1. The reaction of the entire bilayer is expected leading to the formation of a GeTe layer of same thickness as after the entire crystallization of sample #1, with an extra 10 nm-thick layer of a-Ge on the surface.

After deposition, the samples were cut into several pieces. 1.5  1.5 cm 2 specimens were used to performed in situ XRD measurements. The specimens were annealed at constant temperature (isothermal annealing) or following a heating ramp (isochronal annealing) in situ in a XRD setup under a vacuum of ~ 10 -5 mbar. All the XRD measurements were performed in the Bragg-Brentano geometry (-2) on a Panalytical Empyrean diffractometer equipped with an PIXcel detector designed for high speed data collection, using a Cu K- source ( = 0.154 nm). For isochronal annealing, the heating ramp was performed between 100°C and 350°C following a temperature ramp of 10°C per minute steps separated by 4.5 minute-long XRD scans at constant temperature, corresponding to an average heating ramp of  0.9 K min -1 .

Scans were executed every 5°C. Four to five isotherms between 145°C and 190°C were performed on each type of samples, in addition to the ramp annealing.

In situ TEM observations gave complementary information on the growth of the GeTe phase during Ge/Te RD. A TEM lamella has been prepared using the in-situ lift out technique in a FIB-SEM system, Helios450 TM from ThermoFisher, and mounted onto a molybdenum grid.

Final thinning of the lamella has been done using a 8kV acceleration voltage, in order to limit the possible FIB Gallium induced amorphization. Once prepared, the grid has been mounted onto a sample heating holder from Gatan, and then loaded into a Tecnai F20 TEM, from ThermoFisher, operating at a 200kV accelerating voltage in TEM bright field mode. with the factor k = 0.89 in our case. GeTe grain size reaches rapidly a maximum L  12 nm during GeTe growth (T < 200 °C) and stays almost constant, following the diffraction intensity up to 290°C. At this temperature, both α-GeTe volume (XRD integrated intensity) and grain size (L) are found to increase again. Our results are in agreement with similar studies previously reported: the GeTe crystallization temperature (i.e. nucleation temperature) Tn = 170 °C is in agreement with the literature for uncapped layers [START_REF] Raoux | Phase transitions in Ge-Te phase change materials studied by time-resolved x-ray diffraction[END_REF], and the subsequent intensity increase of the peak α-GeTe(202) concurrently with Ge crystallization was already observed [START_REF] Gallard | New insights into thermomechanical behavior of GeTe thin films during crystallization[END_REF]. 110) and (111) of the hexagonal Te phase. The as-deposited Te layer is polycrystalline (poly-Te). No diffraction peaks are observed from the Ge layer, as this layer is amorphous (a-Ge) after deposition. Upon annealing, the vanishing of Te peaks coincides with the simultaneous appearance at T = 175°C of the four diffraction peaks α-GeTe(003), α-GeTe(021), α-GeTe(202), and α-GeTe(024), respectively at 2θ = 25.21°, 25.98°, 29.83°, and 42.16°. Despite the structural and chemical distribution differences between sample #1 and sample #2, the α-GeTe phase is found to form at the same temperature (in the 5°C measurement error) in the two samples. However, despite that the peak α-GeTe(202) is the most intense in the two samples, the samples exhibit different textures, as the peaks α-GeTe(003) and α-GeTe(021) are detected only in sample #2, exhibiting higher diffraction intensities than the peak α-GeTe(024) in this case. Furthermore, Ge crystallization is not observed during Ge/Te RD in sample #2 despite the presence of the 70 nm-thick a-Ge layer, contrasting with GeTe crystallization in sample #1. Fig. 2 shows typical phase volume variations during 2D growth of compounds by RD [START_REF] Nemouchi | A comparative study of nickel silicides and nickel germanides: Phase formation and kinetics[END_REF]: the diffraction intensity of the reacting layers (here only Te since Ge stays amorphous) decreases concurrently with the increase of the growing compound diffraction intensity (i.e. α-GeTe). The superimposition in Fig. 2 of the variations of the diffraction peak α-GeTe(202) in the two samples during the same annealing shows that despite a similar nucleation temperature Tn = 170°C, GeTe growth kinetics are different during crystallization and RD. Indeed, growth kinetics are expected to be different as GeTe growth does not need atomic transport in sample #1 (the composition is homogeneous and corresponds to the GeTe stoichiometry), while GeTe growth during RD depends on atomic transport kinetics [START_REF] Gas | Formation of silicide thin films by solid state reaction[END_REF][START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF]. Average GeTe grain size is larger in sample #2, and its evolution with temperature differs from sample #1: L increases with the GeTe phase volume up to ~ 200 °C during RD; at that temperature, the GeTe volume has reached a maximum, but L is still increasing with temperature following a slower rate, reaching L  36 nm at the end of the experiment. This behavior is in agreement with a growth regime below 200°C, followed by grain growth competition at higher temperature.

Results and discussion

GeTe growth during isochronal annealing

GeTe growth during isothermal annealing

Fig. 3 presents the variations of the α-GeTe(202) XRD normalized integrated intensity recorded during different isothermal annealing of sample #1 (Fig. 3(a)) and sample #2 (Fig. 3(b)) between T = 145 and 190 °C. The signal is proportional to the crystallized -GeTe phase volume, so to the crystallized fraction. For sample #1, the -GeTe crystallization fraction can be fitted with the JMAK model, as expected. However, the GeTe growth in sample #2 cannot be simulated using typical linear-parabolic 2D growth models [START_REF] Gas | Formation of silicide thin films by solid state reaction[END_REF][START_REF] Evans | The relation between tarnishing and corrosion[END_REF]. Instead, GeTe growth in sample #2 can also be fitted with the JMAK model. In this model, the crystallized fraction (or the phase volume) variation with time f(t) follows the law:

𝑓(𝑡) = 1 -exp(-𝜅 × 𝑡 𝑛 ) (2)
 includes the product of the nucleation rate with the growth rate, and n is generally called the Avrami coefficient. According to this model,  is constant if the nucleation rate N = N0 exp(-En/kBT) and the growth rate  = 0 exp(-Eg/kBT) are constant versus time at constant temperature (kB is the Boltzmann constant). This is the case for GeTe growth in the two samples #1 and #2 (Fig. 3). particular, Fig. GeTe growth during crystallization is known to follow a 3D growth [START_REF] Berthier | In situ observation of the impact of surface oxidation on the crystallization mechanism of GeTe phasechange thin films by scanning transmission electron microscopy[END_REF][START_REF] Kolb | Understanding the Crystallization Behavior of Surface-Oxidized GeTe Thin Films for Phase-Change Memory Application[END_REF] , which is in agreement with n = 4 in the JMAK model, considering the homogeneous growth of spherical nucleus in a nucleation-and-growth regime. In this case:

𝜅(𝑇) = 𝜅 0 exp (- 𝐸 𝑎 𝑘 𝐵 𝑇 ) (3)
Ea = En + 3Eg is the activation energy of nucleation and growth. from Kissinger analyses (isochronal annealing) during GeTe crystallization were found to be comprised between 1.7 and 3.9 eV [START_REF] Libera | Time-resolved reflection and transmission studies of amorphous Ge-Te thin-film crystallization[END_REF][START_REF] Matsushita | Measurements of activation energy in the initialization process of amorphous GeTe films[END_REF][START_REF] Sun | Crystallization kinetics of GeTe phase-change thin films grown by pulsed laser deposition[END_REF][START_REF] Navarro | Electrical performances of SiO2-doped GeTe for phase-change memory applications[END_REF][START_REF] Gourvest | Impact of Oxidation on Ge2Sb2Te5 and GeTe Phase-Change Properties[END_REF]. This large energy range could be linked to both the small composition variations reported in the different studies and the effect of uncontrolled incorporation of impurities such as carbon and oxygen, depending on sputtering conditions (target purity, argon gas pressure and purity, deposition speed…) [START_REF] Assaf | Carbon concentration, Curie temperature, and magnetic resonance field of Mn5Ge3(C) thin films[END_REF]. It is interesting to note that the activation energy of nucleation reported in the present work is comprised in the range, 1.7 and 3.9 eV, reported in the bibliography. These observation is in agreement with Navarro et al. [START_REF] Navarro | Electrical performances of SiO2-doped GeTe for phase-change memory applications[END_REF] suggesting that the activation energy determined from Kissinger analyses should correspond to the activation energy of the nucleation process due to the very fast GeTe growth. The interpretation of the GeTe growth following eq.( 2) in sample #2 is less straightforward since n ~ 2.5 in this case. Thus, in situ TEM observations were performed aiming to determine the GeTe growth mode in sample #2.

Figure 5 presents four subsequent TEM images acquired during the in situ heating of a cross-section TEM lamella prepared from sample #2 from RT to T = 225°C. The arrows are used to mark identical locations in the different TEM images. After deposition (Fig. 5(a)), the sample is made of the stack of a 63 nm-thick amorphous Ge layer on top of a 80 nm-thick polycrystalline Te layer. One can note that no intermixing layer is observed at the a-Ge/poly-Te interface. Generally, a 3-to-5 nm-thick intermixing layer is formed between the metal and Si or Ge during RT sputtering [START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF][START_REF] Perrin Toinin | First stages of Pd/Ge reaction: Mixing effects and dominant diffusing species[END_REF]. This layer can be amorphous or crystalline and acts as the initial stage of 2D growth of silicides or germanides [START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF]. During heating (Fig. 5(b) to 5(d)), polycrystalline GeTe starts to grow between Ge and Te. However, GeTe does not form a continuous layer between the a-Ge and poly-Te layers as usually observed for germanide growth [START_REF] Mehrer | Diffusion and Point Defects in Elemental Semiconductors[END_REF][START_REF] Gaudet | Thin film reaction of transition metals with germanium[END_REF][START_REF] Perrin Toinin | First stages of Pd/Ge reaction: Mixing effects and dominant diffusing species[END_REF][START_REF] Perrin Toinin | Origin of the first-phase selection during thin film reactive diffusion: Experimental and theoretical insights into the Pd-Ge system[END_REF][START_REF] Schutter | Texture in thin film silicides and germanides: A review[END_REF][START_REF] Nemouchi | A comparative study of nickel silicides and nickel germanides: Phase formation and kinetics[END_REF]. In sample #2 instead, 3D GeTe grains form both in the a-Ge layer and in the poly-Te layer. These observations are in agreement with the GeTe growth kinetic observed by in situ XRD in this sample: GeTe growth does not follow the usual 2D growth modeled by linear-parabolic growth models, but corresponds to a 3D growth that be modeled by the JMAK model. The fact that n is different from 4 could be linked to different effects. For example, the nucleation rate and nucleus distribution are probably different in the amorphous Ge layer and in the polycrystalline Te layer. Furthermore, n should take into account that the growth rate in sample #2 is dependent of atomic transport kinetics following a t 1/2 law.

According to TEM observations, GeTe growth in sample #2 can be modeled using the JMAK with the data available in the literature. Indeed, quantitative data concerning interfacial reaction for the system Ge-Te are lacking, as well as data concerning self-diffusion, especially in grain boundaries and in amorphous solid solutions. Comparing with silicides growth via RD, which has been more studied than germanides, one can note that GeTe nucleation energy is higher than the usual activation energy of reaction (0.8 eV for Ni2Si [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF], 0.9 eV for Ni5Ge3 [START_REF] Nemouchi | A comparative study of nickel silicides and nickel germanides: Phase formation and kinetics[END_REF] for example), and GeTe growth energy is of same order as for silicides effective self-diffusion during RD (1.5 eV for Ni2Si [START_REF] Nemouchi | Differential scanning calorimetry analysis of the linear parabolic growth of nanometric Ni silicide thin films on a Si substrate[END_REF] for example). The activation energy of nucleation is found ~40% higher in the homogeneous GeTe amorphous solid solution than at the a-Ge/poly-Te interface. Furthermore, the nucleation time is shorter for RD below 175°C (see Fig. 4(b)). This behavior is probably related to the different nature of nucleation in the two cases: homogeneous in amorphous GeTe and heterogeneous at the a-Ge/poly-Te interface. The activation energy of growth is found about 24% higher for GeTe crystallization from the stoichiometric amorphous solid solution compared to RD from the a-Ge/poly-Te bilayer. However, growth kinetics is faster for crystallization (see Fig. 2 and Fig. 3), as expected. Indeed, crystallization growth kinetics should correspond to the "atom ordering" process, while RD should consist of both atomic transport (atom diffusion in grain boundaries and interfaces) and "atomic ordering" (i.e. reaction). One can note that the difference of growth kinetics between crystallization and RD is not that significant (Fig. 2) and En  Eg during GeTe RD, suggesting that nucleation (or reaction) may be the limiting process for GeTe RD, instead of self-diffusion for most of other germanides. The fact that the activation energy of growth is found lower during RD could be linked to point defect injection during RD. Indeed, atomic transport is in general several orders of magnitude faster during RD compared to equilibrium diffusion [START_REF] Portavoce | Influence of interfacial reaction upon atomic diffusion studied by in situ Auger electron spectroscopy[END_REF]. This effect has been suggested to be due to point defect injection during low-temperature RD [START_REF] Portavoce | Influence of interfacial reaction upon atomic diffusion studied by in situ Auger electron spectroscopy[END_REF][START_REF]Self-interstitials[END_REF][START_REF] Abhaya | Silicidation in Pd/Si thin film junction-Defect evolution and silicon surface segregation[END_REF][START_REF] Masse | Point defect creation induced by solid state reaction between nickel and silicon[END_REF].

Conclusion

GeTe crystallization has been compared to Ge/Te reactive diffusion aiming to improve our knowledge of GeTe growth mechanisms using in situ XRD measurements and in situ TEM observations. GeTe crystallization temperature was found to be Tn = 170 °C ± 5 °C during isochronal annealing of  0.9 K min -1 in agreement with the literature. GeTe was found to follow a two-steps-crystallization including Ge crystallization as already reported in former investigations. The Ge crystallization observed during GeTe crystallization is probably linked to a mechanism involving the second step of GeTe crystallization as they occur simultaneously, but also because Ge crystallization is not observed during Ge/Te RD despite the presence of the tenth-of-nanometer-thick a-Ge layer. GeTe formation during isochronal RD occurs at the same temperature as GeTe crystallization (Tn = 175 °C ± 5°C). However, GeTe growth kinetics as well as the GeTe texture are different between crystallization and RD, and GeTe grain size is about the double for RD at the end of annealing. Isothermal annealing shows that GeTe crystallization and RD correspond to a 3D growth that can be modeled by the JMAK model.

This model allowed the activation energies of nucleation and growth to be determined for the two processes. Comparisons between kinetics and activation energies of crystallization and RD show that GeTe is different from usual germanides: it follows a 3D growth instead of a 2D growth during RD, which can be due to a peculiarly fast atomic transport kinetics compared to reaction (i.e. nucleation) kinetics.

Sample CRediT author statement

FIGURE captions

Fig. 1 (

 1 Fig. 1(a) presents series of XRD patterns recorded during the ramp annealing of sample #1 (GeTe layer) as a function of temperature. No diffraction peak is detected in this sample before annealing. The sample is entirely amorphous. Upon heating, two peaks simultaneously appear at 2θ = 29.75° and 42.28° when reaching T = 170°C. They respectively correspond to the (202) and (024) Bragg reflections of the rhombohedral α-GeTe phase. Ge crystallization is detected at T = 290°C with the apparition of the Ge(111) diffraction peak at 2θ = 27.31° and coincides with an intensity increase of the diffraction peak α-GeTe(202).

Fig. 1 :

 1 Fig. 1: XRD patterns ( = 0.154 nm) as a function of temperature acquired during in situ isochronal annealing of ~ 0.9 K min -1 for a) sample #1 (GeTe layer) and b) sample #2 (Ge/Te bilayer).

Fig. 2

 2 Fig.2presents the variations of the normalized integrated XRD intensities versus annealing temperature. The signal is proportional to the corresponding phase diffracting volume[START_REF] Oueldna | Phase transitions in thermoelectric Mg-Ag-Sb thin films[END_REF]. Only the peak α-GeTe(202) (open circles) is presented in the case of sample #1: the Ge(111) peak intensity is very low, and thus, shows significant fluctuations preventing getting reliable data after integration. Fig.2shows also the average grain size L of α-GeTe along the normal of the sample surface, determined from the full width at half maximum (FWHM) w of

Fig. 2 :

 2 Fig. 2: Normalized integrated intensities of diffraction peaks and average grain size (L) during in situ XRD isochronal annealing of ~ 0.9 K min -1 : sample #1 (GeTe layer), α-GeTe(202) (open circles); and sample #2 (Ge/Te bilayer), hexagonal-Te(101) (solid squares) and α-GeTe(202) (open squares).

Fig. 1 (

 1 Fig. 1(b) presents the series of XRD patterns acquired during the ramp annealing of sample #2 (Ge/Te bilayer). Four diffraction peaks are already observed before annealing at 2θ = 27.53°, 38.16°, 40.16° and 43.07°, respectively corresponding to the Bragg reflections (101), (102), (110) and (111) of the hexagonal Te phase. The as-deposited Te layer is

  3(c) and Fig.3(d) show that GeTe growth occurs according to a same regime at low temperature, between 145 and 160°C (n ~ 4) for sample #1 and between 145 et 175°C (n ~ 2.5) for sample #2, while the growth regime changes (i.e. the value of n changes) and eventually cannot be fitted by the JMAK model at high temperatures. The value of n can have different interpretations, including a dependence on the geometry of the growing phase (2D or 3D for example) or a dependence on the nucleation site density and its saturation[START_REF] Starink | The Meaning of the Impingement Parameter[END_REF][START_REF] Ruitenberg | Determination of the isothermal nucleation and growth parameters for the crystallization of thin Ge 2Sb 2Te 5 films[END_REF].

Fig. 3 :

 3 Fig. 3: Normalized integrated intensity f(t) of the α-GeTe(202) diffraction peak versus time recorded during in situ XRD isothermal annealing at various temperatures between 145°C and 190°C: a) and c) sample #1 (GeTe layer); b) and d) sample #2 (Ge/Te bilayer). The solid line in a) and b) corresponds to a fit using the JMAK model with the Avrami exponent n determined from the corresponding signal  = ln(-ln(1f(t))) in c) and d).

Fig. 4 (Fig. 4 (

 44 Fig.4(a) presents the plot of  versus 1/kBT using the logarithmic scale determined from the in situ XRD signals measured from sample #1 (Fig.3(a)) at the two temperatures 145°C and 160°C(n ~ 4), and from sample #2 at three temperatures. Considering sample #1, the activation energy of nucleation and growth was found to be Ea = 6.23 eV. The nucleation time t corresponding to the time needed to detect for the first time the diffraction signal during isothermal annealing informs on the nucleation kinetics, and was described as follows: 1 ∆𝑡 = Ω × exp (-𝐸 𝑛 𝑘 𝐵 𝑇 )

Fig. 4 :

 4 Fig. 4: Variations of a) the parameter ln() and b) the nucleation time ln(t) according to the inverse of temperature for sample #1 (GeTe layer) and sample #2 (Ge/Te bilayer).

  model with the same assumptions as for sample #1, but Ea = En + (n-1)Eg in this case. Similar to the case of sample #1, the variations of  determined from the in situ XRD signals measured from sample #2 (Fig.3(b)) at T = 145°C, 160°C, and 175°C (n ~ 2.5) are presented in Fig.4(a)versus 1/kBT. The activation energy of nucleation and growth was found to be Ea = 3.06 eV.

Fig. 5 :

 5 Fig. 5: Cross-section TEM images acquired during in situ sample #2 (Ge/Te) heating from RT to 225°C: a) RT, b) 180°C, c) 205°C, and d) 225°C. The arrows are used to mark identical locations in the different images. The added color highlights the new phase.

4

  (b). The activation energy of nucleation for GeTe RD was found to be En = 1.25 eV, giving an activation energy of growth Eg = 1.21 eV. Tab. 1 summarizes the activation energies determined from the isothermal in situ XRD measurements performed at low temperature ( 175 °C), using the JMAK model. These activation energies of nucleation and growth are difficult to compare

Guillaume Roland:

  Investigation, Validation, Formal analysis, Data Curation, Visualization, Writing-Original Draft. Alain Portavoce: Conceptualization, Methodology, Writing -Original Draft and review& editing, Supervision, Project Administration, Funding acquisition. Maxime Bertoglio: Investigation, Methodology. Marion Descoins: Methodology, Writing -review& editing. Jacopo Remondina: Investigation, Validation, Formal analysis. Didier Dutartre: Project administration, Validation. Frédéric Lorut: Investigation, Methodology, Project administration. Magali Putero: Conceptualization, Methodology, Writing -review& editing, Supervision, Project Administration, Funding acquisition.

Fig. 1 :

 1 Fig. 1: XRD patterns ( = 0.154 nm) as a function of temperature acquired during in situ isochronal annealing of ~ 0.9 K min -1 for a) sample #1 (GeTe layer) and b) sample #2 (Ge/Te bilayer).

Fig. 2 :

 2 Fig. 2: Normalized integrated intensities of diffraction peaks and average grain size (L) during in situ XRD isochronal annealing of ~ 0.9 K min -1 : sample #1 (GeTe layer), α-GeTe(202) (open circles); and sample #2 (Ge/Te bilayer), hexagonal-Te(101) (solid squares) and α-GeTe(202) (open squares).

Fig. 3 :

 3 Fig. 3: Normalized integrated intensity f(t) of the α-GeTe(202) diffraction peak versus time recorded during in situ XRD isothermal annealing at various temperatures between 145°C and 190°C: a) and c) sample #1 (GeTe layer); b) and d) sample #2 (Ge/Te bilayer). The solid line in a) and b) corresponds to a fit using the JMAK model with the Avrami exponent n determined from the corresponding signal  = ln(-ln(1f(t))) in c) and d).

Fig. 4 :

 4 Fig. 4: Variations of a) the parameter ln() and b) the nucleation time ln(t) according to the inverse of temperature for sample #1 (GeTe layer) and sample #2 (Ge/Te bilayer).

Fig. 5 :

 5 Fig. 5: Cross-section TEM images acquired during in situ sample #2 (Ge/Te) heating from RT to 225°C: a) RT, b) 180°C, c) 205°C, and d) 225°C. The arrows are used to mark identical locations in the different images. The added color highlights the new phase.

Table 1 :

 1 Activation energies determined from in situ XRD isotherms using the JMAK model.The variations of t measured from the same in situ XRD signals are presented in Fig.

	Sample	Ea (eV)	En (eV)	Eg (eV)
	#1-crystallization	6.23	1.72	1.50
	#2-RD	3.06	1.25	1.21

Acknowledgements

This project was supported by the French National Association of Research and Technology (ANRT Project 2020/0907).


Jacopo Remondina

Alain Portavoce

Maxime Bertoglio

Guillaume Roland

Frédéric Lorut

Yannick Le Friec

Daniel Benoit

Magali Putero

Thermal desorption study on possible hydrogen sources and diffusion barriers in CMOS technology

Keywords: Hydrogen, Thermal desorption spectroscopy, CMOS, Nitride, Oxide

Hydrogen passivation of silicon dangling bonds is a key requirement in the realization of reliable CMOS devices. The creation of a hydrogen reservoir in the device near the silicon may be a solution supporting constant passivation during the life-cycle of the device. In this work, we used thermal desorption spectroscopy to evaluate the amount of hydrogen stored in undoped silicon glass (USG) deposited using SiH4 (USG-SiH4) or tetraethyl orthosilicate (USG-TEOS), and silicon (carbon) nitrides, typical dielectrics used in CMOS technology. The goal of the study is to determine if these materials can be used as hydrogen source or diffusion barrier. Our results suggest that USG-SiH4 can act as an efficient source of hydrogen, while Si nitrides can act as reliable diffusion barrier.

Introduction

Si-based complementary-metal-oxide-semiconductor (CMOS) technology is used to produce low power consumption and high noise resistance [START_REF] Skotnicki | Materials and device structures for sub-32 nm CMOS nodes[END_REF][START_REF] Boeuf | An evaluation of the CMOS technology roadmap from the point of view of variability, interconnects, and power dissipation[END_REF] microelectronic transistors for all kind of applications, from the more popular/mainstream (PC, smartphone, wearable, IoT, etc) to the more dedicated/demanding ones (industrial applications, automotive, biological, etc) [START_REF] Szelag | Hybrid III-V/Silicon Technology for Laser Integration on a 200-mm Fully CMOS-Compatible Silicon Photonics Platform[END_REF]4]. However, Si device performances are directly influenced by the number of defects present in the active region of the device. Indeed, despite the use of ultra-pure Si wafers, some intrinsic defects were shown to compromise device performances, such as vacancies and related extended defects (dislocations, interfaces…) [START_REF]Schrimpf, Defects in Microelectronic Materials and Devices[END_REF]. These defects are known to alter the energy band scheme, introducing localized states [START_REF] Weber | Dangling-bond defects and hydrogen passivation in germanium[END_REF] that can negatively influence the properties of a wide range of devices. One of the proposed and implemented solutions consists in saturating Si dangling bonds using hydrogen (H) [START_REF] Dey | Optimal H2-dilution playing key role in accomplishing significant nanocrystallinity with both Si and Ge moieties in SiGe nanocomposite thin film network[END_REF] (or deuterium [START_REF] Lee | The effect of deuterium passivation at different steps of cmos processing on lifetime improvements of cmos transistors[END_REF]). Usually, Si dangling bond saturation by H is performed during growth, exposing the material to a H-rich atmosphere [START_REF] Vereecke | Quantum efficiency and dark current evaluation of a backside illuminated CMOS image sensor[END_REF]. However, this technique cannot be used for all applications, depending on device fabrication process, and the passivated region needs to be encapsulated by an H diffusion barrier for device reliability.

A second technique would consist in including in the surrounding materials an environment supporting H passivation during the entire life-time of the device [START_REF] Benoit | Hydrogen desorption and diffusion in PECVD silicon nitride. Application to passivation of CMOS active pixel sensors[END_REF]. In this work, we used thermal desorption spectroscopy (TDS) to investigate the possibility of using typical CMOS-compatible dielectrics (two Si oxides and two Si nitrides) as H source or H diffusion barrier for such application.

Materials and methods

The materials were investigated in the form of thin films. Deposition was performed on cleaned Si(001) wafers in PECVD reactors. Two 940 nm thick Si dioxides layers were investigated for possible H sources: both are undoped silicon glass (USG) either deposited at 275°C using silane gas as precursor (USG-SiH4), or deposited at 380°C using tetraethoxysilane (USG-TEOS). Nitrogen was used as carrier gas. The two materials expected to act as H diffusion barrier were a 40 nm-thick SiCN layer deposited at 350°C and a 600 nm-thick Si3N4 layer deposited at 380°C. Two types of samples were produced, as the nitride layers were deposited either on a cleaned Si wafer or a possible H source layer (Si3N4/USG-SiH4 and SiCN/USG-SiH4 samples).

After deposition, the samples were placed on a 1 cm diameter PBN furnace and loaded in an ultrahigh vacuum (UHV) system equipped with an Ametek LC-D quadrupole mass spectrometer, allowing TDS measurements to be performed (see SI). The samples were introduced in the UHV chamber (< 10 -10 mbar) through an introduction chamber (base vacuum of 10 -7 mbar). The sample temperature was measured and controlled via an Eurotherm controller thanks to a thermocouple placed close to the sample on the PBN furnace. The samples experienced an isochronal annealing of 4°C.min -1 during mass spectrometry.

Results and discussion

A full desorption spectrum comprising masses up to 45.7 Dalton was first recorded during sample annealing and compared to a same full spectrum recorded in same conditions in the UHV setup without any sample (see Fig. S2 in SI), allowing to select the masses of interest. Fig. 1a presents the TDS measurements of the selected masses performed on the USG-SiH4 layer. CO2 and H2O are regularly detected in vacuum chambers during heating due to the degassing of parts coming from atmosphere, such as the sample holder in our case. They are used as references in Fig. 1a, as their desorption trends are identical with or without sample in the chamber and were found to be similar to all the detected molecules beside H, nitrogen (N) and oxygen (O) related molecules. Indeed, only H1, H2, N2, and O2 molecules show desorption peaks (Fig. 1a) compared to their base line measured in the UHV chamber free of sample (Fig. S2). H and N molecules were expected to leave the Si oxide layers in addition to O2 as they are part of their fabrication process. One can note that H2O desorption shows no peak, it is similar to the H2O desorption base line recorded without sample in the setup. Consequently, the H desorption peak appears to be independent of the H2O molecule dissociation and corresponds to H contained in the bulk of the layer. Besides all the desorption peaks appear at the same temperature, around 300 °C, which is close to USG-SiH4 deposition temperature (275 °C). Same observations were made on the different samples. Thus, further discussions will only focus on molecular H desorption, relative to the present subject of interest. Fig. 1b presents the desorption spectra recorded on both oxide layers USG-SiH4 and USG-TEOS, after subtraction of the chamber background signal and normalization according to the sample surface area size. H is observed to desorb from both layers, but the main desorption peaks occur at different temperatures: T ~ 310 °C from USG-SiH4 and T ~ 360 °C from USG-TEOS. The H content of the USG-SiH4 film is found to be significantly higher. The difference of H content of the two films may be related to their respective growth temperatures, as the USG-SiH4 layer was grown at lower temperature (275 °C). Fig. 2a presents the H2 desorption spectra recorded from the Si3N4 layer compared to that of USG-SiH4. Two desorption peaks are detected around T = 310 °C and T = 375 °C. The H amount leaving this layer is noticeably smaller than from the USG-SiH4 layer (red solid line) but similar to that of the USG-TEOS layer. Furthermore, the desorption spectra recorded if the USG-SiH4 layer is capped with the Si3N4 layer is similar to that of the single Si3N4 layer, meaning that Si3N4 is a weak H source but an efficient H diffusion barrier. Fig. 2b presents the H2 desorption spectra recorded on the SiCN layer and on the USG-SiH4 layer capped with SiCN compared to that of USG-SiH4 alone. H desorption from SiCN is weaker than from USG-TEOS and Si3N4, and thus, SiCN cannot be considered as an efficient source of H. However, despite that more H is found to leave the SiCN/USG-SiH4 sample than from the single SiCN layer, the amount detected in this case being similar to that detected from the Si3N4/USG-SiH4 sample, the SiCN layer can also be considered as an efficient H diffusion barrier. In particular, the thickness of the SiCN layer is more than one order of magnitude smaller than that of the Si3N4 layer. H desorption from H-passivated Si(001) substrates was studied as an attempt to pseudo-quantify the H amount detected in our experiments. The Si substrates were passivated with H by staying 1 minute in a 5% HF solution. In this case, a single monolayer (ML) of H covers the Si surface [START_REF] Ubara | Formation of Si-H bonds on the surface of microcrystalline silicon covered with SiOx by HF treatment[END_REF]. The desorption signal being normalized to the surface area size of the samples, this ML was used as the H amount unit. Table 1 presents the results of the conversion of desorption peak areas to H MLs.

Table 1: TDS peak areas measured on the TDS spectra converted into H ML units using the desorption amount from H-passivated Si(001) substrates as reference * . The error on the ML detection was estimate to be 0.5 ML.

Figure 1 :

 1 Figure 1: TDS measurements performed during 4 °C min -1 ramp annealing of a 940 nmthick layer of USG-SiH4: a) for different masses and b) for H2 mass, compared with USG-TEOS film.

Figure 2 :

 2 Figure 2: (a) H2 desorption spectra recorded on the Si3N4 layer (black line) and the Si3N4/USG-SiH4 sample (blue line) compared to the USG-SiH4 layer desorption (red line), and (b) H2 desorption spectra recorded on the SiCN layer (black line) and the SiCN/USG-SiH4 sample (blue line) compared to the USG-SiH4 layer desorption (red line).

Acknowledgements

This project was supported by the French National Association of Research and Technology (ANRT Project 2020/0907).

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.

Supplementary information