Xuedong Shang
email: xuedong.shang@inria.fr

Emilie Kaufmann
email: emilie.kaufmann@univ-lille1.fr

Michal Valko
email: michal.valko@inria.fr

Adaptive black-box optimization got easier: HCT only needs local smoothness

Keywords: continuously-armed bandits, global optimization, black-box optimization

Hierarchical bandits is an approach for global optimization of extremely irregular functions. This paper provides new elements regarding POO, an adaptive meta-algorithm that does not require the knowledge of local smoothness of the target function. We first highlight the fact that the subroutine algorithm used in POO should have a small regret under the assumption of local smoothness with respect to the chosen partitioning, which is unknown if it is satisfied by the standard subroutine HOO. In this work, we establish such regret guarantee for HCT, which is another hierarchical optimistic optimization algorithm that needs to know the smoothness. This confirms the validity of POO. We show that POO can be used with HCT as a subroutine with a regret upper bound that matches the one of best-known algorithms using the knowledge of smoothness up to a √ log n factor.

Introduction

Global optimization (GO) has applications in several domains including hyper-parameter tuning [START_REF] Jamieson | Non-stochastic best arm identification and hyperparameter optimization[END_REF][START_REF] Li | Hyperband: Banditbased configuration evaluation for hyperparameter optimization[END_REF][START_REF] Samothrakis | Training gradient boosting machines using curvefitting and information-theoretic features for causal direction detection[END_REF]. GO usually consists of a data-driven optimization process over an expensive-to-evaluate function. It is also known as black-box optimization since the inner behavior of a function is often unknown.

In GO, we optimize an unknown and costly-to-evaluate function f : X → R based on n (noisy) evaluations, that can be sequentially selected. This setting is a generalization of multi-armed bandits, where the arm space X is some measurable space [START_REF] Bubeck | X -armed bandits[END_REF]. Each arm x ∈ X gets its mean reward f (x) through the reward function f , which is the function to be optimized. At each round t, the learner chooses an arm x t ∈ X and receives a reward r t . We study the noisy setting in which the obtained reward is a noisy evaluation of f : r t f (x t ) + ε t , where ε t is a bounded noise.

Treating the problem without any further assumption is a mission impossible. A natural solution is to assume a global smoothness of the reward function [START_REF] Agrawal | The continuum-armed bandit problem[END_REF][START_REF] Robert | Nearly tight bounds for the continuum-armed bandit problem[END_REF][START_REF] Kleinberg | Multi-armed bandit problems in metric spaces[END_REF][START_REF] Cope | Regret and convergence bounds for immediate-reward reinforcement learning with continuous action spaces[END_REF][START_REF] Auer | Improved rates for the stochastic continuum-armed bandit problem[END_REF][START_REF] Kleinberg | Bandits and experts in metric spaces[END_REF]. A weaker condition is the local smoothness where only neighborhoods around the maximum are required to be smooth. In fact, local smoothness is sufficient for achieving near-optimality [START_REF] Valko | Stochastic simultaneous optimistic optimization[END_REF][START_REF] Gheshlaghi Azar | Online stochastic optimization under correlated bandit feedback[END_REF][START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF]. Optimistic tree-based optimization algorithms [START_REF] Munos | Optimistic optimization of deterministic functions without the knowledge of its smoothness[END_REF][START_REF] Valko | Stochastic simultaneous optimistic optimization[END_REF][START_REF] Preux | Bandits attack function optimization[END_REF][START_REF] Gheshlaghi Azar | Online stochastic optimization under correlated bandit feedback[END_REF] approach the problem with a hierarchical partitioning of the arm space, and take the optimistic principle. This idea comes from planning in Markov decision processes [START_REF] Kocsis | Bandit-based Monte-Carlo planning[END_REF][START_REF] Munos | From bandits to Monte-Carlo tree search: The optimistic principle applied to optimization and planning[END_REF][START_REF] Grill | Blazing the trails before beating the path: Sampleefficient Monte-Carlo planning[END_REF].

Our work is motivated by POO [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF], which adapts to the smoothness without the knowledge of it. POO is a meta-algorithm on top of any hierarchical optimization algorithm that knows the smoothness, that we call a subroutine. Not only does POO require only the mildest local regularity conditions, it also gets rid of the unnecessary metric assumption that is often required. Local smoothness naturally covers a larger class of functions than global smoothness, yet still assures that the function does not decrease too fast around the maximum. The analysis of POO is modular: Assuming the subroutine has a regret of order R n under a local smoothness assumption with respect to a fixed partitioning [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF], Assumption 1, formally introduced in Section 2), POO run with such subroutine is has a regret bounded by R n √ log n. POO was originally analyzed using HOO as a subroutine. However, unlike what [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF] hypothesize, it is non-trivial to provide a regret bound for HOO under Assumption 1. We elaborate on that in Section 3. In order to validate POO, there needs to exist a subroutine with a regret guarantee that is provable under Assumption 1. This is what we deliver.

In particular, we prove that HCT-iid 1 by Azar et al. ( 2014) satisfies the required regret guarantee, and is, therefore, a desirable subroutine to be plugged in POO. Similar to HOO, HCT is a hierarchical optimization algorithm based on confidence intervals. However, unlike HOO, these confidence intervals are obtained by repeatedly sampling a representative point of each cell in the partitioning before splitting the cell. This yields partition trees that have a controlled depth, which are easier to analyze under a local smoothness assumption with respect to the partitioning. The question whether HOO have similar regret guarantees under the desired assumption remains open.

Smoothness assumptions for black-box optimization

Let X be a measurable space. Our goal is to find the maximum of an unknown noisy function f : X → R of which the cost of function evaluation is high, given a total budget of n evaluations. At each round t, a learner selects a point x t ∈ X and observes a reward r t f (x t ) + ε t , bounded by [0, 1] from the environment where the noise ε t is assumed to be independent from previous observations and such that E[ε t |x t ] 0. After n evaluations, the algorithm outputs a guess for the maximizer, denoted by x(n). We assume that there exists at least one x ∈ X s.t. f (x ) sup x∈X f (x), denoted by f in the following. We measure the performance by the simple regret, also called theoptimization error,

S n f -f (x(n)).
Another related notion is the cumulative regret, defined as

R n nf - n t=1 f (x t ).
1. denoted by HCT in the rest of the paper since we do not consider the correlated feedback setting As observed by [START_REF] Bubeck | Pure exploration in multi-armed bandits problems[END_REF], a good cumulative regret naturally implies a good simple regret: If we recommend x(n) according to the distribution of previous plays, we immediately get

E[S n ] ≤ E[R n ]/n.

Covering tree that guides the optimization

Hierarchical bandits rely on a tree-formed hierarchical partitioning P {P h,i } defined recursively, P 0,1 X , P h,i P h+1,2i-1 ∪ P h+1,2i .

Many of known algorithms depend on a metric/dissimilarity over the search space to define the regularity assumptions that link the partitioning to some near-optimality dimension, that is independent of the partitioning. However, this was shown to be artificial [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF], since (1) the metric is not fully exploited by the algorithms and (2) the notion of near-optimality dimension independent of partitioning is ill-defined. Hence, it is natural to make smoothness assumptions directly related to the partitioning. We now present the only regularity assumption on the target function f that is expressed in terms of the partitioning P. We stress again that requiring only local smoothness assumptions is an improvement since (1) it covers a larger class of functions, (2) it only constrains f along the optimal path of the covering tree which is a plausible property in an optimization scenario, and (3) shows that the optimization is actually easier than it was previously believed.

Assumption 1 (local smoothness w.r.t. P) Let x be a global maximizer and i h be the index of the only cell at depth h that contains x . There exist ν > 0, ρ ∈ (0, 1) s.t.,

∀h ≥ 0, ∀x ∈ P h,i h , f (x) ≥ f -νρ h .
Note that this assumption is the same as the one of [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF]. Multiple maximizers may exist, but this assumption needs to be satisfied only by one of them.

As first observed by [START_REF] Auer | Improved rates for the stochastic continuum-armed bandit problem[END_REF], the difficulty of a GO problem should depend on the size of near-optimal regions, and how fast they shrink. [START_REF] Auer | Improved rates for the stochastic continuum-armed bandit problem[END_REF] use a margin condition that quantifies this difficulty by the volume of near-optimal regions. In this work, we use a similar notion of near-optimality dimension instead, that is directly related to the partitioning.

Definition 1 (near-optimality dimension w.r.t. P) We define the near-optimality dimension of f w.r.t. P as

d(ν, ρ) inf{d ∈ R + : ∃C > 0, ∀h ≥ 0, N h (3νρ h ) ≤ Cρ -d h }, where N h (3νρ h ) 2 is the number of cells P h,i s.t. sup x∈P h,i f (x) ≥ f -3νρ h .
Indeed, N h (3νρ) represents the number of cells that any algorithm needs to sample in order to find the maximum. A smaller d(ν, ρ) implies an easier optimization problem.

HCT under local smoothness w.r.t. P

Analyzing HOO under Assumption 1 is not trivial. A key lemma in the analysis of HOO (Lemma 3 by [START_REF] Bubeck | X -armed bandits[END_REF]) which controls the variance of near-optimal cells is not true under local smoothness assumptions as Assumption 1. Indeed, HOO could induce a very deep covering tree,3 while producing too many nodes that are neither near-optimal nor sub-optimal. The concept of near-optimal and sub-optimal nodes is here characterized by the sub-optimality gap of the node which measures the distance between the local maximum of the node and the global maximum. Intuitively, nodes that are neither near-optimal nor sub-optimal represent the nodes of whom the sub-optimality gap is neither too large nor too small. To control the regret due to these nodes, [START_REF] Bubeck | X -armed bandits[END_REF] use global smoothness (weakly Lipschitz) assumption. Assumption 1 is weaker, only local, and does not offer such comfort. If we want to control the regret due to these nodes without Lemma 3 of [START_REF] Bubeck | X -armed bandits[END_REF], one possible way is to control the depth of the covering tree to ensure that we do not have too many of them. In particular, another algorithm known as HCT implies a controlled depth of the tree which allows it to be analyzed under Assumption 1 as opposed to HOO. We now give a brief description of HCT and the present a new analysis of it.

Description of HCT

The pseudocode of HCT (Algorithm 1) and a detailed snippet (Algorithm 2) describes the process of traversing the covering tree. The algorithm stores a finite subtree T t at each round t which is initialized by T 0 = {(0, 1)}. Each cell is associated with a representative point x h,i and the algorithm keeps track of some statistics regarding this point. One of these statistics is the empirical mean reward µ h,i (t) which is the average on the first T h,i (t) rewards received when querying x h,i . The HCT algorithm also keeps track of an upper confidence bound U -value for the cell (h, i),

U h,i (t) µ h,i (t) + νρ h + c log(1/ δ(t + )) T h,i (t)
, where t + 2 log(t) , δ(t) min{c 1 δ/t, 1/2}, and its corresponding B-value,

B h,i (t) min {U h,i (t), max {B h+1,2i-1 (t), B h+1,2i (t)}} if (h, i) is a leaf U h,i (t) otherwise,
which is designed to be a tighter upper confidence bound than the U -value. Here, c and c 1 are two constants, and νρ h represents the resolution 4 of the region P h,i . At each round t, the algorithm traverses the current covering tree along an optimistic path P t before choosing a point. This optimistic path P t is obtained by repeatedly selecting cells that have a larger B-value until a leaf or a node that is sampled less than a certain number of times is reached. If a leaf is reached, then this leaf is sampled and expanded. Otherwise, the node that is not sampled enough is re-sampled. More precisely, HCT samples one node a certain number of times τ h (t) in order to sufficiently reduce the uncertainty Algorithm 1: High confidence tree (HCT, Azar et al. 2014)

Input : ν > 0 ρ ∈ (0, 1), c > 0, tree partition {P h,i }, confidence δ Initialize: T 1 ← {(0, 1), (1, 1), (1, 2)}, H(1) ← 1, U 1,1 (1) ← U 1,2 (1) ← +∞ 1 for t ← 1 to n do 2 if t = t + then 3 for (h, i) ∈ T t do 4 U h,i (t) ← µ h,i (t) + νρ h + c log(1/ δ(t + )) T h,i (t) 5 end 6 for (h, i) ∈ T t backward from H(t) do 7 if (h, i) is a leaf then 8 B h,i (t) ← U h,i (t) 9 else 10 B h,i (t) ← min{U h,i (t), max{B h+1,2i-1 (t), B h+1,2i (
τ h (t) c 2 log(1/ δ(t + )) ν 2 ρ -2h .

Analysis of HCT under a local metricless assumption

We now state our main theorem. We prove that HCT achieves an expected regret bound under Assumption 1 which matches the regret bound given by Azar et al. ( 2014) up to constants. Moreover, compared to that result, the near-optimality dimension d featured in

Algorithm 2: OptTraverse Input : T t Initialize: (h, i) ← (0, 1); P ← {(0, 1)}; T 0,1 (t) = τ 0 (t) = 1 1 while (h, i) is not a leaf and T h,i (t) ≥ τ h (t) do 2 if B h+1,2i-1 (t) ≥ B h+1,2i (t) then 3 (h, i) ← (h + 1, 2i -1) 4 else 5 (h, i) ← (h + 1, 2i) 6 end 7 P ← P ∪ {(h, i)} 8 end 9 return (h, i) and P
Theorem 2 is the one of Definition 1 that is defined with respect to the partitioning and not with respect to a metric.

Theorem 2 Assume that function f satisfies Assumption 1. Then, setting δ 1/n, the cumulative regret of HCT after n rounds is bounded as

E[R n ] ≤ O (log n) 1/(d+2) n (d+1)/(d+2) .
Then, by applying simply the recommendation strategy that follows the distribution of previous plays, we get the following simple regret bound.

Corollary 3 The simple regret of HCT is bounded as

E[S n ] ≤ O (log n) 1/(d+2) n -1/(d+2) .
We now sketch the proof. The full proof follows the analysis of [START_REF] Gheshlaghi Azar | Online stochastic optimization under correlated bandit feedback[END_REF] and is detailed in Appendix A. As mentioned above, HCT has a controlled depth. Indeed, given the threshold τ h (t) required at depth h, in Appendix A.2, we prove that the depth of the covering tree is bounded as stated in the following lemma.

Lemma 4 The depth of the covering tree produced by HCT after n rounds is bounded as

H(n) ≤ H max (n) 1 2(1 -ρ) log nν 2 c 2 ρ 2 •
Next, we introduce a favorable event under which the mean reward of all expanded nodes is within a confidence interval,

ξ t ∀(h, i) ∈ L t , ∀T h,i (t) = 1 . . . t : | µ h,i (t) -µ h,i | ≤ c log(1/ δ(t))/T h,i (t) ,
where L t is the set of all possible nodes in trees of maximum depth H max (t).

We split the regret into two parts depending on whether ξ t holds or not. In Appendix A.4, we prove that the failing confidence term is bounded by √ n with high probability. In the case when ξ t holds, we bound the regret in Appendix A.5 by treating separately the two parts, ∆ ht,it and ∆ t , of the instantaneous regret ∆ t ,

∆ t f -r t = f -f (x ht,it ) + f (x ht,it ) -r t = ∆ ht,it + ∆ t .
Next, we bound ∆ t by Azuma-Hoeffding concentration inequality [START_REF] Azuma | Weighted sums of certain dependent random variables[END_REF]. Then, we bound ∆ ht,it with the help of the following lemma, which is the major difference compared to the original HCT analysis by [START_REF] Gheshlaghi Azar | Online stochastic optimization under correlated bandit feedback[END_REF]. In particular, the lemma states that if Assumption 1 is verified then f is upper-bounded by the U -value of any optimal node.

Lemma 5 Under Assumption 1 and under event ξ t , we have that for any optimal node (h , i ), U h ,i (t) is an upper bound on f .

Proof Since t + ≥ t, we have

U h ,i (t) µ h ,i (t) + νρ h + c log(1/ δ(t + )) T h ,i (t) ≥ µ h ,i (t) + νρ h + c log(1/ δ(t)) T h ,i (t) •
Moreover, as we are under event ξ t , we also have

µ h ,i (t) + c log(1/ δ(t)) T h ,i (t) ≥ f (x h ,i ). Therefore, U h ,i (t) ≥ f (x h ,i ) + νρ h ≥ f .
With the help of Lemma 5, we upper bound ∆ ht,it as

∆ ht,it ≤ 3c log(2/ δ(t)) T ht,it (t) •
To bound the total regret of the all nodes selected, we divide them into two categories, depending on whether their depth is smaller or equal than H (to be optimized later) or not.

For the nodes in depths h ≤ H, we use Lemma 5 again, now to show that OptTraverse only selects nodes that have a parent which is (3νρ ht-1 )-optimal. For the nodes for which h > H, we bound the regret using the selection rule of HCT.

The sums of the regrets from the two categories are proportional and inversely proportional to an increasing function of H. By finding the value of H for which the sum of the two terms reaches its minimum and adding the regret coming from the situations where the favorable event does not hold, gives us the following cumulative regret of HCT with probability

1 -δ, R n ≤ O (log(n/δ)) 1/(d+2) n (d+1)/(d+2) .
However, the analysis of POO requires the expected bound of the underlying subroutine. For that purpose, we simply set δ 1/n and that gives us the statement of Theorem 2.

POO (HCT) that does not need to know the smoothness

In this section, we present a new instantiation of POO, more precisely POO (HCT), which is POO using HCT as a subroutine. POO (parallel optimistic optimization) is a meta-algorithm that uses any hierarchical optimization algorithm that knows the smoothness as a subroutine. In POO (HCT), several instances of HCT are run in parallel. Each instance of HCT is run with a different pair of parameters (ν, ρ) in a well-chosen grid denoted by G. In the end, POO (HCT) chooses the instance that performs the best. In particular, POO(HCT) first selects the instance that has the largest empirical mean reward and then returns one of the points evaluated by this instance, chosen uniformly at random.

Modularization of POO (HCT) Let (ν , ρ ) be the optimal pair of parameters for POO, a pair for which Assumption 1 is true, and let ρ > ρ and ν > ν be the instance in G such that d(ν, ρ) is the closest to d(ν , ρ )-see its formal definition in Appendix B.1 of [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF]. Since the regret bound obtained in the previous section matches the one of HOO up to constants, we can directly plug the simple regret bound into the second step of POO's analysis (Appendix B.2 of [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF]) and obtain the following bound for the simple regret S ν,ρ n of the HCT instance run with ρ,

S ν,ρ n ≤ α((log 2 n)/n) 1/(d(ν,ρ)+2) ,
where α is a constant depending on ν . This bound matches the regret bound of HCT up to a √ log n factor. The empirically best instance that is picked by POO may not always be the instance that is run with this ρ. However, as shown by [START_REF] Grill | Black-box optimization of noisy functions with unknown smoothness[END_REF], the error is negligible w.r.t. the simple regret. We stress that the last statement is true because the underlying subroutine has a cumulative regret guarantee and POO chooses its best instance according to the empirical mean reward. Fortunately, this happens to be the case for HCT.

Discussion

We studied POO (HCT), a new instantiation of POO on top of HCT. We proved that HCT is a plausible subroutine for POO by adapting the analysis of HCT under a new assumption w.r.t. a fixed partitioning. However, whether it is possible to weaken the assumptions of HOO in the same way while keeping similar regret guarantees remains open.

A.2 Maximum depth of the tree (proof of Lemma 4) Lemma 4 The depth of the covering tree produced by HCT after n rounds is bounded as

H(n) ≤ H max (n) 1 2(1 -ρ) log nν 2 c 2 ρ 2 •
Proof The deepest tree that can be constructed by HCT is a linear one, where at each level one unique node is expanded. In such case,

|I + h (n)| = 1 and |I h (n)| = 2 for all h < H(n). Therefore, we have n = H(n) h=0 i∈I h (n) T h,i (n) ≥ H(n)-1 h=0 i∈I + h (n) T h,i (n) ≥ H(n)-1 h=0 i∈I + h (n) T h,i (t h,i ) ≥ H(n)-1 h=0 i∈I + h (n) τ h (t h,i ) definition of t h,i ≥ H(n)-1 h=0 c 2 ν 2 ρ -2h
Eq. 1

≥ (cρ) 2 ν 2 ρ -2H(n) H(n) since h ≤ H(n) -1 ≥ (cρ) 2 ν 2 ρ -2H(n) .
By solving this expression, we obtain

H(n) ≤ 1 2 log nν 2 c 2 ρ 2 / log(1/ρ) ≤ 1 2(1 -ρ) log nν 2 c 2 ρ 2 follows from log(1/ρ) ≥ 1 -ρ ≤ 1 2(1 -ρ) log nν 2 c 2 ρ 2 H max (n).

A.3 High-probability event

In Section 3.2, we described the favorable event ξ t . We now define it precisely. We first define a set L t that contains all possible nodes in trees of maximum depth H max (t),

L t T :depth(T )≤Hmax(t)
Nodes(T )

and we introduce the favorable event

ξ t    ∀(h, i) ∈ L t , ∀T h,i (t) = 1 . . . t : | µ h,i (t) -µ h,i | ≤ c log(1/ δ(t)) T h,i (t)    •
Next, we prove that our favorable event holds with high probability.

Lemma 6 With c 1 and c defined in Section A.1, for any fixed round t,

P [ξ t ] ≥ 1 - 4δ 3t 6 •
Proof We upper-bound the probability of the complementary event ξ c t as

P [ξ c t ] ≤ (h,i)∈Lt t T h,i (t)=1 P   | µ h,i (t) -µ h,i | ≥ c log(1/ δ(t)) T h,i (t)   union bound ≤ (h,i)∈Lt t T h,i (t)=1 2 exp -2c 2 log(1/ δ(t)) Chernoff-Hoeffding inequality = 2 exp -2c 2 log(1/ δ(t)) t|L t | = 2( δ(t)) 2c 2 t|L t | ≤ 2( δ(t)) 2c 2 t2 Hmax(t)+1 = 2( δ(t)) 2c 2 t2 1 2(1-ρ) log nν 2 c 2 ρ 2 +1 Lemma 4 ≤ 8t( δ(t)) 2c 2 tν 2 c 2 ρ 2 1 2(1-ρ) ≤ 8t δ t (ρ/(3ν)) 1/8 ) 8 1-ρ tν 2 (1 -ρ) 4ρ 2 1 2(1-ρ) plugging in values of c and c 1 = 8t δ t 8 1-ρ ρ 3ν 1 1-ρ t 1 2(1-ρ) ν √ 1 -ρ 2ρ 1 1-ρ ≤ 4 3 δt -2ρ-13 2(1-ρ) ≤ 4δ 3t 6 • A.4 Failing confidence bound
We decompose the regret of HCT into two terms depending on whether ξ t holds. Let us define ∆ t f -r t . Then, we decompose the regret as

R n = n t=1 ∆ t = n t=1 ∆ t 1 ξt + n t=1 ∆ t 1 ξ c t = R ξ n + R ξ c n .
The failing confidence term R ξ c n is bounded by the following lemma.

Lemma 7 With c 1 and c defined in Section A.1, when the favorable event does not hold, the regret of HCT is with probability 1 -δ/(5n 2 ) bounded as

R ξ c n ≤ √ n.
Proof We split the term into rounds from 1 to √ n and the rest,

R ξ c n = n t=1 ∆ t 1 ξ c t = √ n t=1 ∆ t 1 ξ c t + n t= √ n+1 ∆ t 1 ξ c t .
The first term can be bounded trivially by √ n since |∆ t | ≤ 1. Next, we show that the probability that the second term is non zero is bounded by δ/(5n 2 ).

P   n t= √ n+1 ∆ t 1 ξ c t > 0   = P   n t= √ n+1 ξ c t   ≤ n t= √ n+1 P [ξ c t ] union bound ≤ n t= √ n+1 δ t 6 Lemma 6 ≤ ∞ √ n δ t 6 dt = δ 5n 5/2 ≤ δ 5n 2 • A.5 Proof of Theorem 2
Theorem 2 Assume that function f satisfies Assumption 1. Then, setting δ 1/n, the cumulative regret of HCT after n rounds is bounded as d+2) . and the same holds for the parent of the selected node,

E[R n ] ≤ O (log n) 1/(d+2) n (d+1)/(
U h p t ,i p t (t) ≤ f (x h p t ,i p t ) + νρ h p t + 2c log(1/ δ(t + )) T h p t ,i p t (t)
• By Lemma 5, we know that U h ,i (t) is a valid upper bound on f . If an optimal node (h , i ) is a leaf, then B h ,i (t) = U h ,i (t) is also a valid upper bound on f . Otherwise, there always exists a leaf which contains the maximum for which (h , i ) is its ancestor. Now, if we propagate the bound backward from this leaf to (h , i ) through Eq. 4, we have that B h ,i (t) is still a valid upper bound on f . Thus for any optimal node (h , i ), at round t under ξ t , we have

B h ,i (t) ≥ f . (8)
We combine Eq. 8 with Eq. 5, and Eq. 7 to obtain

∆ ht,it f -f (x ht,it ) ≤ νρ ht + 2c log(1/ δ(t + )) T ht,it (t) •
The same result holds for its parent,

∆ h p t ,i p t f -f (x h p t ,i p t ) ≤ νρ h p t + 2c log(1/ δ(t + )) T h p t ,i p t (t)
•

We now refine the two above expressions. The subroutine OptTraverse tells us that HCT only selects a node when T h,i (t) < τ h (t). Therefore, by definition of τ ht (t), we have

∆ ht,it ≤ 3c log(2/ δ(t)) T ht,it (t) • (9)
On the other hand, OptTraverse tells us that

T h p t ,i p t (t) ≥ τ h p t (t), thus ∆ h p t ,i p t ≤ 3νρ h p t ,
which means that every selected node has a parent which is (3νρ ht-1 )-optimal.

Step 3: Bound on the cumulative regret. We return to term R ξ n and split it into different depths. Let 1 ≤ H ≤ H(n) be a constant that we fix later. We have

R ξ n n t=1 ∆ ht,it 1 ξt ≤ H(n) h=0 i∈I h (n) n t=1 ∆ h,i 1 (ht,it)=(h,i) 1 ξt ≤ H(n) h=0 i∈I h (n) n t=1 3c log(2/ δ(t)) T h,i (t) 1 (ht,it)=(h,i)
Eq. 9

= H h=0 i∈I h (n) n t=1 3c log(2/ δ(t)) T h,i (t) 1 (ht,it)=(h,i) + H(n) h=H+1 i∈I h (n) n t=1 3c log(2/ δ(t)) T h,i (t) 1 (ht,it)=(h,i) ≤ H h=0 i∈I h (n) τ h (t h,i ) s=1 3c log(2/ δ(t h,i )) s + H(n) h=H+1 i∈I h (n) T h,i (n) s=1 3c log(2/ δ(t h,i )) s ≤ H h=0 i∈I h (n) τ h (t h,i ) 1 3c log(2/ δ(t h,i )) s ds + H(n) h=H+1 i∈I h (n) T h,i (n) 1 3c log(2/ δ(t h,i )) s ds ≤ H h=0 i∈I h (n) 6c τ h (t h,i ) log(2/ δ(t h,i )) + H(n) h=H+1 i∈I h (n) 6c T h,i (n) log(2/ δ(t h,i )) = 6c        H h=0 i∈I h (n) τ h (t h,i ) log(2/ δ(t h,i )) (a) + H(n) h=H+1 i∈I h (n) T h,i (n) log(2/ δ(t h,i )) (b)        .
We bound separately the terms (a) and (b). Since t h,i ≤ n, we have

(a) ≤ H h=0 i∈I h (n) τ h (n) log(2/ δ(n)) ≤ H h=0 |I h (n)| τ h (n) log(2/ δ(n)).
Notice that the covering tree is binary and therefore

|I h (n)| ≤ 2|I h-1 (n)|.
Recall that HCT only selects a node (h t , i t ) when its parent is 3νρ ht-1 -optimal. Therefore, by definition of the near-optimality dimension,

|I h (n)| ≤ |2I h-1 (n)| ≤ 2Cρ -d(h-1) ,
where d is the near-optimality dimension. As a result, for term (a), we obtain that d+1) .

(a) ≤ H h=0 2Cρ -d(h-1) τ h (n) log(2/ δ(n)) = H h=0 2Cρ -d(h-1) c 2 log(2/ δ(n)) ν 2 ρ -2h log(2/ δ(n)) Eq. 1 = 2Cρ d c log(2/ δ(n)) ν H h=0 ρ -h(
Consequently, we bound (a) as

(a) ≤ 2Cρ d c log 2/ δ(n) ν ρ -H(d+1) 1 -ρ • (10)
We proceed to bound the second term (b). By the Cauchy-Schwartz inequality,

(b) ≤ H(n) h=H+1 i∈I h (n) log 2/ δ t h,i H(n) h=H+1 i∈I h (n) T h,i (n) ≤ n H(n) h=H+1 i∈I h (n) log 2/ δ t h,i ,
where we trivially bound the second square-root factor by the total number of pulls. Now consider the first square-root factor. Recall that the HCT algorithm only selects a node when T h,i (t) ≥ τ h (t) for its parent. We therefore have T h,i ( t h,i ) ≥ τ h ( t h,i ) and the following sequence of inequalities, In the above derivation, the last equality relies on the fact that for any h > 0, I + h (n) covers all the internal nodes at level h and therefore its children cover I h+1 (n). We thus obtain

n = H(n) h=0 i∈I h (n) T h,i (n) ≥ H(n)-1 h=0 i∈I + h (n) T h,i (n) ≥ H(n)-1 h=0 i∈I + h (n) T h,i ( t h,i ) t h,i well defined for i ∈ I + h (n) ≥ H(n)-1 h=0 i∈I + h (n) τ h ( t h,i ) ≥ H(n)-1 h=H i∈I + h (n) τ h ( t h,i ) = H(n)-1 h=H i∈I + h (n) c 2 log(1/ δ( t + h,i ))) ν 2 ρ -2h H(n)-1 h=H i∈I + h (n) c 2 log(1/ δ( t + h,i ))) ν 2 ρ -2h ≥ H(n)-1 h=H i∈I + h (n) c 2 log(1/ δ( t + h,i ))) ν 2 ρ -2H = c 2 ρ -2H ν 2 H(n)-1 h=H i∈I + h (n) log(1/ δ( t + h,i ))) = c 2 ρ -2H ν 2 H(n)-1 h=H i∈I + h (n)

  log(1/ δ( max(t h+1,2i-1 , t h+1,2i ) + )) since t h,i = max(t h+1,2i-1 , t h+1,2i )

	=	c 2 ρ -2H ν 2	H(n)-1 h=H i∈I + h (n)	log(1/ δ(max(t + h+1,2i-1 , t + h+1,2i )))	∀t 1 , t 2 , [max(t 1 , t 2 )] + = max(t + 1 , t + 2 )
	=	c 2 ρ -2H ν 2	H(n)-1 h=H i∈I + h (n)	max(log(1/ δ(t + h+1,2i-1 )), log(1/ δ(t + h+1,2i )))
		≥	c 2 ρ -2H ν 2	H(n)-1 h=H i∈I + h (n)	log(1/ δ(t + h+1,2i-1 )) + log(1/ δ(t + h+1,2i )) 2
	=	c 2 ρ -2H ν 2	H(n) h=H+1 i∈I + h-1 (n)	log(1/ δ(t + h,2i-1 )) + log(1/ δ(t 2	+ h,2i ))	change of variables
	=	c 2 ρ -2H 2ν 2	H(n) h=H+1 i∈I + h (n)	log(1/ δ(t

+ h,i )).

This definition is slightly different from the original POO paper, where a coefficient 3 is present instead of 2 due to a technical detail.

This can, however, bring good empirical performance in hyper-parameter tuning.

The term resolution refers to the maximum variance of the cell. If it is too large, then we need to shrink the volume, thus increase the resolution.

Acknowledgments

The research presented was supported by European CHIST-ERA project DELTA, French Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council, a doctoral grant of École Normale Supérieure in Rennes, Inria and Otto-von-Guericke-Universität Magdeburg associated-team north-European project Allocate, and French National Research Agency projects ExTra-Learn (n.ANR-14-CE24-0010-01), BADASS (n.ANR-16-CE40-0002), and BoB (n.ANR-16-CE23-0003).

Appendix A. Detailed regret analysis

A.1 Preliminaries

Before starting the proof, we first fix some constants and introduce some additional notation required for the analysis in Section 3.

• c 1 (ρ/(3ν)) 1/8 , c 2 1/(1 -ρ)

• ∀1 ≤ h ≤ H(t) and t > 0, I h (t) denotes the set of all nodes created by HCT at level h up to step t

• ∀1 ≤ h ≤ H(t) and t > 0, I + h (t) denotes the subset of I h (t) which contains only the internal nodes

• At each step t, (h t , i t ) denotes the node selected by the algorithm.

• t h,i max t∈C h,i t denotes the last time (h, i) has been selected

t denotes the last time when one of its children has been selected

• For any t, let y t (r t , x t ) be a random variable, we define the filtration F t as a σ-algebra generated by (y 1 , . . . , y t ).

Another important notion in HCT is the threshold τ h on the number of pulls needed before a node at level h can be expanded. The threshold τ h is chosen such that the two confidence terms in U h,i are roughly equivalent, that is,

Since t + is defined as 2 log(t) , we have t ≤ t + ≤ 2t. In addition, log is an increasing function, thus we have

where the first inequality follows from the fact that 0 < δ(t) ≤ 1/2. We begin our analysis by bounding the maximum depth of the trees constructed by HCT.

We study the regret under events {ξ t } t and prove that

holds with probability 1 -δ. We decompose the proof into 3 steps.

Step 1: Decomposition of the regret. We start by further decomposing the instantaneous regret into two terms,

The regret of HCT when confidence intervals hold can thus be rewritten as

We notice that the sequence { ∆ t } n t=1 is a bounded martingale difference sequence since E ∆ t |F t-1 = 0 and | ∆ t | ≤ 1. Thus, we apply the Azuma's inequality on this sequence and obtain

with probability 1 -δ/(4n 2 ).

Step 2: Preliminary bound on the regret of selected nodes and their parents. Now we proceed with the bound of the first term R ξ n . Recall that P t is the optimistic path traversed by HCT at round t. Let (h , i ) ∈ P t and (h , i ) be the node which immediately follows (h , i ) in P t . By definition of B-values and U -values, we have

where the last equality follows from the fact that the subroutine OptTraverse selects the node with the largest B-value. By iterating the previous inequality along the path P t until the selected node (h t , i t ) and its parent (h p t , i p t ), we obtain

Since the root, which is an optimal node, is in P t , there exists at least one optimal node (h , i ) in path P t . As a result, we have

We now expand Eq. 5 on both sides under ξ t . First, we have

By plugging Eq. 11 into above expression, we get

Now if we combine Eq. 12 with Eq. 10, we bound R ξ n as

We choose H to minimize the above bound by equalizing the two terms in the sum and obtain

which after being plugged into Eq. 13 gives

Finally, combining Eq. 15, Eq. 3, and Lemma 7, we obtain with probability 1 -δ.


L Courtade

Ch Turquat

Ch Muller

J G Lisoni

L Goux

D J Wouters

D Goguenheim

P Roussel

L Ortega

Oxidation kinetics of Ni metallic films: formation of NiO-based resistive switching structures

Keywords: oxidation kinetics, resistive switching, bi-stable oxide films, microstructural analysis PACS number(s): 72.80.Ga, 61.10.Nz, 81.65.Mq, 68.55Ac, 68.55.Jk, 68.37.Lp, 73.40.Rw, 84.37.+q, 85.90.+h

Resistive switching controlled by external voltage has been reported in many Metal/Resistive oxide/Metal (MRM) structures in which the resistive oxide was simple transition metal oxide thin films such as NiO or TiO 2 deposited by reactive sputtering. In this paper, we have explored the possibility to form NiO-based MRM structures from the partial oxidation of a blanket Ni metallic film using a Rapid Thermal Annealing route, the remaining Ni layer being used as bottom electrode. X-ray diffraction was used to apprehend the Ni oxidation kinetics while transmission electron microscopy enabled investigating local microstructure and film interfaces. These analyses have especially emphasized the predominant role of the asdeposited Ni metallic film microstructure (size and orientation of crystallites) on (i) oxidation kinetics, (ii) NiO film microstructural characteristics (crystallite size, texture and interface roughness) and (iii) subsequent electrical behavior. On this latter point, the as-grown NiO films were initially in the low resistance ON state without the electro-forming step usually required for sputtered films. Above the threshold voltage varying from 2 to 5 V depending on oxidation conditions, the Pt/NiO/Ni MRM structures irreversibly switched into the high resistance OFF state. This irreversibility is thought to originate in the microstructure of the NiO films that would cause the difficulty to re-form conductive paths.

Introduction

Resistive switching phenomena controlled by external currents or voltages attract a lot of attention for future high-density non-volatile memory devices. The resistive switching-based memory concept enables the integration of the memory element in contacts and interconnects resulting in very small memory cells. Additionally, this concept makes possible a 3-dimensional stacking as recently demonstrated by Baek et al. on a multi-layer cross point structured binary oxide resistive random access memory (so-called OxRRAM) for high-density data storage applications [1].

Resistive memory materials range from organic (e.g. copper tetracyano quinodimethane) to inorganic materials (e.g. chalcogenide alloys, perovskite-type oxides or transition metal oxides) with e ither metallic or conductive oxide electrodes [2,3]. For inorganic materials, bi-stable switching phenomena have been reported in many simple metal oxides such as Nb 2 O 5 , Al 2 O 3 , Ta 2 O 5 , TiO 2 , NiO or ZrO 2 [4][5][6][START_REF] Courtade | IEEE Proceedings of the Non Volatile Memory Technology Symposium[END_REF]. Typical current-voltage I(V) characteristic of Metal/Resistive oxide/Metal (MRM) structures exhibits a drastic change in resistance between a high resistance state (i.e. OFF state) and a low resistance state (i.e. ON state). Among transition metal oxides, NiO is a promising material due to its compatibility with standard complementary metal-oxidesemiconductor process, its high ON/OFF ratio and simple constituents [4,[START_REF] Seo | [END_REF]. Resistance switching in crystalline nickel oxide films was observed in 1964 by Gibbons et al. [4] and the switching mechanism was explained by a reversible formation/rupture of filamentary conductive paths from the interfaces throughout the NiO film [4,[START_REF] Inoue | IEEE Proceedings of the Non Volatile Memory Technology Symposium[END_REF][START_REF] Kim | [END_REF].

In literature, most of the polycrystalline NiO films are deposited by dc (direct current) reactive sputtering on conductive substrates to fabricate MRM structures [START_REF] Kim | [END_REF]11]. In this paper, an alternative way was explored using a rapid thermal annealing (RTA) route. Hence, NiO-films were produced from the partial oxidation of a blanket Ni metallic film used as bottom electrode, this approach being similar to the one proposed by Chen et al. to form bi-stable CuO x /Cu stacks [12].

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Several process parameters, such as oxidation time, annealing temperature and oxygen partial pressure, were tested to control the growth of NiO films. In addition to the electrical testing, x-ray diffraction experiments enabled apprehending the Ni oxidation kinetics whereas cross-sectional transmission electron microscopy (TEM) was used to observe both the stack microstructure and films interfaces.

Experimental details

Ni metallic films with 24 or 100 nm thickness were deposited at room temperature by dc physical vapor deposition (Applied Materials sputtering tool) on SiO 2 /Si 3 N 4 /Si substrates. The film deposition was performed in a base pressure of few mTorr with a dc power source of 400 or 2 000 W for thinner and thicker films respectively (corresponding deposition rates: 10 and 55 nm/min).

Different RTA treatments were evaluated to oxidize the Ni films: several process parameters such as oxidation time (from 10 seconds to 30 minutes), annealing temperature (200, 300 and 400°C) and oxygen partial pressure (pure O 2 ; 20 and 500 ppm O 2 ) were tested. After the Ni oxidation, a 100 nm thick Pt top electrode (TE) was sputtered through a shadow mask with feature sizes ranging from 1 000 to 160 µm in order to test the electrical behavior of elementary MRM structures.

Local TEM observations coupled with energy dispersive spectroscopy (EDS) analyses were carried out to investigate the microstructure of Ni and NiO films. TEM samples were prepared by focused ion beam (FIB) technique (Ga + ions accelerated at 30 kV; a low current ion beam was used to clean the surface at the end of the preparation) and imaging was performed using a Tecnai G 2 (FEI Company) at 200 kV (wavelength of 0.0251 Å). EDS analyses were performed with an EDAX instrument using a Si(Li) detector with a thin beryllium window enabling analysis of elements with Z > 11.

X-ray texture analyses were performed on a Siefert 4-circles diffractometer in the Schulz reflection geometry. Copper radiation emitted from an x-ray tube was monochromatized (λ Kα (Cu)

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--5 = 1.5418 Å) and collimated using a bi-dimensional multilayer optical system from Xenocs.

Stereographic pole figures were measured by scanning the tilt angle χ between 0 and 90° in steps of 3°, and the azimuthal angle Φ between 0 and 360° in steps of 3°. The pole figures are the projection of the diffraction intensity of a given Bragg reflection as the sample is rotated. The experimental data were corrected for background. X-ray diffraction technique was used to apprehend the oxidation kinetics of Ni metallic films in different experimental conditions. After each thermal treatment, ex situ analyses were performed to check the presence of Ni and NiO phases. Diffraction patterns were registered at room temperature on a Siemens-Bruker D5000 diffractometer configured in Bragg-Brentano geometry, operating with CuK α radiation and equipped with a diffracted-beam monochromator and Soller slits. Data were collected in the angular range 36 -46° in 2θ with steps of 0.02° to measure the intensities of (111) Bragg reflections of Ni and NiO phases. Using a profile fitting procedure described elsewhere [13], a peak broadening analysis was performed to extract average size of Ni and NiO crystallites (experimental data were corrected for instrumental broadening).

Besides, in situ x-ray diffraction experiments were performed on a Bruker D8 Advance x-ray diffractometer equipped with an Anton-Paar HTK 1200N furnace enabling temperature-dependent measurements under controlled atmosphere. The experimental conditions were as follows: after heating under vacuum from 25°C to the nominal temperature with heating rate of 18°C/min, an isothermal treatment at 400 or 500°C was applied in pure oxygen. Data were collected every 10 seconds using a linear detector (Bruker Vantec 1) over the same angular domain 36 -46° in 2θ with steps of 0.00741°. A peak broadening analysis was also performed to apprehend the time and temperature-dependent evolution of Ni and NiO crystallite sizes.

Current-Voltage I(V) characteristics were measured on MRM structures with AixACCT TF Analyzer 2000 system using either triangular waveform (typically ± 5 V and 10 Hz; corresponding voltage sweep: 0,2 V/msec) or staircase voltage ramp (voltage step: 0.1 V; dwell time: 1 sec per step). From these measurements, the bi-stable resistive behavior of NiO films was checked and the

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--6 switching threshold voltage V th was determined (when a switching occurred). The MRM structures being initially in ON state, the threshold voltage corresponded to the switching from low to high resistance state. Moreover, low (R ON ) and high (R OFF ) resistances were deduced from the slope of I(V) characteristics. Values of V th , R ON and R OFF were then correlated with process parameters.

Microstructure and growth kinetics of NiO films

Microstructural analysis of Ni and NiO films

Preliminary θ-2θ diffraction patterns were collected on 24 and 100 nm thick Ni layers before the oxidation step. As seen in Fig. 1a,x TEM experiments were performed on stacks obtained in various oxidation conditions. For instance, Fig. 2a and 2b presents the results of TEM experiments performed on sample coated with a 100 nm thick Ni layer, annealed in pure O 2 at 400°C for 30 seconds and covered with a Pt top electrode. A cross-sectional observation shows a stack of several layers with different contrasts (Fig. 2a). In situ x-ray energy dispersive spectrometry (EDS) analysis enabled checking the chemical composition of each film and evidenced a thick Ni-rich layer above Si-rich layer, the upper top layer being Pt. Unfortunately, the EDS detector used was not sensitive to light elements such as oxygen. Consequently, the presence of NiO layer cannot be clearly established from EDS analyses.

To prove the presence of NiO layer, selected area electron diffraction experiments were performed using the smallest available aperture. Although this aperture was still too large to select only one layer, the presence of NiO was confirmed. Indeed, some NiO d -spacings (Powder Diffraction File, PDF no. 47-1049) such as d (111) [NiO] differ enough from the d-spacings of the surrounding layers to unambiguously identify the nickel oxide. Using a similar approach with Ni dspacings (PDF no. 04-0850), the presence of Ni layer was checked. Furthermore, dark field imaging experiments were performed using the diffraction spot corresponding to either d (111) [NiO] = 2.412 Å or d (200) [Ni] = 1.762 Å (Fig. 2b). These experiments have then definitely confirmed the presence of NiO layer and have enabled locating both Ni and NiO layers within the stack (Fig. 2a).

As shown on TEM cross-section in Fig. 2a, there is a large spread of NiO thicknesses typically ranging from 40 to 75 nm. The Ni layer is dense and constituted by columnar grains with an average size up to 350 nm parallel to the surface. Micro-diffraction experiments (not shown here) suggested that the Ni layer is preferentially oriented with the [111] crystallographic direction

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--8 parallel to the substrate's normal, in agreement with the previous x-ray texture analyses. Besides, the Ni/SiO 2 interface is flat and clear of defects whereas Pt/NiO and NiO/Ni interfaces are rough and enclose some voids. The NiO layer is constituted of smaller grains as compared to Ni with a maximum size 120 nm.

Oxidation kinetics of Ni metallic layer

Ex situ x-ray diffraction to apprehend RTA oxidation

A detailed analysis of NiO/Ni stack formation was performed from x-ray diffraction patterns collected on a 100 nm thick Ni film after RTA oxidation at 400°C in pure O 2 with annealing times ranging from 10 seconds to 30 minutes. In Fig. 3a, concomitant diffracted intensity variations of both Ni (decrease) and NiO (increase) phases are observed with increasing oxidation times. This feature indicates that the Ni metallic film is progressively consumed as the NiO film grows. In addition, the peak profiles were individually adjusted using the program Winplotr [14] to extract the time-dependent evolution of integrated intensity and full width at half maximum (FWHM) of both (111) NiO and (111) Ni reflections. The time-dependent variation of the intensities indicates that the initial extremely rapid oxidation is followed by a slower oxidation rate (not shown here). In addition, the integral breadth β (linked to FWHM) of the (111) NiO Bragg peak decreases with increasing oxidation time (Fig. 3b). The integral breadth β may be interpreted in terms of crystallite size using the Scherrer formulae β = λ / (D × cos θ), where D is the apparent crystallite size (in the present case, the local strain effect was not taken into account). Consequently, the decrease of β is interpreted as a growth of NiO crystallites along with oxidation times (Fig. 3b).

Oxidation kinetics from in situ time-dependent x-ray diffraction

To complement previous ex situ analyses, in situ temperature and time-dependent x -ray diffraction experiments were performed on 100 nm thick Ni films annealed at 400 or 500°C in pure ACCEPTED MANUSCRIPT --9 O 2 . Before the oxidation step, the Ni metallic films were heated under vacuum in the furnace mounted on the diffractometer and x-ray diffraction patterns were collected from 25°C to 400 or 500°C. As shown in Fig. 4a, the temperature-dependent evolution of the (111) Ni Bragg reflection exhibits a drastic intensity increase. Using an individual profile fitting procedure, the integrated intensity (Fig. 4b) and integral breadth β (Fig. 4c, left) were extracted. Both evolutions indicate a further crystallization of Ni layer during heating and the decrease of β above 100°C may be interpreted as an increase of the average Ni crystallite size (Fig. 4c, right). The same tendency was observed for sample heated up to 500°C. In a second step, time-dependent x-ray diffraction patterns were collected in pure O 2 at constant temperature for 15 hours. The scanned angular range was limited to 36-46° in 2θ, domain that contains the (111) Ni and (111) NiO reflections. Whatever the isotherm, concomitant variation between both Ni and NiO intensities was observed with increasing oxidation times as illustrated in Fig. 5a for isotherm at 400°C. Figure 5b presents the time-dependent variation at 400°C of integrated intensity for (111) NiO and (111) Ni Bragg peaks. These evolutions indicate that the Ni metallic film is progressively consumed as the NiO film grows up to the total consumption of the Ni layer. The same kind of evolution was observed at 500°C with faster oxidation kinetics. Figure 6a shows the time-dependent variation, at 400 and 500°C, of the (111) NiO integrated intensity ratio I(t) / I ∞ (t → ∞) between time t and final time (i.e. t → ∞) after total Ni consumption. These ratios qualitatively point out that the Ni oxidation rate increases with increasing annealing temperature.

Figure 6b shows, at 400 and 500°C, the time-dependent increase of NiO average crystallite size D deduced from the decrease of the (111) NiO integral breadth β. Isothermal nucleation/growth process of particles is usually described by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory [15][16][17]. This model gives the time-dependent evolution of the volume V(t) of a nucleating and growing phase. In the present case, the evolution in time of NiO crystallite size D has been fitted using a JMAK-type equation:

D(t) = D ∞ [1 -exp(-k × t n )]
, where D ∞ corresponds to the final ACCEPTED MANUSCRIPT --10 crystallite size when t → ∞. The satisfactory fitting of experimental data shown in Fig. 6b clearly indicates that the growth of NiO crystallites follows the JMAK model. Table 1 summarizes the parameters of the JMAK model for each temperature. The parameters n and k are respectively the JMAK coefficient and the growth rate constant depending on temperature. The growth rate constant k is around 0.96 at 400°C while at 500°C, k equals 1.42. Besides, the crystallite size is also influenced by the temperature: at 400°C, D ∞ ≈ 26 nm while D ∞ ≈ 32 nm at 500°C. These results clearly show that growth of NiO crystallites is significantly accelerated at 500°C.

Comparison of RTA and in situ oxidation conditions

Inset in Fig. 6b illustrates the growth of NiO crystallites versus oxidation times (< 2 hours) (i) for pre-treated Ni films at 400 or 500°C (in situ diffraction experiments) and (ii) for a Ni layer directly subjected to the oxidizing atmosphere (ex situ diffraction experiments). At 400°C, the rather different growth rate constants k indicate that the oxidation kinetics is much faster for RTA route (Table 1). The main difference between these two sets of conditions stands in the pre-treatment of Ni metallic layer prior to oxidation. Furthermore, at 400°C after 30 minutes in pure O 2 , the crystallite size is about 14 nm when the Ni film is pre-treated under vacuum whereas it is of 21 nm without pre-treatment. Hence, these analyses show that the oxidation kinetics of Ni layer strongly depends on its own microstructure, which is modified by the thermal pre-treatment. → -4 V → 0, no switching occurs and the high current response is indicative of the initial low resistance state. As the voltage is ramped up to 4.5 V, the current abruptly decreases at 4.2 V pointing out a resistance switching of the MRM structure from a low (R ON = 55 Ω) to a high resistance state (R OFF = 2 360 Ω). In this latter case, the switching was irreversible, the structure remaining in OFF state whatever the bias imposed.

Electrical behavior of NiO films obtained in various conditions

Typical switching characteristic

The Pt top electrode area dependence of resistance switching was also studied: in reducing TE diameter from 540 to 160 µm, the current in the low resistance (LRS) state remains quite unchanged whereas a decrease of one order of magnitude is observed in the high resistance state (HRS). These results indicate that the conduction in HRS is strongly dominated by the area of top electrode whereas the formation of filamentary paths in the oxide can be proposed to explain the conduction in LRS. Besides, it has been mentioned in the previous paragraph that the MRM structures systematically turn off around 4.2 V in applying a positive voltage to the bottom electrode with a sweep of 0.2 V/msec. In decreasing the voltage sweep, the current through the structure increases whereas the threshold voltage decreases (e.g. for a sweep of 0.02 V/msec, the switching is observed at 3.7 V). Consequently, the opposite variations of current level and threshold voltage seem to be in favor of a heating Joule effect with a subsequent destruction of filaments in LRS and a switching to HRS.

Influence of annealing time and temperature under pure oxygen

I(V) measurements were performed on samples coated with a 24 or 100 nm thick Ni layer, oxidized in pure O 2 at different temperatures (200, 300 and 400°C) with RTA times ranging from 10 seconds to 5 minutes. The observed electrical behavior was systematically correlated with ex situ x-ray diffraction patterns enabling detection of Ni and/or NiO phases. Fig. 8a shows that the threshold voltage V th increases along with annealing time whatever the initial thickness of Ni film.

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For samples with a 100 nm thick Ni layer, an annealing at 300°C or above in pure O 2 is required to at least partially oxidize the metallic film and to form NiO-based structures exhibiting a resistance switching with threshold voltages ranging from 3.5 to 5 V (Table 2a). Indeed, the observation of (111) NiO Bragg reflection for RTA at 300 and 400°C is in agreement with the formation of a bi-stable NiO layer whereas the absence of NiO phase for RTA at 200°C explains the lack of switching behavior. Moreover, the threshold voltage V th increases along with the annealing temperature: for instance after 180 seconds of oxidation, V th raises from 4.3 V at 300°C to 5 V at 400°C.

For samples with initial 24 nm thick Ni layer, V th follows the same tendency since it increases along with annealing time and temperature (Fig. 8a). For instance, after 180 seconds of annealing, V th increases from 3.1 V at 200°C to 3.8 V at 300°C. Furthermore, it is observed that no switching occurs when the Ni films were annealing at 400°C for times larger than 30 seconds. X-ray diffraction experiments have shown that, in these conditions, the Ni metallic film is fully consumed (with subsequent disappearance of bottom electrode). In this latter case, electrical measurements are impossible. Besides the threshold voltage, the evolution of R ON and R OFF resistances deduced from the slope of I(V) curves measured on samples annealed at 300°C shows a very slight increase with annealing time although the R OFF /R ON ratio remains quite unchanged (Fig. 8b). The average R OFF /R ON ratio is around 55 for samples with an initial 100 nm thick Ni layer annealed at 300°C, whereas it is only around 15 for samples with 24 nm thick Ni layer. Therefore, in the perspective of memory devices, 100 nm thick Ni layers provide a large memory window enabling a better discrimination between ON and OFF states. In contrast, the lower threshold voltages measured on 24 nm thick Ni layers oxidized at moderate temperature with short annealing times would match the low-voltage application requirements.

Influence of oxidizing atmosphere

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Different oxidizing atmospheres (20 and 500 ppm of O 2 ; pure O 2 ) were tested on samples coated with initial 24 or 100 nm thick Ni layer and annealed using RTA at 200 and 400°C for 30 seconds and 3 minutes. Once again I(V) characteristics were systematically correlated with x-ray diffraction patterns. Figure 9 presents the distribution of threshold voltages V th measured in different experimental conditions. The main trend is the augmentation of V th with increasing oxygen partial pressure: V th varies from 2.2 V (24 nm thick Ni layer annealed at 200°C for 30 seconds under 20 ppm of O 2 ) to 5 V (100 nm thick Ni layer annealed at 400°C for 180 seconds in pure O 2 ). It is worth to note that in few cases no switching was observed for 24 nm thick Ni layers (Table 2b). For instance at 400°C, whatever the oxygen partial pressure, the annealing time has to be limited to 30 seconds to prevent the total oxidation of Ni layer (checked by x-ray diffraction). In contrast, for lower temperatures (i.e. 200 and 300°C), the Ni layer is only partially oxidized whatever the annealing time and oxygen partial pressure and a switching is observed.

As partial conclusion, Fig. 9 helps to establish the experimental conditions required to form bistable NiO films with switching voltages which may be easily tailored in monitoring annealing time and/or O 2 partial pressure.

Discussion

In this section, both microstructural and electrical characteristics of Pt/NiO/Ni MRM structures are discussed in correlation with the experimental conditions. Tables 2a and 2b gather the experimental conditions, the presence of Ni and/or NiO layers (detected by x-ray diffraction) and the threshold voltage of the MRM structures when switching was observed.

Impact of Ni microstructure on oxidation kinetics

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Electrical testing presented in previous section revealed major differences in the switching behavior which may be linked to the Ni microstructure before oxidation step. Indeed, a resistive switching occurs on a 24 nm Ni layer oxidized at 200°C whereas it is not observed on a 100 nm Ni film annealed in the same conditions (cf. Tables 2a and2b). This peculiar behavior may find its origin in the thickness-dependent microstructure of Ni layer which influences the oxidation kinetics.

As shown in Fig. 1a, the 24 nm thick Ni layer presents much smaller crystallites and a lower texture along [111] Ni direction as compared to the 100 nm thick Ni layer. Consequently, the growth kinetics of NiO film seems to depend on the texture and crystallite size of the initial Ni metallic layer.

Several arguments may support this hypothesis:

-Firstly, x-ray texture analyses (Fig. 1d) showed a fibre texture of NiO film along the [111] direction (cf. section 3.1.). This result is in good agreement with the work of Czerwinski et al. -Finally, Graham et al. demonstrated that the oxidation rate is strongly dependent on the Ni surface morphology [23,24]. Moreover, at lower temperatures, the influence of oxygen transport increases through short circuit paths, such as grain boundaries and dislocations [23,25,26]. As a result, smaller Ni crystallites result in a larger amount of grain boundaries which may facilitate the oxygen diffusion and accelerate the NiO formation.

Consequently, the non-oxidation of 100 nm thick Ni film at 200°C (with subsequent lack of switching) may find its origin (i) in a stronger [111] Ni texture that increases the resistance to ACCEPTED MANUSCRIPT --15 oxidation and (ii) in larger crystallites that decrease the amount of grain boundaries and limit oxygen diffusion.

Besides, as seen in inset of Fig. 6b, the comparison of time-dependent growth of NiO crystallites (extracted from ex situ and in situ x-ray diffraction experiments) confirms again the influence of Ni microstructure on the NiO growth. When the Ni metallic film is annealed under vacuum prior to oxidation, a further crystallization occurs with an enhancement of [111] Ni texture and a growth of Ni crystallites (cf. Figs. 4b and4c). In contrast, the RTA conditions lead to a rather different situation with two competitive mechanisms that simultaneously occur: Ni crystallization with subsequent changes in the microstructure; growth of NiO film with progressive consumption of the Ni layer. Thus, the different "history" of Ni layer before oxidation may explain the faster kinetics (cf. Table 1) observed in RTA conditions in which metallic layer is directly subjected to the oxidizing atmosphere without pre-treatment.

To conclude, in agreement with previous published works [23][24][25][26], the Ni surface morphology and its thermal treatment prior to the oxidation radically change the growth kinetics of the NiO film.

Consequently, the present experiments emphasize the importance of thermal pre-treatment in order to avoid the competition between Ni crystallization and oxidation.

Initial ON state

As compared to sputtered films reported in literature, the NiO films obtained from Ni oxidation are initially in ON state without special electro-forming generally required to reach the conductive state. Oxygen content in oxide films may explain such a behavior. Varying oxygen content in sputtering gas mixture, several authors have shown a drastic modification of the electrical properties of Ni x O films from a metallic behavior (highly Ni-excess films) at low oxygen content (< 5 % of oxygen in sputtering gas) to a monostable threshold switching (Ni-deficient films) at high oxygen content (> 20 %), the memory switching region being limited to intermediate oxygen

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--16 contents (10-17%) [27,28]. The comparison of the cubic cell parameter of films obtained in pure oxygen (a = 4.18 Å) with those of Ni-deficient (a = 4.23 Å) and Ni-excess (a = 4.17 Å) films suggests that nickel oxide films obtained from Ni oxidation are close to the Ni-excess region within memory switching area [27,28]. Moreover, the vicinity of the metallic behavior region may certainly explain the initial conductive state.

Irreversible switching

On the basis of the previous discussion, one could expect a reversible switching of NiO-based structures. On the contrary the MRM structures remained in the OFF state after the first and unique switching. This irreversibility certainly originates in the roughness of NiO/Ni and Pt/NiO interfaces seen on TEM cross-section images (Fig. 2a). Indeed, the TEM observations have revealed that the Ni metal surface is covered by a continuous but irregular oxide layer with thicknesses ranging from 40 to 75 nm. This latter characteristic may have a crucial role on the reversibility of switching since several authors have attributed the switching to the formation/rupture of conductive paths near the interface [4,[START_REF] Inoue | IEEE Proceedings of the Non Volatile Memory Technology Symposium[END_REF]. During oxidation, the growth of the NiO oxide film is rather non-uniform as a consequence of different growth rates for different Ni grain orientations. Indeed, Peraldi et al. have demonstrated the important role of the crystallographic orientation of nickel on the oxidation kinetics [21,22]. For instance, oxidation of the (100) nickel face is more rapid than oxidation of (110), (111), and (112) faces. Once again, it appears crucial to stabilize the Ni film microstructure before oxidation step in order to decrease the roughness of the NiO film.

Finally, in previous works, Haugsrud mentioned the existence of microfissures within the NiO oxide obtained from Ni oxidation at high temperatures [26,29]. Several main routes were suggested to account for the presence of microfissures [29]. Based on these results, one can propose the existence of such microfissures within the NiO film which may explain the difficulty to re-form the conductive paths after switching. These microfissures may also lead to a partial delamination at the ACCEPTED MANUSCRIPT --17 interface between Ni and NiO films, such a delamination being sometimes observed during TEM cross-section sample preparation by mechanical polishing (Tripod technique).

Conclusion

In this paper, the oxidation kinetics of Ni metallic films were apprehended in order to form NiO layers exhibiting a voltage-induced resistive switching within Pt/NiO/Ni MRM structures. Process parameters (temperature, annealing time and oxidizing atmosphere) of Rapid Thermal Annealing route were tested to achieve oxidation. Different thermal treatments were selected to oxidize the metallic film with conditions (i) preventing the complete consumption of Ni film used as bottom electrode and (ii) producing bi-stable oxide films. With these experimental conditions, the as-grown NiO films were initially in the low resistance ON state without special electro-forming usually required for sputtered NiO films. The obtaining of Ni-excess oxide films may explain the initial metallic behavior. Nevertheless, above a threshold voltage, varying from 2 to 5 V depending on the oxidizing conditions, the MRM structures irreversibly switched into a high resistance OFF state. This irreversibility may be linked to the roughness of NiO/Ni and Pt/NiO interfaces due to a nonstabilized Ni film microstructure prior to oxidation.

In the perspective of memory devices, further microstructural and electrical analyses are required to decrease the threshold voltage of reversible memory elements. Currently, new experimental conditions are evaluated in order to control the Ni crystallization prior to oxidation and alternative routes for Ni oxidation such as plasma treatment under N 2 O or O 2 is explored. Finally, fully integrated bi-stable NiO/Ni stacks in via structures will be fabricated and the stability of the MRM structures in conditions close to those used in the back-end process will be checked.

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Figure captions

  -ray diffraction profiles of (111) Ni reflection show that the initial 100 nm Ni layer presents a stronger texture along [111] direction as compared to the 24 nm Ni layer. Moreover, interpreting the width of (111) Ni diffraction peak in terms of crystallite size, the 24 nm Ni layer presents much smaller crystallites (≈ 10 nm) as compared to those of the 100 nm Ni layer (≈ 40 nm). Further x-ray texture analyses were performed on a non-oxidized 100 nm thick Ni layer on {111} Ni and {200} Ni Bragg reflections. On one hand, the {111} Ni pole figure shown in Fig. 1b gives a qualitative description of the orientation of the as-deposited Ni film. The maximum intensity of the (111) Ni Bragg reflection is at the center of the pole figure, the [111] directions being dispersed over a revolution cone centered on the substrate's normal and with about 5° of full width at half maximum. On the other hand, the {200} Ni pole figure (Fig. 1c) shows a drastically lower texture with a random orientation of [200] directions around the substrate's normal. Thus, both {111} Ni and {200} Ni pole figures indicate a strong [111] fibre texture of the Ni layer.Similar texture analysis was performed on the NiO film obtained from Ni oxidation (100 nm thick layer) at 400°C for 120 seconds in pure O 2 . On the {111} NiO pole figure (Fig.1d), the maximum intensity is centered and the rest of the intensity is spread over a wide angular range. This indicates a texture of the NiO layer along[111] direction even if this texture is much lower than the one observed on Ni metallic film. Consequently, the crystallographic orientation of NiO grains appears to be conditioned by the texture of the underlying Ni film since the NiO films preferentially grow with the [111] direction parallel to the substrate's normal.

Figure 7

 7 Figure 7 shows a typical current-voltage I(V) switching characteristic measured with a staircase

  which showed that the stronger the[111] Ni substrate texture, the stronger the [111] NiO oxidized layer texture became after oxidation[18]. In contrast, López-Beltrán et al. obtained texture-free polycrystalline NiO layer from the oxidation of a weakly textured Ni film[19]. -Second, concerning the Ni film texture, Czerwinski et al. and Peraldi et al. have demonstrated the important role of crystallographic orientation of nickel on its oxidation kinetics [18,20-22]. Hence, a strong [111] texture of Ni leads to formation of highly protective and slow growing NiO film, i.e. a weaker Ni texture significantly decreases the resistance to the oxidation.

Fig. 1 .

 1 Fig. 1. (a) Comparison of (111) Ni Bragg reflections for 24 and 100 nm thick Ni layers prior to

Fig. 2 .

 2 Fig. 2. (a) TEM cross-section of the stack Pt/NiO/Ni/SiO 2 /Si 3 N 4 /Si: the NiO film was obtained from

Fig. 3 .

 3 Fig. 3. (a) Evolution versus oxidation time of (111) NiO (on left) and (111) Ni (on right) Bragg reflections. (b) Annealing time-dependent evolution of integral breadth β of (111) NiO Bragg

Fig. 4 .

 4 Fig. 4. (a) Temperature-dependence of diffraction patterns recorded under vacuum from 25 to

Fig. 5 .Fig. 6 .

 56 Fig. 5. (a) Time-dependent evolution of diffraction patterns recorded at isotherm 400°C in pure O 2

Fig. 7 .

 7 Fig. 7. Typical I(V) switching characteristics of the Pt/NiO/Ni MRM structures: the NiO film was

Fig. 8 .Fig. 9 .

 89 Fig. 8. (a) Evolution of threshold voltage (V th ) as a function of the annealing time for Pt/NiO/Ni

Table captions Table 1 .

 captions1 Parameters of the Johnson-Mehl-Avrami-Kolmogorov (so-called JMAK) equation: D(t) = D ∞ [1 -exp(-k × t n )] which may describe the nucleation/growth process of NiO crystallites in various experimental conditions (RTA conditions or in situ oxidation).

Table 2 .

 2 Sum up of electrical behavior correlated with the presence of NiO and/or Ni phases checked by x-ray diffraction. The initial Ni thickness is 100 nm (a) or 24 nm (b) (NSW = no switching; SW = switching; corresponding V th is given when a resistive switching is observed).

		T (°C)	D ∞ (nm)	k	n
	RTA conditions	400	19.6	8.8	0.35
	In situ	400	26.5	1.0	0.41
	oxidation	500	32.4	1.4	0.22

Table 1

 1

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Acknowledgments

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This work is partially supported by EC through the Marie Curie project APROTHIN ("Advanced PROcess Technologies for Horizontal INtegration", FP6-514221).


A Tsopela

A Laborde

L Salvagnac

I Séguy

R Izquierdo

P Juneau

P Temple-Boyer

J Launay

LIGHT EMITTING DEVICES AND INTEGRATED ELECTROCHEMICAL SENSORS ON LAB-ON-CHIP FOR TOXICITY BIOASSAYS BASED ON ALGAL PHYSIOLOGY

Keywords: Lab-on-chip, microelectrode, OLED, O2-related algal metabolism, herbicide detection, water toxicity analysis

In the frame of water toxicity analysis, a portable, glass-based, lab-on-chip was developed, integrating threeelectrode electrochemical microsensors and organic lightemitting diodes (OLED). The basic detection principle consists in monitoring electrochemically O2-related, algal metabolism in presence of herbicides. Thus, aiming on Diuron herbicide detection, a concentration-dependent inhibition effect on photosynthetic oxygen production rate was evidenced in the [0 -1 M] range. Finally, OLEDbased integrated system demonstrates higher detection characteristics than those using external white light source (sensitivity: 0.48 versus 0.26 nA/s/M) and is highly promising for further integration of optical and electrochemical sensors enabling double complementary detection.

INTRODUCTION

Assessment of water quality has become an increasingly important environmental issue in the past few years. Water pollution induced by agricultural and industrial activities is known to have a severe impact on all living organisms. This is associated to the excessive use of herbicides and pesticides to protect crops. These toxicants can easily penetrate the soil and are finally detected in ground and/or river waters. It is therefore a major challenge to monitor their different quantities. Although water toxicant detection is most commonly performed in laboratories, it is important to find alternative methods to existing analysis tools, more adapted to field detection. Biosensors and bioassays can eventually meet the implemented requirements and have attracted wide attention [START_REF] Jang | State-of-the-art lab chip sensors for environmental water monitoring[END_REF]. For instance, optical [START_REF] Lefèvre | Algal fluorescence sensor integrated into a microfluidic chip for water pollutant detection[END_REF] and electrochemical [START_REF] Shitanda | Amperometric screen-printed algal biosensor with flow injection analysis system for detection of environmental toxic compounds[END_REF] algalbased systems are proposed in literature for the detection of herbicides as their metabolism is highly influenced by such toxic species.

In this study, a portable, glass-based, lab-on-chip was developed to monitor algal metabolism and detect targeted herbicides. It involves three-electrode (Pt -Pt -Ag/AgCl) electrochemical microcells (ElecCell) to perform electrochemical analysis, as well as organic light-emitting diodes (OLED) to trigger algal photosynthetic activities.

EXPERIMENTAL

Electrochemical microcells (ElecCell) and organic light-emitting diodes OLED were assembled on a lab-onchip. Six independent detection chambers were designed on each platform enabling simultaneous processing of different assays. Three fluidic chambers were dedicated to electrochemical detection and three to future fluorescencebased optical detection. The lab-on-chip structure was fabricated on glass substrate in order to be compatible with optical technology and associated measurement.

Fabrication procedure included electrochemical microcell integration on the glass substrate, fluidic microsystem implementation, and device packaging. A 3D model of lab-on-chip sensor platform, including 6 independent chambers is presented in Figure 1.

Required parameters are missing or incorrect.

Integration of electrochemical microcells

Concentric, three-electrode (Pt -Pt -Ag/AgCl) electrochemical microcells (ElecCell) were integrated on glass substrate through photolithography patterning and physical vapor deposition (PVD) according to a previously developed process [START_REF] Ben Amor | Enhanced detection of hydrogen peroxide with platinized microelectrode arrays for analyses of mitochondria activities[END_REF][START_REF] Tsopela | Integrated electrochemical biosensor based on algal metabolism for water toxicity analysis[END_REF]. Three PVD steps were performed in a row. First, a 200nm platinum layer was deposited on a 20nm titanium underlayer to form platinum-based working and counter microelectrodes. Then, a 400nm silver layer was deposited on top and was later oxidized in chloridebased solutions to form the integrated Ag/AgCl pseudoreference microelectrode. Finally, a Si3N4 passivation layer, deposited by chemical vapor deposition (CVD), was patterned in order to insulate electrically the different microelectrodes and define precisely their active surface. To improve detection properties, specific attention was brought on working microelectrode design and functionalization. Thus, a microelectrode array involving 25 disks of 10m diameter was realized and later functionalized thanks to a black platinum (Bl-Pt) electrodeposition process.

Implementation of the fluidic microsystem and device packaging

The fluidic microsystem was structured by patterning 250m-thick SU8 photoresist. Dimensions were specified according to the ElecCell geometry as well as to requirements related to the optical system. Consequently, circular-shaped fluidic analysis microchambers were designed with a 8mm diameter and therefore a 12.5L volume. Finally, the fluidic platform was sealed by wafer bonding using a glass cover. Finally, device packaging and electrical connection were performed in order to obtain a plug-in, easy to use, lab-on-chip device for algal solution analysis (Fig. 2).

Required parameters are missing or incorrect.

Bioassays and electrochemical analysis of algal photosynthetic activities

Chlamydomonas reinhardtii green algae were cultivated at 25°C in high salt medium (HSM) thanks to time-regulated light/dark (16h/8h) cycles. Algal cell concentration was measured by a multisizer Coulter counter in order to ensure a constant value for the different experiments. After leaving algae in dark for a sufficiently long time period, herbicide detection tests were finally conducted by mixing algal solutions with different Diuron concentrations (range: 0 -1 M, in agreement with maximum acceptable value implied in Canada: 0.64 m). Using the (Pt/Bl-Pt -Pt -Ag/AgCl) ElecCell, O2 monitoring was finally performed by chronoamperometry with an applied potential of -0.7 V versus integrated Ag/AgCl. All the experiments were performed at ambient temperature (~ 22°C).

Thanks to its realization on a glass substrate, the glassbased lab-on-chip was fully compatible with optical technology and associated measurement (Fig. 2). As a result, it allowed herbicide detection while using a nonintegrated, halogen, white light source. However, in order to eventually obtain portable and autonomous analysis microsystem, the light source used for the stimulation of algal photosynthetic activities should be integrated on the final lab-on-chip device. Consequently, blue OLED devices were fabricated on a glass substrate through multilayer evaporation of PEDOT:PSS/NPB/PCAN/Alq3/Al and placed on top of the detection chamber, in direct contact with the lab-on-chip glass cover (Fig. 3). It was specifically designed in order to have a good overlap between the OLED electroluminescence spectrum and the Chlamydomonas reinhardtii algae absorption band (Fig. 4).

RESULTS AND DISCUSSION

O2 measurement in control algal solutions

Using the halogen white light source, dark and light periods were alternated in control algal solutions and O2 concentration was monitored by chronoamperometry (Fig. 5). When light is on, photosynthesis is responsible for oxygen production and therefore for the O2-related cathodic reduction current increase. On the contrary, when light is off, algae are consuming oxygen due to respiration and the cathodic reduction current decreases. Prior to further analysis, it was checked, using non-algal control solutions that current variations were not related to illumination interferences (result not shown).

Thanks to these first experiments, it appeared that current follows roughly linear variations with time and therefore that algal metabolisms can be studied according to to the slope of the current versus time.

Furthermore, a compromise was found concerning algal concentration in order to yield well-defined differences between oxygen production and consumption and high signal-to-noise ratio. This leads to an optimal concentration around 13 × 10 6 cells/mL. This value was chosen for the following experimental campaigns.

Required parameters are missing or incorrect.

Diuron detection using halogen white light source

First experiments were performed using a halogen white source illumination. Light-induced oxygen production was monitored in real time and O2 production rate decrease with increasing Diuron concentration points out the toxicant effect (Fig. 6). Nevertheless, respiration rates before turning on the light were found to be not identical. Since Diuron herbicide targets only photosynthetic activity, the causes of this variability are difficult to determine. They can be related to algae culture intrinsic properties, sampling protocol and/or biofouling phenomena. As a result, a normalization step was conducted in order to eliminate this variability. It consists in setting an average "reference" value ͞ x. for the different respiration slopes (xi) and applying a correction coefficient to all the photosynthesis slopes (yi).

y i * = y i x x i ae è ç ö ø ÷ (1)
Through this approach, sensor calibration curves were defined by studying the photosynthetic O2 production rate for different Diuron concentrations (Fig. 7). For the two light intensities tested, a linear decrease was obtained between both parameters, allowing to define the Diuron detection sensitivity. Values of 0.26 and 0.1 nA/s/M were calculated for light intensities of 600 and 1800 E/m 2 /s respectively.

Required parameters are missing or incorrect.

Figure 7: Calibration curves (normalized oxygen production rates versus Diuron concentration) for two different light conditions using halogen white light source.

Light intensity is known to play an important role in algal metabolism. At low intensity levels, photosynthesis is enhanced when light intensity is increased as more photons are captures by pigments and processed through the chloroplast reaction centers [START_REF] Cho | Microalgal photosynthetic activity measurement system for rapid toxicity assessment[END_REF]. However, at high intensity levels, reaction centers of chloroplast becomes saturated or even damaged. An inhibition of photosynthetic activity is thus observed [START_REF] Barber | Too much of a good thing: light can be bad for photosynthesis[END_REF]. Our result demonstrates that, when illumination conditions are too high, they are responsible for such inhibition phenomenon.

Finally, it should be noted that herbicide detection was also performed using natural lake water to cope with real samples analysis. Thus, Diuron detection sensitivity around 0.26 nA/s/M was still obtained for a 600 E/m 2 /s light intensity (Fig. 8).

Diuron detection using blue OLED as light source

Given the fact that blue OLED devices were integrated on the ElecCell-based lab-on-chip at the system level, a non-algal control measurement of current monitoring through light/dark periods was conducted. In contrast with previous control measurement carried out with the nonintegrated halogen white light source, an increase in the cathodic reduction current was observed when the light was turned on (result not shown). This current increase was attributed to the temperature increase induced by OLED heating. Indeed, the temperature on the lab-on-chip backside was measured around 35°C after a two minutes blue OLED illumination. In order to compensate this phenomenon, the illumination rate recorded for the non-algal control solution was subtracted from the photosynthetic O2 production rate determined for different Diuron concentrations. Experiments were performed in natural lake water samples in the [0 -0.6 M] Diuron concentration range (Fig. 8). Detection sensitivity obtained when using the integrated blue OLED light source was estimated at around 0.48 nA/s/M that corresponds to almost double the value obtained while using the non-integrated halogen white light source. This sensitivity improvement was related to the better overlap between the blue OLED emission spectrum with the major Chlamydomonas reinhardtii algae absorption band (Fig. 4).

Figure 8: Calibration curves obtained in lake water samples using as light sources, either an integrated blue OLED (blue points) or a halogen white light source (red points).

CONCLUSION

A portable device for in-situ herbicide detection, based on algal physiology, was developed. The fabricated lab-onchip platform consists in three fluidic chambers integrating (Pt/Bl-Pt -Pt -Ag/AgCl) electrochemical microcells (ElecCell) and three chambers dedicated to further optical fluorescence-based detection. The effect of Diuron herbicide was validated using the lab-on-chip devices with culture medium solutions. Illumination was first supplied through a non-integrated, halogen white light source. Thus, a Diuron concentration-dependent decrease in the photosynthetic oxygen production rate was demonstrated for two different light intensities: 600 and 1800 E/m 2 /s. Diuron detection sensitivity of approximately 0.26 and 0.1 nA/s/M were respectively obtained, evidencing photosynthesis inhibition phenomena due to too high intensity levels. Then, in order to obtain an autonomous system, the same experiments were successfully carried out in natural lake water samples, with an integrated blue OLED used as light source for the excitation of photosynthetic mechanism. Taking into account undesired heating phenomena, Diuron detection was found to gain a factor two (value around 0.48 nA/s/M in the [0 -0.6 M] concentration range) evidencing that photosynthetic apparatus was more effective when the blue OLED light source is used compared to the halogen white light source.

These first results demonstrate that the proposed labon-chip biosensor can effectively follow the change in photosynthetic activity induced by Diuron herbicide and reflected through a modification in oxygen production rate. It can therefore be an efficient indicator of water pollution. Nevertheless, further studies should be conducted to still improve detection properties while optimizing the OLED photoluminescence spectrum, illumination power and integration process.

Figure 1 :

 1 Figure 1: 3D model of the lab-on-chip sensor platform.

Figure 2 :

 2 Figure 2: Algal test solution inserted in one of the detection chambers of the lab-on-chip device.

Figure 3 :

 3 Figure 3: Integration of blue OLED devices on the lab-onchip glass cover. Required parameters are missing or incorrect.

Figure 4 :

 4 Figure 4: Comparison of emission spectra of OLED and halogen white light source with algal absorption spectrum.

Figure 5 :

 5 Figure 5: Current measurements of algal respiration and photosynthesis using a (Pt/Bl-Pt -Pt -Ag/AgCl) ElecCell device integrated on a lab-on-chip.

Figure 6 :

 6 Figure 6: Algal O2-amperometric responses for various Diuron concentrations using a (Pt/Pt-Bl -Pt -Ag/AgCl) ElecCell-based lab-on-chip device.

ACKNOWLEDGEMENTS

The authors would like to thank the French "Agence nationale de la Recherche" (ANR, project DOLFIN, n° ANR-13-JS03-0005-01) and the "Fonds France Canada pour la Recherche" (FFCR) for financial support. Finally, microfabrication procedure was partly supported by the French RENATECH network.