Thai Ha-Huy
email: thai.hahuy@univ-evry.fr

Cuong Le Van
email: cuong.le-van@univ-paris1.fr

Manh-Hung Nguyen
email: mhnguyen@toulouse.inra.fr

Arbitrage and asset market equilibrium in infinite dimensional economies with short-selling and risk-averse expected utilities *

Keywords: asset market equilibrium, individually rational attainable allocations, individually rational utility set, no-arbitrage prices, no-arbitrage condition. JEL Classification: C62, D50, D81, D84, G1

We consider a model with an infinite number of states of nature, von Neumann -Morgenstern utilities, where agents have different probability beliefs and where short sells are allowed. We show that no-arbitrage conditions, defined for finite dimensional asset markets models, are not sufficient to ensure existence of equilibrium in presence of an infinite number of states of nature. However, if the individually rational utility set U is compact, we obtain an equilibrium. We give conditions which imply the compactness of U. We give examples of non-existence of equilibrium when these conditions do not hold.

Introduction

In finite dimensional markets with short-selling, conditions on agents' utilities insuring the existence of equilibria (or equivalent to the existence of equilibria) are well understood. In particular they can be interpreted as no-arbitrage conditions. In an uncertainty setting, where agents have different beliefs and different risk aversions, as originally shown by [START_REF] Hart | On the Existence of an Equilibrium in a Securities Model[END_REF], the no-arbitrage condition may be interpreted as compatibility of agent's risk adjusted beliefs.

There is a huge literature on sufficient and necessary conditions for the existence of equilibria in the line of Hart's model, where the market is complete.

In finite dimension, one can refer, for instance to [START_REF] Page | On equilibrium in Hart's securities exchange model[END_REF], [START_REF] Werner | Arbitrage and the Existence of Competitive Equilibrium[END_REF], [START_REF] Nielsen | Asset market equilibrium with short-selling[END_REF], [START_REF] Page | On equilibrium in Hart's securities exchange model[END_REF]Wooders (1995, 1996), Allouch (1999), Allouch et alii (2000), Won and Yannelis (2008). When the number of assets is infinite, and the market is complete, the noarbitrage condition used for finite dimension do not imply existence of equilibrium. The standard assumption is to assume that the individually rational utility set is compact ( see e.g. [START_REF] Cheng | Asset market equilibrium in infinite dimensional complete markets[END_REF], Brown and Werner (1995), Dana and Le Van (1996), [START_REF] Dana | General Equilibirum in asset markets with or without short-selling[END_REF], Dana and Le Van (2000), Le [START_REF] Van | Asset market equilibrium in L p spaces with separable utilities[END_REF]). 1 In this paper, we consider a model with an infinite number of states of nature, a finite number of agents and Von Neumann -Morgenstern utilities with different expectations.

More precisely, we consider a model where the utility of agent i is

U i (x i ) = ∞ s=1 π i s u i (x i s )
where π i is her belief and x i is her consumption. The commodity space is l p (π) with p ∈ {1, 2, . . . , +∞}. 2 1 Chichilnisky and Heal (1993) give a condition which implies the boundedness of the individually rational feasible set in L 2 . Since the feasible set is closed, it is therefore weakly compact in L 2 . 2 We use the model proposed by Hart. Investors are interested only in their wealth in the second period. We suppose that the market is complete with an asset system (r 1 , r 2 , . . . , r k , . . . ).

For each portfolio z = (z 1 , z 2 , . . . , z k , . . . ), the wealth or investor if state s occurs is

ws = k z k r k s .
Her expected utility is:

V (w 1 , w 2 , . . . , w s , . . . ) = s πsu(ws).

Since the market is complete, the choice of portfolio is equivalent to the choice of wealth. As in Hart's pioneer paper, we consider the expected utility function on wealth.

When the number of states is infinite, the no-arbitrage condition (à la Werner or à la Dana-Le Van) only ensures the boundedness of the individually rational utility set. We give examples where this condition is satisfied and no equilibrium exists. The strategy is therefore to give assumptions which imply the compactness of the individually rational utility set and hence existence of equilibrium. Our conditions might be considered as among the weakest since we give also examples of non existence of equilibrium when these conditions do not hold.

The paper is organized as follows. In Section 2, we set up the model and define the equilibrium. In section 3, we introduce no-arbitrage conditions and give conditions for the existence of equilibrium. Proofs are put in Section Appendix, Section 4. We mention that our methods of proofs are inspired by the ones in Le [START_REF] Van | Asset market equilibrium in L p spaces with separable utilities[END_REF]. However, their model rules out the risk-neutral agents. That is not the case in our model.

The model

There are m agents indexed by i = 1, . . . , m. The belief of agent i in state s is π i s ≥ 0, and ∞ s=1 π i s = 1 . Let us denote by π the mean probability 1 m i π i . We first assume: A0: π i is equivalent to π j for any i, j i.e. there exists a number h > 0 such that h

≤ π i s π j s ≤ 1
h for all i, j, s. 3 Under A0, without loss of generality, one can assume that π i s > 0 for any i, any s. In this paper, we always suppose that the condition A0 is satisfied and π i s > 0 for any i, any s. The consumption set of agent i is X i = l p (π) with p ∈ {1, 2, . . . , +∞} and agent i has an endowment e i ∈ l p (π). We assume that for each agent i there exists a concave, strictly increasing, differentiable 4 function u i : R → R , such that, for any i, the function

U i (x i ) = ∞ s=1 π i s u i (x i s )
is real-valued for any x i ∈ X i . Agent i has l p (π) as consumption set, e i as initial endowment and U i as utility function, with i = 1, . . . , m.

Definition 1 An equilibrium is a list (x * i ) i=1,...,m , p * ) such that x * i ∈ X i for every i and p * ∈ l q + (π) \ {0} and 3 With A0, the consumptions set of the agents belong to same topological space l p (π).

We observe that when all agents have the same belief as in [START_REF] Cheng | Asset market equilibrium in infinite dimensional complete markets[END_REF], then A0 is satisfied. 4 For simplicity, we assume differentiability. The results still hold by using super-gradients.

But the proofs will become tedious.

(i) For any i, U i (x) > U i (x * i ) ⇒ ∞ s=1 p * s x s > ∞ s=1 p * s e i s , (ii) m i=1 x * i = m i=1 e i .
Definition 2 A quasi-equilibrium is a list (x * i ) i=1,...,m , p * ) such that x i * ∈ X i for every i and p * ∈ l q + (π) \ {0} and (i)

For any i, U i (x) > U i (x * i ) ⇒ ∞ s=1 p * s x s ≥ ∞ s=1 p * s e i s , (ii) m i=1 x * i = m i=1 e i .
In this paper, since the consumption set is the whole space, and the utility functions U i are continuous, any quasi-equilibrium is an equilibrium (see e.g. Geistdoerfer-Florenzano (1982), Proposition 3). Define

a i = inf x u i (x) = u i (+∞), b i = sup x u i (x) = u i (-∞).
Let I 1 be the set of indices i such that a i < b i , and I 2 be the set of indices such that a i = b i . I 1 is the set of risk averse agents, I 2 is the set of risk neutral ones.

We now give some definitions.

Definition 3

1. The individually rational attainable allocations set A is defined by

A = {(x i ) ∈ (l p (π)) m | m i=1 x i = m i=1
e i and U i (x i ) ≥ U i (e i ) for all i}.

2. The individually rational utility set U is defined by

U = {(v 1 , v 2 , ..., v m ) ∈ R m | ∃x ∈ A s.t U i (e i ) ≤ v i ≤ U i (x i ) for all i}.

No-arbitrage condition and existence of equilibrium

We will first strengthen the definition of useful vectors introduced by Werner.

We then introduce a notion of no-arbitrage price based on these strong useful vectors.

A vector w is strongly useful for agent i if, for any x i ∈ X i , we have

(i) ∞ s=1 π i s u i (x i s + λw s ) ≥ ∞ s=1 π i s u i (x i s ), ∀λ ≥ 0, and (ii)
∞ s=1 π i s u i (x i s + λw s ) > ∞ s=1 π i s u i (x i s ) for some λ > 0.
It is easy to check that a vector w is strongly useful for agent i if, and only if

∀x i ∈ X i , ∃ λ > 0, ∀λ ≥ λ, ∞ s=1 π i s u i (x i s + λw s ) > ∞ s=1 π i s u i (x i s ).
This condition turns out to be equivalent to

∞ s=1 π i s u i (x i s )w s > 0, ∀x i ∈ X i .
Following Dana and Le Van [8], we now introduce a no-arbitrage condition.

(NA) There exist (x i ) m i=1 ∈ Π i X i , such that

λ i π i s u i (x i s ) = λ j π j s u j (x j s ) = λ k a k π k s , ∀s, ∀i ∈ I 1 , ∀j ∈ I 1 , ∀k ∈ I 2 ,
and ∀i ∈ I 1 , a i < inf s u i (x i s ) < sup s u i (x i s ) < b i .
Observe that if we define a price p ∈ l ∞ + (π) by

∀s, p s = λ i π i s u i (x i s ) = λ k a k π k s , i ∈ I 1 , k ∈ I 2
then we have p • w > 0 for any strongly useful vector w. Condition (NA) is useful to prove the boundedness of U. That is the statement of the following proposition. The proof is given in Appendix.

Proposition 1 If (NA) holds, then U is bounded.

The next proposition and its corollary give sufficient conditions to obtain (NA). Proposition 2 Assume either u i (-∞) = +∞ for all i ∈ I 1 or u i (+∞) = 0 for all i ∈ I 1 . Then (NA) holds if, and only if, π i = π j , for any i ∈ I 2 , any j ∈ I 2 .

The following corollary is immediate.

Corollary 1 If I 2 = ∅, then (NA) holds if (i) either u i (-∞) = +∞ for all i ∈ I, (ii) or u i (+∞) = 0 for all i ∈ I.
In the case of finite number of states, condition (NA) implies compactness of U.

In infinite dimension, with a vector space L as commodity space, Brown and Werner [START_REF] Brown | Arbitrage and existence of equilibrium in infinite asset markets[END_REF], Dana, Le Van and Magnien [10] assume the compactness of U and get existence of equilibrium with prices in L . In our model if the number of states is finite, condition (NA) implies compactness of U and is a sufficient condition for the existence of equilibrium (see e.g. Dana and Le Van [8]). In this paper, the number of states is infinite, we give an example which shows that (NA) is not sufficient to ensure existence of equilibrium if the conditions in Corollary 1 do not hold. We exhibit a model with two agents. Agent 1 has a 1 = 0, b 1 < +∞. Agent 2 has a 2 > 0, b 2 = +∞. The assumptions of Propositions 4 and 6 are clearly not satisfied. We still have (NA) but no equilibrium exists in this model.

Example 1 Consider an economy with two agents (i = 1, 2), with endowments equal to 0. The probabilities are equivalent:

π 1 s = 1 2 s , π 2 s = 1 Sα 1-α s 2 s , where 1 < α < 1, and S α = s 1-α s 2 s .
The reward utilities satisfy

u 1 (x) = b 1 , ∀x ≤ 0, u 1 (+∞) = 0, u 1 (0) = 0, u 2 (x) = a 2 , ∀x ≥ 0, u 2 (-∞) = +∞, u 2 (0) = 0.
There exists z > 0 with u 1 (z) < b 1 . Let x 1 s = z, ∀s. Since u 2 (-∞) = +∞, there exists x 2 s < 0 which satisfies

u 2 (0) = (1 -α s )u 2 (x 2 s ).
One can check that

λπ 1 s u 1 (x 1 s ) = π 2 s u 2 (x 2 s ), ∀s, with λ = u 2 (0) u 1 (z) × 1 Sα . Since 0 = u 1 (+∞) < u 1 (z) = u 1 (x 1 s ) < b 1 , a 2 = u 2 (0) < u 2 (0) 1 -α s = u 2 (x 2 s ) < u 2 (-∞) = +∞,
no-arbitrage condition (NA) is satisfied. We now show that no equilibrium exists. Assume there exists an equilibrium

(p, (x 1 , x 2 )) with x 1 s = x s = -x 2 s . We have ∀s, λ 1 π 1 s u 1 (x s ) = λ 2 π 2 s u 2 (-x s ), or λπ 1 s u 1 (x s ) = π 2 s u 2 (-x s ), with λ = λ 1 λ 2 .
For all s:

λ 1 2 s u 1 (x s ) = 1 -α s 2 s u 2 (-x s ), or equivalently λ = (1 -α s ) u 2 (-x s ) u 1 (x s )
, ∀s with λ = λS α . Since s p s x s = 0 and p s > 0 for any s, one must have s 0 with x s 0 ≤ 0. In this case

λ = (1 -α s 0 ) a 2 b 1
, and for any s = s 0 , x s > 0. We then obtain

u 2 (-x s 0 +1 ) u 1 (x s 0 +1 ) > u 2 (0) u 1 (0) = a 2 b 1 since x s 0 +1 > 0. Now, because 1 -α s 0 +1 > 1 -α s 0
we obtain, on the one hand:

λ = (1 -α s 0 +1 ) u 2 (-x s 0 +1 ) u 1 (x s 0 +1 ) > (1 -α s 0 ) u 2 (-x s 0 +1 ) u 1 (x s 0 +1 ) > (1 -α s 0 ) a 2 b 1
and on the other hand

λ = (1 -α s 0 ) a 2
b 1 which is a contradiction. Then there exists no equilibrium.

We now state our main results. Their proofs are given in Appendix. The ideas are to prove that the set A is l 1 (π)-compact and hence the set U is also compact when b i = +∞ for all i, and when a i = 0 for every i and b i < +∞ for some i, the set A is not necessary l 1 (π)-compact but the set U is still compact.

Theorem 1 Suppose one of these two conditions holds:

(i) a i = 0 for every i.

(ii) b i = +∞ for every i.

Then there exists an equilibrium with equilibrium price in l q (π) with

1 p + 1 q = 1, 1 ≤ p ≤ +∞.
Theorem 2 Suppose that I 2 = ∅. Suppose that these two conditions hold:

(i) a i = 0 for every i ∈ I 1 .

(ii) b i = +∞ for every i ∈ I 1 .

Then we have

(NA) ⇔ ∃ Equilibrium.

The equilibrium price is in l q (π)

with 1 p + 1 q = 1, 1 ≤ p ≤ +∞.
We can be surprised that in presence of risk-neutral agents we have to impose u i (+∞) = 0 and u i (-∞) = +∞ for any agent i ∈ I 1 . We give two examples with two agents, one of them is risk neutral while the second one is risk averse. In Example 2, the risk averse agent has the marginal utility at -∞ equal to +∞. In Example 3, her the marginal utility at +∞ is 0. In both examples, there exists no equilibrium.

Example 2 Consider an economy with two agents (i = 1, 2), with endowments equal to 0. The probabilities are equivalent:

π 1 s = 1 2 s , π 2 s = 1 Sα 1-α s 2 s , where 1 < α < 1, and S α = s 1-α s 2 s . Agent 1 is risk neutral.
The reward utilities satisfy

u 1 (x) = 1, ∀x ∈ R, u 2 (x) = a 2 , ∀x ≥ 0, u 2 (-∞) = +∞, u 2 (x) > a 2 , ∀x < 0.
Assume there exists an equilibrium (p, (x 1 , x 2 )). Then x 1 s = -x 2 s = x s for any s. There exists λ > 0 such that

λ 2 s u 1 (x s ) = 1 -α s S α 2 s u 2 (-x s )∀s ⇔ u 2 (-x s ) = λ S α 1 -α s ∀s.
Since s p s x s = 0 and p s > 0 for all s, there exists x s 0 ≤ 0, i.e -x s 0 ≥ 0. Then

a 2 = λ S α 1 -α s 0 and ∀s = s 0 , x s > 0. Hence u 2 (-x s 0 +1 ) > a 2 . This implies λ S α 1 -α s 0 +1 > λ S α 1 -α s 0 ⇒ α s 0 +1 > α s 0 . A contradiction.
Example 3 Agent 2 is risk neutral. The reward utilities satisfy

u 1 (x) = b 1 , ∀x ≤ 0, u 1 (+∞) = 0, u 1 (x) < b 1 , ∀x > 0, u 2 (x) = 1, ∀x ∈ R.
Assume there exists an equilibrium (p, (x 1 , x 2 )). Then x 1 s = -x 2 s = x s for any s. There exists λ > 0 such that

u 1 (x s ) = λ 1 -α s S α ∀s.
Since s p s x s = 0 and p s > 0 for all s, there exists x s 0 ≤ 0, i.e -x s 0 ≥ 0.

Then b 1 = λ 1 -α s 0 S α and ∀s = s 0 , x s > 0. Hence u 1 (x s 0 +1 ) < b 1 . This implies λ 1 -α s 0 +1 S α < λ 1 -α s 0 S α ⇒ α s 0 +1 > α s 0 .
A contradiction.

Remark 1 When there are short-sales constraints, the allocation set A is compact in l p (π) for any p. Indeed, suppose we have short-sales constraints x i s ≥ c s for every i, s, with c = (c 1 , c 2 , . . . , c s , . . . ) ∈ l p (π). We also have for every (x 1 , x 2 , . . . , x m ) which belongs to A:

x i s = e s - j =i x j s ≤ e s - j =i c s ≤ |e s | + (m -1)|c s |.
A is hence compact for the product topology. Suppose that the sequence (x 1 (n), x 2 (n), . . . , x m (n)) belong to A converges to (x 1 , x 2 , . . . , x m ) for the product topology when n tends to infinity. We have for every i,

c s ≤ x i s ≤ |e s | + (m -1)|c s |.
This implies that (x 1 , x 2 , . . . , x m ) ∈ l p (π). The set of individually rational allocations A is compact in l p (π). That implies the compactness of U and the existence of a quasi-equilibrium. This explain why in our paper we assume no short-sales constraints.

Concluding remarks (a) In our model with an infinitely countable number of states of nature, the sufficient conditions for the existence of an equilibrium are:

(i) Either the marginal utilites at +∞ equal zero for all the agents, (ii) or the marginal utilites at -∞ equal +∞ for all the agents.

We give examples where agents do not satisfy the conditions above. Their utility functions satisfy the no-arbitrage condition à la Werner. However there is no equilibrium.

(b) Our sufficient conditions differ from the ones in Le Van and Truong Xuan (Assumption H4) [START_REF] Van | Asset market equilibrium in L p spaces with separable utilities[END_REF], and Daher, Martins-da-Rocha, Vailakis, [START_REF] Daher | Asset market equilibrium with short-selling and differential information[END_REF] (Assumption S4). Put in our model, these conditions are stronger than ours. Indeed, they imply, for our model, that for all the agents, the marginal utilities at +∞ equal zero and the marginal utilities at -∞ equal +∞.

Appendix

Proof of Proposition 1

Take any NA-price p. There exists (x i ), λ 1 , λ 2 , . . . , λ m positive such that for all i, s: p s = λ i π i s u i (x i s ). (i) For i, j ∈ I 2 we have λ i a i π i s = λ j a j π j s , ∀s. Take the sum we have:

∞ s=1 λ i a i π i s = ∞ s=1 λ j a j π j s .
This implies λ i a i = λ j a j , hence π i = π j . (ii) We firstly prove that there exists C > 0 such that:

i∈I 1 ∞ s=1 π i s |x i s | ≤ C. Define e = e -i∈I 2 x i = i∈I 1 x i ∈ l 1 (π). For i ∈ I 1 , since a i < inf s u i (x i s ) ≤ sup s u i (x i s ) < b i , we have x i ∈ l ∞ (π). Observe that p ∈ l ∞ (π).
Choose η > 0 such that

a i < u i (x i s )(1 + η) < b i for
for all i ∈ I 1 . Then we define the price q as follows: ∀i, j ∈ I 1 ,

q s = p s (1 + η) = λ i π i s u i (x i s )(1 + η) = λ j π j s u j (x j s )(1 + η).
It follows that, for each i ∈ I 1 , there exist z i such that ∀s,

q s = λ i π i s u i (z i s ). Observe that a i < inf s u i (z i s ) ≤ sup s u i (z i s ) < b i , so z i ∈ l ∞ (π).
Observe also that ∀s, p s < q s . Denote

x + : = x if x > 0, 0 if x ≤ 0 x -: = -x if x < 0, 0 if x ≥ 0 Observe that x = x + -x -, |x| = x + +x -and u(x) = u(x + )+u(-x -)-u(0).
Now we fix N ∈ N. For i ∈ I 1 , from the concavity of the utility function u i we have

λ i N s=1 π i s u i (x i s ) -λ i N s=1 π i s u i (x i+ s ) ≥ λ i N s=1 π i s u i (x i s )(x i s -x i+ s ), λ i N s=1 π i s u i (z i s ) -λ i N s=1 π i s u i (-x i- s ) ≥ λ i N s=1 π i s u i (z i s )(z i s + x i- s ).
Therefore,

λ i N s=1 π i s u i (z i s )x i- s ≤ λ i N s=1 π i s [u i (z i s ) + u i (x i s ) -u i (x i+ s ) -u i (-x i- s )] -λ i N s=1 π i s u i (z i s )z i s + λ i N s=1 π i s u i (x i s )x i+ s -λ i N s=1 π i s u i (x i s )x i s . Define U i N (x) := N s=1 π i s u i (x s ). Note that lim N →∞ U i N (x) = U i (x). The above inequality implies N s=1 q s x i- s ≤ λ i [U i N (z i ) + U i N (x i ) -U i N (x i ) -U i N (0)] - N s=1 q s z i s + N s=1 p s x i+ s - N s=1 p s x i s ≤ λ i [U i N (z i ) + U i N (x i ) -U i N (x i ) -U i N (0)] - N s=1 p i s x i s - N s=1 q s z i s + N s=1 p s x i+ s = C i N + N s=1 p s x i+ s where C i N = λ i [U i N (z i ) + U i N (x i ) -U i N (x i ) -U i N (0)] -N s=1 p i s x i s -N s=1 q s z i s . Observe that since x i et z i belong to l ∞ (π), the limit lim N C i N exists. Hence, ∀i N s=1 (q s -p s )x i- s ≤ C i N + N s=1 p s x i s .
Thus, we have

i∈I 1 N s=1 (q s -p s )x i- s ≤ i∈I 1 C i N + i∈I 1 N s=1 p s x i s = i∈I 1 C i N + N s=1 p s e s .
Since e ∈ l 1 (π), i∈I 1 C i N + N s=1 p s e s converges. Now let N tends to infinity. Notice that U i N (x) → U i (x) for all x, and recall that

U i (x i ) ≥ U i (e i ), 11
Documents de travail du Centre d'Economie de la Sorbonne -2016.62 with x i , z i ∈ l ∞ (π). We then have lim sup

N →∞ C i N ≤ λ i [U i (z i ) + U i (x i ) -U i (e i ) -U i (0)] - ∞ s=1 q s z i s - ∞ s=1 p s x i s =: C i .
Thus,

i∈I 1 ∞ s=1 (q s -p s )x i- s ≤ i∈I 1 C i + ∞ s=1 p s e s =: C 1 + ∞ s=1 p s e s .
We also have

i∈I 1 ∞ s=1 (q s -p s )(x i+ s -x i- s ) = i∈I 1 ∞ s=1 (q s -p s )x i s = ∞ s=1 (q s -p s )e s which implies i∈I 1 ∞ s=1 (q s -p s )x i+ s = ∞ s=1 (q s -p s )e s + i∈I 1 ∞ s=1 (q s -p s )x i- s ≤ C 1 + ∞ s=1 p s e s + ∞ s=1 (q s -p s )e s = C 1 + ∞ s=1
q s e s .

Thus for

i ∈ I 1 ∞ s=1 (q s -p s )|x i s | ≤ 2C 1 + ∞ s=1 (p s + q s )e s = 2C 1 + (2 + η) ∞ s=1 p s e s = 2C 1 + (2 + η) ∞ s=1 p s e s -(2 + η) i∈I 2 ∞ s=1 p s x i s = 2C 1 + (2 + η) ∞ s=1 p s e s -(2 + η) i∈I 2 λ i a i ∞ s=1 π i s x i s = 2C 1 + (2 + η) ∞ s=1 p s e s -(2 + η) i∈I 2 λ i U i (x i ) ≤ 2C 1 + (2 + η) ∞ s=1 p s e s -(2 + η) i∈I 2 λ i U i (e i ) = C 2 , then η ∞ s=1 p s |x i s | ≤ C 2 .
Let µ i := inf s u i (x i s ) > 0, and µ :

= min i µ i . Then ∞ s=1 p s |x i s | ≥ µ ∞ s=1 π i s |x i s | which implies for all i ∈ I 1 ∞ s=1 π i s |x i s | ≤ D 1 with D 1 = C 2 /(ηµ).
For I 2 we have:

∞ s=1 π I s | i∈I 2 x i s | ≤ ∞ s=1 π I s |e s | + i∈I 1 ∞ s=1 π i s |x i s | ≤ D 2 with D 2 = ∞ s=1 π I s |e s | + |I 1 |D 1 . We take C = max{D 1 , D 2 }.
(iii) By Jensen inequality, the utility set U is bounded.

Proof of Proposition 2

(i) Consider the case where I 2 = ∅. (a) Assume u i (-∞) = +∞, for any i. Let z satisfy a 1 < u 1 (z). For i > 1, let

ζ i s = π 1 s π i s u 1 (z), ∀s. Then 1 h u 1 (z) ≥ ζ i s ≥ hu 1 (z). One can find λ i s.t. ζ i s λ i ≥ α i > a i . Define u i (x i s ) = ζ i s λ i , ∀s. Then a i < α i ≤ u i (x i s ) ≤ 1 λ i h u 1 (z), ∀s
Since b i = +∞, we have x i ∈ l ∞ (π). Obviously

λ i π i s u i (x i s ) = π 1 s u 1 (z), ∀s. (b) Assume u i (+∞) = 0 for all i. Let z satisfy 0 < u 1 (z) < b 1 . Define ζ i s as before. We have ζ i s ≤ 1 h u 1 (z). Choose λ i s.t. ζ i s λ i ≤ β i < b i .
Then define x i s as before. Using the same arguments, we have

λ i π i s u i (x i s ) = π 1 s u 1 (z), ∀s.
(ii) Now we consider the case where I 2 = ∅. If (NA) holds, let p be a (NA)-price. There exists (x i ), λ 1 , λ 2 , . . . , λ m positive such that for all i, s we have p s = λ i π i s u i (x i s ). For i, j ∈ I 2 we have λ i a i π i s = λ j a j π j s , ∀s. Taking the sum we have:

∞ s=1 λ i a i π i s = ∞ s=1
λ j a j π j s .

13 Documents de travail du Centre d'Economie de la Sorbonne -2016.62

This implies λ i a i = λ j a j , hence π i = π j . Conversely, assume that π k = π l = π , ∀k ∈ I 2 , ∀l ∈ I 2 . Assume 1 ∈ I 2 . For i ∈ I 2 , i = 1, choose λ i such that λ i a i = a 1 . Consider the case u i (-∞) = +∞, ∀i ∈ I 1 .

For i ∈ I 1 , define ζ i s :

1 h a 1 ≥ ζ i s = a 1 π 1 s π i s ≥ a 1 h
There exists λ i s.t.

ζ i s λ i ≥ α i > a i and a i < u i (x i s ) = ζ i s λ i ≤ 1 h a 1 .
Since b i = +∞, we have x i ∈ l ∞ (π). Obviously

λ i π i s u i (x i s ) = π 1 s u 1 (z), ∀s.
We use the same argument as before if u i (+∞) = 0, ∀i ∈ I 1 .

Proofs of Theorems 1, 2

The proofs require many intermediary steps. Claims 1, 2 and 3 are required for the proof of Claim 4.

Claim 1 Assume A0. Our model has an equilibrium if we add the assumption that U is compact. If X i is l p (π) with 1 ≤ p < +∞ then the equilibrium price p * is in l q (π), 1 p + 1 q = 1. If p = +∞, then p * ∈ l 1 (π).

Proof : Since U is compact and X i is l p (π) there exists an equilibrium ((x * i ), p * ) (see Dana and al [START_REF] Dana | General Equilibirum in asset markets with or without short-selling[END_REF]) with x * i ∈ l p (π). When 1 ≤ p < +∞, the price p * belongs to l q (π). When p = ∞ we will show that the equilibrium price belongs to l 1 (π). The equilibrium price can be written as p * + φ where p * ∈ l 1 (π) and φ is a purely finitely additive function. For any i, the equilibrium allocation x * i solves the problem:

max ∞ s=1 π i s u i (x i s ), s.t. ∞ s=1 p * s x i s + φ(x i ) = ∞ s=1 p * s e i s + φ(e i ).
From Theorem V.3.1, page 91, in Arrow-Hurwicz-Uzawa in [START_REF] Arrow | Programming in Linear spaces[END_REF], for any i, there exists

ζ i > 0 s.t. ∞ s=1 π i s u i (x * i s ) -ζ i ( ∞ s=1 p * s x * i s + φ(x * i )) ≥ ∞ s=1 π i s u i (x s ) -ζ i ( ∞ s=1 p * s x s + φ(x i )).
Suppose that φ = 0. Since φ ≥ 0, then φ(1) > 0 , with 1 = (1, 1, 1, . . .). Define x i (N ) as:

x i s (N ) = x * i s with s = 1, 2, . . . , N, x i s (N ) = x * i s -1 with s ≥ N + 1.
Observe that x i (N ) ∈ l ∞ (π). We have:

∞ s=1 π i s u i (x * i s )-ζ i ( ∞ s=1 p * s x * i s +φ(x * i )) ≥ ∞ s=1 π i s u i (x i s (N ))-ζ i ( ∞ s=1 p * s x i s (N )+φ(x i (N ))) ⇒ s≥N +1 π i s u i (x * i s )-ζ i ( s≥N +1 p * s x * i s +φ(x * i )) ≥ s≥N +1 π i s u i (x * i s -1)-ζ i ( s≥N +1 p * s (x * i s -1)+φ(x * i -1)) ⇒ s≥N +1 π i s u i (x i * s ) - s≥N +1 π i s u i (x * i s -1) -ζ i s≥N +1 p * s ≥ ζ i (φ(x * i ) -φ(x * i -1)) = ζ i φ(1).
Let N → ∞, the left-hand-side converges to 0. This implies φ(1) ≤ 0: a contradiction. Hence φ = 0.

Claim 2 A closed, bounded set B in l 1 (π) is compact if and only if B satisfies the following property: For all > 0, there exists N ∈ N such that for all x ∈ B we have

s≥N π s |x s | < .
Proof : Suppose that B is compact and there exists a subsequence {x(n

)} n of B, > 0 such that ∞ s=n π s |x s (n)| > , ∀n.
Without loss of generality, we can assume that x(n) converges to x in l 1 (π) or x(n) -x l 1 (π) → 0. By choosing N large enough such that x(n)-x l 1 (π) < 2 and s≥n π s |x s | < 2 for all n ≥ N . And for all n ≥ N , we have

∞ s=n π s |x s | ≥ ∞ s=n π s |x s (n)| - ∞ s=n π s |x s (n) -x s | > - 2 = 2 .
Now we suppose that for any > 0, there exists N such that ∀x ∈ B,

∞ s=N π s |x s | < .
We have to prove that for any sequence x(n) ∈ B, there exists a convergent subsequence of x(n) in l 1 (π).

Since B is bounded in l 1 (π), there exists a > 0 such that,

∀ x ∈ B, s≥1 π s |x s | ≤ a.
Since {x(n)} n belong to a compact set for the product topology, there exists a subsequence {x(n k )} k which converges to x for the product topology. In particular for all s, x s (n k ) converges to x s when k → ∞.

Fix > 0. We will prove that for k, l big enough,

x(n k ) -x(n l ) l 1 (π) < . Choose N > 0 such that for all x ∈ B, s≥N π s |x s | < 4 . Choose M such that for all n k > M we have M s=1 π s |x s (n k ) -x s | < 4 . For all n k ≥ N 0 , n l ≥ N 0 where N 0 = max{N, M } we have N 0 s=1 π s |x s (n k ) -x s (n l )| ≤ N 0 s=1 π s |x s (n k ) - x s | + N 0 s=1 π s |x s (n k ) -x s | < 2 . Then s≥1 π s |x s (n k ) -x s (n l )| ≤ N 0 -1 s=1 π s |x s (n k ) -x s (n l )| + s≥N 0 π s |x s (n k )| + s≥N 0 π s |x s (n l )| < Hence {x(n k )} k is a Cauchy sequence, then it converges in l 1 (π) topology. So B is compact in l 1 (π) topology.
The following claim is a corollary of Claim 2.

Claim 3 1. A closed, bounded set B in l 1 (π) is compact for l 1 (π)-topology if and only if it is compact for the weak topology σ(l 1 (π)), l ∞ (π)). 2. A closed, bounded set B in l p (π), p > 1 is compact for l 1 (π).

Proof : 1. Since Lemma 2 is equivalent to the Dunford-Pettis criterion, the result follows. 2. For p > 1, a closed bounded set is σ(l p (π), l q (π))-compact. But it is also σ(l 1 (π), l ∞ (π))-compact, since l p (π) ⊂ l 1 (π) and l ∞ (π) ⊂ l q (π). Apply statement 1.

Claim 4 Assume A0. If b i = +∞ for all i, then the allocation set A is l 1 (π)compact. The set U is also compact.

Proof : The idea of the proof is that, if the attainable allocation sequence does not belong to a weakly compact set, then for some state s, there will be an agent i such that x i s tends to +∞ and an agent j such that x j s tends to -∞. Then by reducing x i s and increasing x j s , the value of U i (x i ) does not diminish very much. Because b j = +∞, the value of U j (x j ) will become very large, even tends to infinity, and that leads us to a contradiction with the bounded property of U. Assume the contrary: A is not compact. Then from Claim 2, there exists a sequence {(x 1 (n), x 2 (n), ..., x m (n))} n ⊂ A, an agent i and a constant > 0 such that

∀ n, ∞ s=n π i s |x i s (n)| > . Denote for all k, v k := lim sup n→∞ U k (x k (n)). By Proposition 1, A is bounded in l 1 (π). We can suppose, without loss gen- erality, that ∞ s=n π i s |x i s (n)| → c i > 0 when n → ∞. This implies lim n ∞ s=n π i s x i+ s (n)- lim n ∞ s=n π i s x i- s (n) = c i .
The limits of these two sums exist because x i ∈ l 1 (π). We know that j =i x i s (n) = e s -x i s (n). So, for every s, ∃j such that

x j s (n) ≤ -x i s (n)-|es| m-1 .
Since there is a finite number of agents j = i, we can assume that, for simplicity, there exist i and j which satisfy two properties:

(i) ∃ E i n ⊂ N ∩ {s ≥ n}, x i s > 0 for all s ∈ E i n and lim n s∈E i n π i s x i s (n) = c i > 0.
(ii) For all s ∈ E i n x j s (n) ≤ -

x i s (n) -|e s | m -1 .
With each M > 0, define the set S i n ⊂ E i n as follows

S i n = {s : x i s (n) > |e s | + M (m -1)}.
We have an observation:

lim n E i n \S i n π i s x i s (n) = 0. Indeed lim n→∞ s∈E i n \S i n π i s x i s (n) ≤ s∈E i n \S i n π i s (|e s | + M (m -1)) ≤ ∞ s=n π i s |e s | + M (m -1) ∞ s=n π i s
which tends to zero, since e ∈ l 1 (π). Hence we have S i n = ∅ for all n big enough, and lim

n→∞ s∈S i n π i s x i s (n) = c i .
We have

x j s (n) ≤ |e s | -x i s (n) m -1 < -M.
Since π i and π j are equivalent, we can assume that lim n s∈S i n π j s x i s (n) = c j > 0. Notice that these limits do not depend on M . Define

α := min(v k , v i -u i (0)c i m-1 ) -1, (k = 1, . . . , m). Define A α the set of (x k ) ∈ l 1 (π) satisfies U k (x k ) ≥ α ∀k and
x k = e. Using the same proof as in Proposition 1, we show there exists C > 0 such that U j (x j ) < C for all (x 1 , . . . , x m ) ∈ A α . Notice that our sequence (x k (n)) belongs to A α for n large enough. Since b j = +∞ we can choose M very big such that

v j + u j (-M )c j m -1 > C.
Now consider the sequence (y 1 (n), y 2 (n), . . . , y m (n)) defined as follows

y i s (n) := x i (n) - x i s -|e s | m -1 + M with s ∈ S i n , y j s (n) := x j s (n) + x i s -|e s | m -1 -M with s ∈ S i n .
Let y k s = x k s with every k = i, j or s / ∈ S i n . Notice that i y i (n) = e, and y i s (n) ≤ x i s (n), y j s (n) ≥ x j s (n) for all s. We will prove that {U l (y l (n))} l=1,m is bounded below by α, but U j (y j (n)) is not bounded above by C. And this is a contradiction. Indeed,

U i (y i (n)) -U i (x i (n)) = s∈S i n π i s (u i (y i s (n)) -u i (x i s (n))) ≥ s∈S i n π i s u i (x i s (n) - x i s (n) -|e s | m -1 + M )(- x i s (n) -|e s | m -1 + M ) ≥ s∈S i n π i s u i (M )(- x i s (n) m -1 ) + u i (M )( |e s | m -1 + M ) s∈S i n π i s ≥ - u i (M ) m -1 s∈S i n π i s x i s (n) + u i (M )( |e s | m -1 + M ) s∈S i n π i s .
When n → ∞, the second term of the right hand side term in the inequality above tends to zero while first term tends to

-u i (M )c i m-1 . Thus, lim sup n→∞ U i (y i (n)) ≥ v i - u i (M )c i m -1 ≥ v i - u i (0)c i m -1 > α.
For n large enough, U k (y k (n)) is bounded below by α,∀k = j. Then we can estimate the limit of U j (y j (n)) when n → ∞,

U j (y j (n)) -U j (x j (n)) = s∈S i n π j s (u j (y j s (n)) -u j (x j s (n))) ≥ s∈S i n π j s u j (x j s (n) + x i s (n) -|e s | m -1 -M )( x i s (n) -|e s | m -1 -M ) ≥ s∈S i n π j s u j (-M )( x i s (n) -|e s | m -1 -M ) ≥ u j (-M ) m -1 s∈S i n π j s x i s (n) -M u j (-M ) s∈S i n π j s - u j (-M ) m -1 s∈S n |e s |π j s .
Take the limit

lim sup n→∞ U j (y j (n)) ≥ v j + u j (-M )c j m -1 > C. A contradiction. Hence A is l 1 (π)-compact.
The proof of Claim 6 requires Claim 5

Claim 5 Suppose that A is bounded and (v 1 , v 2 , . . . , v m ) is in the closure of U. Suppose that there exists a sequence {x(n)} n ⊂ A such that there exists i such that lim n U i (x i (n)) > v i , and for all j = i , lim n U j (x j (n)) ≥ v j . Then (v 1 , v 2 , . . . , v m ) ∈ U.

Proof : Fix t ∈ N arbitrarily. Let C > 0 be the upper bound of A in l 1 (π), we know that |x j t (n)| < C π j t for all j and all n. Fix some j. We define the sequence {y k (n)} k=1,...,m as follows

y k (n) = x k (n) if k = i, j, y i s (n) = x i s (n) if s = t, y i t (n) = x i t (n) -, y j t (n) = x j t (n) + . For k = i, j, lim n U k (y k (n)) = v k .
And we have

U i (y i (n)) -U i (x i (n)) = π i t (u i (y i t (n)) -u i (x i t (n))) ≥ π i t (-)u i (x i t (n) -) ≥ -π i t u i (- C π i t -)
and

U j (y j (n)) -U j (x j (n)) = π j t (u j (y j t (n)) -u j (x j t (n))) ≥ π j t u j (x j t (n) + ) ≥ π j t u j ( C π j t + ).
Since lim inf n U i (x i (n)) > v i , by choosing small enough, the sequence {y(n)} n will satisfy lim inf n U i (y i (n)) > v i and lim inf n U j (y j (n)) > v j . By induction we can find a sequence {z k (n)} n ⊂ A which satisfies lim n U k (z k (n)) > v k for all k = 1, 2, . . . , m. Hence (v 1 , v 2 , . . . , v m ) ∈ U.

Claim 6 Assume A0. If a i = 0 for all i, then U is compact.

Proof : Since the (NA) condition holds, from Proposition 1, we know that U is bounded. We will prove that U is closed. Suppose that (v 1 , . . . , v m ) belong to the closure of U and the sequence {x(n

)} n ⊂ A such that lim n U i (x i (n)) = v i .
If the sequence {x(n)} n belongs to a compact set of l 1 (π), without loss of generality, we can suppose that lim n x i (n) = x i in this topology. Since U i is continuous, we have U i (x i ) ≥ v i for all i. Thus (v 1 , . . . , v m ) ∈ U.

If the sequence {x(n)} n does not belong to a compact set, we can suppose that there exists c > 0 such that for an agent i

lim n→∞ ∞ s=n π i s |x i s (n)| = c.
As in the proof of Claim 4, we can choose a pair (i, j) which satisfies the two properties:

(i) ∃ E i n ⊂ N ∩ {s ≥ n}, x i s > 0 for all s ∈ E i n and lim n s∈E i n π i s x i s (n) = c i > 0. (ii) For all s ∈ E i n x j s (n) ≤ - x i s (n) -|e s | m -1 .
With each M > 0, define the set S i n ⊂ E i n as follows

S i n = {s : x i s (n) > |e s | + M (m -1)}.
We have an observation:

lim n E i n \S i n π i s x i s (n) = 0. Indeed lim n→∞ s∈E i n \S i n π i s x i s (n) ≤ s∈E i n \S i n π i s (|e s | + M (m -1)) ≤ ∞ s=n π i s |e s | + M (m -1) ∞ s=n π i s
which tends to zero, since e ∈ l 1 (π). Hence we have S i n = ∅ for all n big enough, and lim

n→∞ s∈S i n π i s x i s (n) = c i .
We have

x j s (n) ≤ |e s | -x i s (n) m -1 < -M.
Since π i and π j are equivalent, we can assume that lim n s∈S i n π j s x i s (n) = c j > 0. Notice that these limits do not depend on M .

Define

α := min(v k , v i -u i (0)c i m-1 ) -1, (k = 1, . . . , m). Define A α the set of (x k ) ∈ l 1 (π) satisfies U k (x k ) ≥ α ∀k and
x k = e. From Proposition 1 we know that there exists C > 0 such that U j (x j ) < C for all (x 1 , . . . , x m ) ∈ A α . Notice that our sequence (x k (n)) ∈ A α for n large enough. Fix > 0. Since u i (+∞) = 0 we can choose M > 0 such that u i (M ) < (m -1) /c. By similar arguments as in the proof of Claim 4, we can construct the sequence (y k (n)) such that:

lim n→∞ U i (y i (n)) ≥ v i - u i (M )c i m -1 lim n→∞ U j (y j (n)) ≥ v j + u j (-M )c j m -1 lim n→∞ U k (y k (n)) = v k for all k = i, j
with c i , c j > 0 and c i < c and c i and c j do not depend on M .

So, for n large enough, U i (y i (n)) > v i -, and for all k = i, j, U k (y k (n)) > v k -whereas lim n U j (y j (n)) = v j + u j (-M )c j m-1 > v j + u j (0)c j m-1 > v j . Let → 0 and by applying Claim 5, we have (v 1 , v 2 , . . . , v m ) ∈ U.

Proof of Theorem 1

This is a direct consequence of Claims 4, 6 and 1.

The proof of the last theorem, Theorem 2, requires Claims 7 and 8.

Claim 7 Assume A0. Assume that there is only one risk neutral agent, I 2 = {i 0 }. If a i = 0 and b i = +∞ for all i ∈ I 1 , then (NA) holds, and U is compact.

Proof : Condition (NA) holds as a direct consequence of Claim 1. Now we prove that U is compact. Suppose that the feasible sequence {x n } satisfies lim n U i (x i n ) = v i for every i. We prove that (v 1 , . . . , v m ) ∈ U.

First case: the sequence {x n } belongs to a compact set, we have (v 1 , v 2 , . . . , v m ) belongs to U.

Second case: the sequence {x n } does not belong to a compact set. By using the same arguments in the proof of Proposition 4, there exist i and j which satisfy two properties: (i) ∃ E i n ⊂ N ∩ {s ≥ n}, x i s > 0 for all s ∈ E i n and lim

n s∈E i n π i s x i s (n) = c i > 0.
(ii) For all s ∈ E i n x j s (n) ≤ -

x i s (n) -|e s | m -1 .
If j = i 0 , since b j = +∞, by using the same arguments as in the proof of Proposition 4 we have a contradiction. This implies j = i 0 . Hence i = i 0 , and we have a i = 0. By using the same arguments in the proof of Proposition 6, we have (v 1 , . . . , v m ) ∈ U.

Hence U is compact.

Claim 8 Let f 1 , . . . , f n , be n vectors in l p (π), and p ∈ l ∞ (π). Take any x ∈ l p (π) such that p • x = p • i f i . Then there exists x 1 , . . . , x n in l p (π) such that i x i = x and p • x i = p • f i for all i.

Proof : This is true for n = 1. Suppose that the claim is true for n -1. Take any x 1 such that p • x 1 = p • f 1 . We have p • (x -x 1 ) = p • n i=2 f i . Using the hypothesis of induction, there exists x 2 , . . . , x n such that n i=2 x i = x -x 1 and p • x i = p • f i for all i.

Proof of Theorem 2

In the proof of Proposition 2, we have π i = π j = π I and λ i a i = λ j a j = ζ, for all i, j ∈ I 2 . For x I ∈ l 1 (π) satisfying x I = i∈I 2 x i , with x i ∈ l 1 (π), define

U I (x I ) = i∈I 2 λ i U i (x i ).
We consider now the economy E I with |I 1 | + 1 agents, |I 1 | agents who are risk averse, with endowment e i , utility function U i , and the last agent (denoted by I) with endowment e I = i∈I 2 e i , utility function U I . It is easy to verify that agent I is risk neutral, with

U I (x I s ) = ζ ∞ s=1 π I s x I s ,
and the new economy satisfies (NA) condition. By Claim 7, this economy has a compact rational utility set and have an equilibrium, denote by (p * , x * ). For all i ∈ I 1 , x * i is the solution to By Claim 8, for all i ∈ I 2 there exists x * i ∈ l p (π) such that i∈I 2 x * i = x * I , and p * • x * i = p * • e i , ∀i.

Fix i ∈ I 2 . Take x i such that U i (x i ) > U i (x * i ). We prove that p * • x i > p • e i . Indeed, we have

U i (x i ) = ∞ s=1 a i π i s x i s = 1 λ i ∞ s=1 λ i a i π i s = 1 λ i ∞ s=1 p * s x i s .
Hence U i (x i ) > U i (x * i ) implies p * • x i > p * • x * i = p * • e i . From Claim 1, the equilibrium price p * is in l q (π) with 1 ≤ p < +∞, 1 p + 1 q = 1 and if p = +∞, then p * ∈ l 1 (π). We have proved that (p * , (x * i ) i ) is an equilibrium of the model.

Conversely, suppose that the model has an equilibrium, then by using Proposition 2, we have (NA). The proof of Theorem 2 is complete.

  If U I (x i ) > U I (x * I ), then p * • x i > p * • e I .By the same arguments as in the proof of Claim 1, there existsλ * i > 0, λ I > 0 such that for i ∈ I 1 , p * s = λ i π i s u i (x * i s ). For I, we have p * s = ζπ I s = λ i a i π I s , ∀i ∈ I 2 , ∀s. The function U I is strictly increasing, so p * • x * I = p * • e * I = p * • i∈I 2 e i .

Documents de travail du Centre d'Economie de la Sorbonne -2016.62

Documents de travail du Centre d'Economie de la Sorbonne -2016.62


N Jamond

P Chrétien

F Houzé

L Lu

L Largeau

O Maugain

L Travers

J C Harmand

F Glas

E Lefeuvre

M Tchernycheva

N Gogneau
email: n.gogneau@lpn.cnrs.fr

Piezo-Generator Integrating Vertical Array of GaN Nanowires

Keywords: GaN nanowires, Piezoelectricity, Piezo-generator, Energy harvesting

We demonstrate the first piezo-generator integrating a vertical array of GaN NWs. We perform a systematic multi-scale analysis, going from single wire properties to macroscopic device fabrication and characterization, which allows to establish for GaN NWs the relationship between the material properties and the piezo-generation and to propose an efficient piezo-generator design. The piezo-conversion of individual MBE-grown p-doped GaN NWs in a dense array is assessed by atomic force microscopy equipped with Resiscope module yielding an average output voltage of 228 ± 120 mV and a maximum value of 350 mV generated per NW. In the case of p-doped GaN NWs, the piezo-generation is achieved when a positive piezo-potential is created inside the nanostructures, i.e. when the NWs are submitted to a compressive deformation. The understanding of the piezo-generation mechanism in our GaN NWs gained from AFM analyses is applied to design a piezogenerator operated under compressive strain. The device consists of NW arrays of several square millimeters in size embedded into spin-on glass with a Schottky contact for rectification and collection of piezo-generated carriers. The generator delivers a maximum power density of ∼12.7 mW/cm 3 . This value settles the new state of the art for piezogenerators based on GaN NWs and more generally on Nitride NWs, and offers promising prospects for the use of GaN NWs for high-efficiency ultra-compact energy harvesters.

Introduction

A new worldwide challenge consists in the development of autonomous micro-devices, such as sensors for environment or nomad electronics. Today, most of these micro-devices are still powered by batteries, which present the issues of a limited lifetime and a difficulty of miniaturization. Thanks to recent technological progress leading to the reduction of the power consumption of the micro-devices down to the mW or even to the µW level, the energy harvesting systems appear as a promising power source. Among the alternative sustainable energy resources, the mechanical deformations (e.g. vibrations, friction, body movements or sound) present the advantages of being ubiquitous, available at all time and more accessible than solar and thermal energies. The generators harvesting these ambient energies, found in the vicinity of the device, are based on piezoelectric materials [1][2].

One-dimensional nanostructures, such as nanowires (NWs) presenting piezoelectric properties, have recently emerged as excellent candidates to fabricate novel ultra-compact and efficient piezoelectric generators. Thanks to their superior mechanical properties [3][4] (larger elastic deformation, higher flexibility and resistance), higher sensitivity to applied force [4][5] and higher piezoelectric response [6][7][8] with respect to the bulk material, the NWs present key advantages to fundamentally improve the generator performance.

Since the first demonstration of electrical energy generated by bending single ZnO NWs [9], the mechanical to electrical energy conversion has been demonstrated for other piezoelectric 1D-nanostructures, such as CdS [10], CdSe [11], PZT [12], BaTiO 3 [13], KNBO 3 [14] etc. These studies of nanogeneration of individual semiconductor NWs have naturally lead to the development of micro-and millimetric NW-based piezo-generators. To avoid confusion existing in the literature, in this paper we define a "nanogenerator" as a single NW having the ability to convert a mechanical deformation into electrical energy; while we call "NW-based piezo-generator" a macroscopic device integrating an array of 1D nanostructures (i.e. an array of nanogenerators) and generating piezoelectric energy in response to its deformation.

Piezo-generators integrating 1D-nanostructures have been intensively studied with the development of different designs, such as generators based on vertically aligned NWs [15][16][17][18] or horizontal NWs [17,19] array, either on flexible [15,19] or hard [16][17][18] substrates. [15][16][17][18][19], PVDF nano-materials [20], PZT nanofibers [21] and GaN NWs [22][23] have been developed with power density in the µW-mW/cm 3 range. From ZnO NWs based piezo-generator prototypes, self-powered nano-systems without external battery have been demonstrated including pH and UV [24], chemical [25] and biological sensors [26], commercial light-emitting diodes (LED) [27], liquid-crystal-display (LCD) screen [28] or wireless radio frequency data transmitter [29].

Generators based on ZnO NWs

Compared to ZnO, the III-nitride NWs have been much less studied for piezo-generation in spite of their high piezoelectric coefficients [30]. The concept of nanogenerator for IIInitride nanostructures has been demonstrated for the first time in 2007 by bending GaN nanorods [31]. More recently, their strong piezo-generation responses have been reported, with output voltages ranging from several millivolts to 440 mV for GaN [22] and to 1 V [33] for InN single NWs. These outputs largely exceed those generated by other piezoelectric nanostructures and especially by single ZnO NWs (e.g. 45 mV [34] and 80 mV [35] for respectively n-doped and p-doped ZnO, which are the most widespread nanomaterials integrated into NW-based piezo-generators). To date, several groups have presented AFM characterizations of single piezoelectric properties of nitride NWs [32][33][36][37][38]. On the basis of these nanogenerator demonstrations, obtained by atomic force microscopy, maximum power densities reaching few hundred of mW/cm 2 have been estimated [32,36], which demonstrates the high promise of the III-Nitride nanostructures for efficient piezo-generators. However, despite these promising results, only few NW-based piezo-generators integrating nitride NWs have been reported in the literature, delivering power density not exceeding few hundred of µW/cm 3 (calculated from the published data) [22][23].

In this paper, we demonstrate the first piezo-generator integrating dense array of vertical GaN NWs as the converting medium. The array of self-assembled NWs was grown by PA-MBE on Si(111) substrate. The NW piezo-generation was first characterized at the nanoscale by using AFM microscope equipped with a modified Resiscope module yielding an average output voltage of 228 ± 120 mV generated per NW (the maximum value being 350 mV). We show that in the studied case of N-polar p-doped GaN NWs, the piezo-generation occurs when the nanostructures are submitted to a compressive strain, i.e. when a positive piezopotential is created inside the NWs. The fine understanding of the relationship between the material properties and the piezoelectric generation gained from AFM analyses has been applied to design a piezo-generator operating under compressive strain. The fabricated device consists of NW arrays of several square millimeters in size embedded into spin-on glass with a Schottky contact for rectification and collection of piezo-generated carriers. The maximum power density produced by this prototype is 12.7 mW/cm 3 , which largely exceeds the one delivered by the previously reported piezo-generators based on Nitrides NWs [22][23]. The delivered power is reproducible (i.e. it is obtained repeatedly for each loading cycle and for various frequencies). This result establishes the new state of the art for GaN NW-based piezogenerator performance.

Methods

Self-assembled GaN NWs were grown by plasma-assisted molecular beam epitaxy (PA-MBE) on n-type doped Si(111) substrate. The NW growth was performed at 760 °C under Nrich conditions with an N/Ga ratio of about 1.36, the active nitrogen being supplied by a radio-frequency plasma cell. During the growth, the Mg cell was kept at 400 °C in order to induce a p-type residual doping of GaN NWs. Prior to the growth of GaN NWs, a 2.5 nmthick AlN buffer layer was deposited, allowing for a better control of the NW nucleation, density and orientation [39][40][41]. This AlN interlayer could be considered as a potential barrier preventing the circulation of piezoelectric charges, however as discussed below this is not the case. To form this thin AlN interlayer, the deoxidized Si(111) substrate is firstly exposed at 600°C to an Al flux and secondly only to a nitrogen flux (by switching the Al/N shutters), forming then the AlN layer [42]. The pre-deposition of the Al adatoms on the Silicon surface induces the formation of the Al-Si eutectic of composition x = 0.126 [START_REF] Massalski | Binary Alloy Phase Diagrams[END_REF]. During the nitridation phase, the Si adatoms are incorporated into the AlN layer, producing an efficient ndoping and thus yielding a conductive behavior (which is supported by the output voltages measurement presented below). The growth of this AlN interlayer following this procedure is beneficial for piezo-conversion since it prevents the formation of the insulating SiN layer between the GaN NWs and the Si(111) substrate, which annihilates the piezoelectric generation.

Figure 1a shows the typical p-doped GaN NW morphology. The GaN NWs grown under these conditions present a hexagonal cross section bounded by {10-10} planes [44] and they are vertically aligned as demonstrated by the XRD pole figure (Fig. 1c). Pole figures, which provide the in-plane orientation of NWs, were recorded by using the Bragg reflection of GaN (10)(11) planes which are inclined by 62° from the [0001] direction (instrumental resolution of the measurements was 0.5°). The high diffracted intensity at the sample inclination for α = 62° established that the growth direction of the wire (c-axis) is normal to the Si(111) substrate surface.

The GaN NWs are characterized by a height of 1 µm ± 120 nm, a diameter of 45 nm ± 20 nm and a density of the order of 3x10 9 NWs/cm². The NW heights have been determined from AFM measurements in tapping mode, while the diameter and the density have been extracted from SEM images in top-view.

It is important to remind, that the GaN NWs are characterized by a wurtzite crystal structure, which does not have inversion symmetry. Within wurtzite material, the (0001) and (000-1) surfaces, namely metal polarity and N polarity respectively, are not equivalent and present different properties, such as different surface morphologies, optical and kinetic properties [45][46][47]. The polarity of the material is of crucial importance for the piezo-generation properties, since it sets the orientation of the internal polarization. The latter consists of the sum of the spontaneous polarization (P SP ) and of the piezoelectric polarization (P PZ ), which results from the relative displacement of the Ga cations with respect to the N anions along the deformation direction when a stress is applied to the NW. In order to determine the polarity of our p-doped GaN NWs, we have dipped the nanostructures into a KOH aqueous solution. This etching treatment is known to reveal the polarity of GaN surfaces. In fact, hydroxide ions etch the Ga atoms but are repelled by the electronegativity of the nitrogen dangling bond. In consequence, the morphology of Ga-polar surfaces is preserved [48], while the N-polar surfaces are modified with the appearance of tilted {10-1-1} planes [49]. Figure 1d presents SEM image of the morphology of vertical array of p-doped GaN NWs etched for 5 min in 0.5 M KOH solution at 40°C. The observed pyramidal features on the top facet of all the NWs are the typical signature of etched N-polar surfaces. We can thus conclude that in our case, the GaN NWs are characterized by a Nitrogen polarity as previously evidenced by CBED observations and KOH etching [42]. In these conditions, P SP , which is oriented from N to Ga atoms, is parallel to the growth direction and P PZ is anti-parallel (parallel) to the Z axis under compression (stretching) along the c-axis. Consequently, the resulting piezoelectric potential created inside the nanostructures is positive under compressive strain and negative under tensile strain [50]. The piezoelectric properties of GaN NWs have been characterized with an atomic force microscope (AFM) equipped with a modified Resiscope module [START_REF] Schneegans | French patent[END_REF]. This technique combines the advantages of the nanometer scale resolution of the AFM and the real-time electrical measurements of the Resiscope. During scanning over the array of vertical NWs, the conductive AFM tip induces a local bending of the nanostructures. In response to this deflection and due to the piezoelectric effect, the NWs generate a voltage which is detected through the conductive AFM tip. We note that the used AM tip was coated with PtIr since this materials brings the best harvesting efficiency among the others available coatings [START_REF] Jamond | [END_REF].

Throughout the measurements, both the topographic and the electrical signals were recorded continuously and simultaneously. In this measurement method, the external electrical circuit is connected via an ohmic contact formed between the NWs [START_REF] Jacopin | [END_REF] and the substrate, and a Schottky contact formed between the NW top and the AFM tip [50]. No external voltage is applied and the voltage waveform generated by the NW is observed across a load resistance

R L of 1 GΩ.
The piezo-generator device integrating the vertical array of GaN NWs has been tested under bending. The bending was applied by a cylinder localized below the center of the device inducing a vertical movement, while the extremities of the device were clamped. The bending has been measured with a Keyence laser sensor, which allows a real-time monitoring of the displacement of the piezo-generator center, while the force applied by the cylinder on the sample has been computed by simulation from the displacement measurement. Finally, the electrical measurements have been performed with a Keysight MSOX3034T Oscilloscope, with an internal impedance of 1 MΩ, by connecting respectively the ground and the input of the oscilloscope to the ohmic and Schottky contacts of the piezo-generator.

Results and Discussion

Piezo-generator performance depends primarily on the piezoelectric properties of the active material, which in the present case consists of a layer of vertical GaN NWs. In order to take full advantage of the piezo-conversion capacity of the NWs and to choose the correct design, it is of crucial importance to assess the properties of these nanostructures. To this end, the piezoelectric properties of the N-polar GaN NWs have been characterized with the AFM-Resiscope [START_REF] Schneegans | French patent[END_REF]. Figure 2 presents the topographic image and the corresponding 3D image of output voltage peaks from an array of GaN NWs recorded for a constant normal AFM tip force of 149 nN. We note here that this constant normal force is much higher than the one used for characterizing free standing ZnO NWs (5 nN) [9]. In fact, our GaN NWs are characterized by a high flexibility. In order to avoid folding them completely down to the substrate which would lead to a leakage of piezoelectric charges and thus to an inhibition of the piezo-conversion, we have partly mechanically consolidated them [32]. Consequently, a higher deflection force is needed to bend the partly consolidated NWs to a similar degree of deformation as the free-standing nanostructures. Each peak corresponds to the piezo-response of an individual NW. We note that since the GaN NWs are probed with the same constant normal AFM tip force during a scan, the output signal distribution reflects the dimensional dispersion of the NWs (resulting from the self-assembled growth mode used to synthesize them). Each output peak exhibits a positive voltage that indicates a p-type conductivity of the nanostructures [54] (in agreement with the growth conditions). From the statistical analysis of the 3D output voltage mapping, we measure that the GaN NWs generate an average output voltage of 228 ± 120 mV and a maximum of 350 mV per NW. These values largely exceed the maximum voltage generated by single ZnO NWs (45 mV [34] and 80 mV [35] respectively for n-doped and p-doped ZnO NWs), and demonstrates the promise of GaN nanostructures to develop high-efficient piezo-generators.

Figure 2: (a) Topographic image and (b) corresponding 3D output voltages collected by AFM-Resiscope on p-type GaN NWs for a constant normal force of 149 nN. The average and maximum output voltages generated by the GaN NWs are respectively equal to 228 ± 120 mV and 350 mV.

The development of high-efficient NW-based piezo-generators requires a fine understanding of the piezoelectric conversion mechanisms in individual NWs. With the nanometer scale resolved AFM technique, which allows the simultaneous recording of topographic and electrical signals, it is possible to directly correlate the morphology of a given bended NW with its generated voltage peak. This correlation has been established by Su et al. [31] and Gogneau et al. [50], then highlighting that the piezo-generation of N-polar ndoped GaN nanostructures is always observed when the AFM tips gets in contact with the stretched side of the bended GaN nanostructures, i.e. when a negative piezo-potential is created inside the nanostructures. By considering this mechanism, the piezo-generation of Npolar GaN NWs but with a p-type doping should be detected on the compressive side of the NWs. In order to verify this point, we have superimposed topography and voltage profiles together with a schematic representation of the tip position on the NW (Fig. 3a). Hence, we observe that the output signal is generated only when the AFM tip is in contact with the compressed side of the NW. This observation indicates that in the case of N-polar p-type GaN NWs, the Schottky diode formed at the NW top/AFM tip interface is positively biased when the nanostructure is characterized by a compressive strain and thus by a positive piezopotential [50,54]. Hence, only under compression, the piezoelectric NW (nitrogen-polar and p-doped) acts as a generator. Now we transpose this piezo-conversion mechanism to a NWbased piezo-generator, in which the array of vertical NWs is submitted to an axial compression. The underlying piezo-generation mechanism is schematized in figure 3b. When the axial compression is applied (straining state), the Schottky diode formed at the NW/electrode interface is forward biased. The electrons from the external circuit can thus flow into the semiconductor in order to compensate the positive piezo-potential. By contrast, when the strain is released, the Schottky contact is reverse biased, preventing the electrons to flow back through the top electrode and force them to the external circuit via the ohmic contact, insuring electrical neutrality of the unstrained semiconductor crystals.

Figure 3: (a) Corresponding topographic and electric profiles of GaN NWs characterized by AFM-Resiscope. The output signal (in red) is detected when the tip is in contact with the compressed side of the NW (represented in blue). The diode is positively biased when the nanostructure is characterized by a compressive strain inducing a positive piezo-potential inside the NW. We note here that on topographic profile, the NW diameter is oversized in comparison with the real one due to the tip/surface convolution. (b) Schematic illustration of the electrical generation mechanism for an array of vertical N-polar p-doped GaN NWs involved in a piezo-generator device, in the case of strained state (axial compression) and strain releasing.

The performance of NW-based piezo-generators is determined by the ability of the active layer (array of NWs) to efficiently convert a mechanical deformation into an electrical energy, by the efficiency of the energy harvesting, and more generally by the design of the device.

Concerning the first point, in this paper we use high density N-polar p-type GaN NWs, which are characterized by a high piezoelectric response (with outputs reaching 350 mV per NW in figure 2). Regarding the second point, in a recent study we have shown that the use of Platinum-based Schottky contact is optimal to harvest the energy piezo-generated by GaN NWs [START_REF] Jamond | [END_REF]. Therefore, in the present paper we have used Pt-based contact, which we think contributes to increase the output power of the demonstrated device. Regarding the last point, we have established the relationship between the material properties and the piezo-generation mechanism. Namely in the case of our N-polar p-type GaN NWs, the piezo-generated energy is detected when the nanostructures are submitted to a compression strain state (figure 3).

Hence, the device architecture, integrating N-polar p-type GaN array and relying on a compressive strain, was chosen. Figure 4a describes the fabrication process of a NW-based piezo-generator integrating a dense array of N-polar p-doped GaN NWs. First, the as grown GaN NW array was embedded into a hydrogen silsesquioxane resist (HSQ) [55] by spincoating and then thermally cured at 400 °C for 1h under nitrogen at atmospheric pressure. The choice of the HSQ is motivated by the fact that the silica matrix has a strong mechanical adherence to the NWs. It thus insures the transmission of the applied deformation to the nanostructures in a homogeneous way, while conferring a mechanical stability to the ensemble. Second, reactive ion etching (RIE) with SF 6 and CHF 3 reactive gases was used to etch-back the HSQ and to uncover the top portion of the NWs (∼100 nm). Then, the Schottky contact, consisting of 5 nm of Cr and 200 nm of Pt, was deposited through a shadow-mask on the top of the NWs. The Pt based-Schottky contact was chosen since it provides a diode low reverse leakage while the barrier height remains moderate, which allows to efficiently harvest the generated piezo-energy, as previously demonstrated [START_REF] Jamond | [END_REF]. Finally, the ohmic contact, consisting of a bilayer Ti/Au (10 nm/150 nm), has been deposited directly on the Si(111) substrate. Figure 4a The piezo-generator device integrating the vertical array of GaN NWs has been tested by compressive solicitation. In order to homogeneously strain the active layer, the generator is bended by inducing a vertical movement via a cylinder with a controlled displacement. We would like to stress that the solicitation force used to characterize the NW-based piezogenerator (maximum 1.5 N) does not induce any damage in the Si(111) substrate. No cracks or cleavage marks have been observed by optical microscopy and SEM after solicitation tests. We also note that no degradation of electrical properties has been evidenced after about 100 bending cycles. The figure 5a schematizes the solicitation mode. Due to the difference of thickness between the substrate (350 µm) and the active layer (∼ 1 µm) and due to the strong mechanical adherence of the HSQ matrix to the NWs, the lateral bending of the device induces stretching of the matrix and, as a consequence of the volume conservation, a thinning of the active layer. This matrix deformation is transmitted to the embedded GaN NWs leading to their homogeneous compression, as illustrated on figure 5b. We note that under these mechanical solicitation conditions, one cannot exclude that in addition to the axial compression induced by the mechanical thinning of the HSQ matrix a small lateral bending of the nanostructures can be induced. The lateral NW curvature leads to a piezo-potential distribution evolving between a negative value on the stretched side of the NW and a positive value on the compressed side of the NW [9,50]. However, the Schottky contact of the piezo-generator covering the entire NW top, a conducting channel would be created between the both NW sides, leading to the neutralization of the piezoelectric charges and thus no output voltage could be measured. Therefore, we can assume that the piezo-potential induced by small lateral NW bending does not contribute to the generation in the present device architecture [9].

Figure 5c presents the output voltage generated by the GaN NW-based piezo-generator synchronized with the magnitude of the force applied to bend the device. The piezo-generator has sustained a large amount of bending cycles (more than 10 3 ), with a frequency ranging from 1 to 6 Hz and a maximum applied force of about 1.5 N. The piezo-generated voltage is clearly correlated with the applied deformation.

For low frequency solicitation, a signal drop is observed when the applied force reaches its maximum. This behavior may be due to the transitory nature of the generation and to the low frequency of the solicitation. The current crossing the device (translated into the measured output voltage) is proportional to d(ɛ)/dt, where ɛ is the deformation of the device.

When the deformation reachs its maximum, d(ɛ)/dt becomes zero and thus the measured output should decrease due to discharging of the generator through the load circuit. This hypothesis is supported by the fact that at 3 Hz the signal drop is more pronounced than at 6 Hz. Further investigations and modeling are currently in progress in order to explain the observed signal time variation. The observed square-like signal in response to a sinusoidal force can be understood by the piezoelectric mechanisms into play. At the beginning of the bending cycle, for small deformations, the compression of the active layer is not sufficient to create the internal piezoelectric potential necessary for carriers to cross the Schottky barrier [37]. For higher deformation, the piezo-potential becomes sufficient to open the Schottky barrier and thus piezoelectric signal can be measured. However, the generation does not increase proportionally to the NW deformation as demonstrated by X. Xu et al. [37], but grows faster than the deformation (square output signal versus sinusoidal signal), until reaching the saturation. This behavior is directly correlated to the increase of the piezo-potential amplitude inside the nanostructure with the deformation state and to its influence on the Schottky diode.

In fact, the positive piezo-potential at the semiconductor side effectively reduces the Schottky barrier height, as illustrated by the energy band diagram in figure 5d [56][57]. This phenomenon gives rise to an increase of voltage collection at the interface and thus to a higher piezoelectric energy harvested. To explain the observed saturation of the output voltage, a saturation of the internal electric field inside the NWs resulting from the saturated rotation of electric dipoles [START_REF] Donald | Engineering Analysis of Smart Material Systems[END_REF] or a possible limitation of the harvested energies though the Schottky contact as observed previously [START_REF] Jamond | [END_REF] can be considered. Further investigations are in progress to clarify this point.

Finally, whatever the solicitation conditions (number of cycles and frequency in the 1 to 6 Hz range), the GaN NW-based piezo-generator always exhibits a saturated maximal output voltage of about 200 mV. By considering the square-like signal of the output, we determine the average output power generated by our device in one stimulation event by using the following relation:

L i Average R V 2 2 = Ρ
, while the maximum power can be calculated following the equation:

L i Max R V 2 = Ρ
, where i V is the output voltage generated by the active layer and L R is the load resistance used in the measurement ( Ω = M R L 1

). Knowing that the solicited piezoelectric active region has a cylinder shape with a diameter of 0.2 cm and a height of 1 µm (Fig. 4c), the effective volume,V PZ , is of the order of 3.14x10 -6 cm 3 . The piezoelectric output being equal to 200 mV by this active volume, we estimate an average output power density of 6.35 mW/cm 3 , and a maximum output power density of about 12.7 mW/cm 3 . These two values exceed largely the power density generated by the previously reported piezogenerators based on nitride NWs [22][23] and establish the new state of the art for piezogenerators based on III-N NWs.

These very promising results enable to envision various practical applications of nitride NW-based energy harvesting devices. In particular, they would be very interesting in medical applications such active implants, in which they could profitably replace batteries by bringing significant lifetime extension of the implants [START_REF] Deterre | [END_REF]. The very small power needs of active implants, about 1 µW in the case of the last generation leadless pacemakers, would be attainable using only 0.157 mm3 nitride NW active material. As shown here, nitride NWs are capable to generate power from very low mechanical frequencies, making this active material perfectly compatible with the typical human mechanical frequency spectrum [60]. Moreover, the generated power exhibits very interesting stability versus the frequency variations.

According to state of the art of nanopower energy harvesting electronic interface circuits, piezoelectric voltages as low as 200 mV can be stepped-up to applications requirements, typically in the range of 1.2 V to 5 V, with more than 80% energy efficiency [61].

Conclusion

In conclusion, we have demonstrated the fabrication of the first piezo-generator integrating vertical array of GaN NWs. The piezoelectric properties of the GaN NWs array have been investigated by AFM Resiscope, yielding an average output voltage of 228 ± 120 mV and a maximum voltage of 350 mV per NW. The comparison between the topography and output voltage profiles allowed us to highlight that the piezoelectric conversion in the Npolar p-doped GaN NWs is achieved when a compressive strain and thus a positive piezopotential are created inside the nanostructures. Based on the single NW characterizations, we have designed a NW-based piezo-generator integrating vertical array of GaN NWs operating under compressive strain. This prototype delivers a maximum power density of 12.7 mW/cm 3 for a large number of compressive cycles using different frequencies from 1 to 6 Hz. This

Figure 1 :

 1 Figure 1: Tilted (a) and cross-section (b) SEM images of Mg-doped GaN NWs grown by PA-MBE on Si(111) substrate. (c) X-ray pole figures for GaN NWs recorded by using the Bragg reflection of GaN (10-11) planes. The high diffracted intensity at the sample inclination for 62° from the [0001] direction demonstrates that the growth direction of the wire is normal to the substrate surface. (d) SEM image of Mg-doped GaN NWs after dipping into KOH etchant. The pyramidal top facet of the NWs is a signature of N-polar surfaces.

  illustrates the above-describe piezo-generator fabrication steps. The effective size of the devices and SEM images illustrating the different fabrication steps are shown in figures 4b and 4c.

Figure 4 :

 4 Figure 4: (a) Schematization of the piezo-generator fabrication process. (b) Optical image of the processed piezo-generator. (c) Schematic of the device and SEM images illustrating each step of the fabrication process.

Figure 5 :

 5 Figure 5: (a) Schematic of the setup used to characterize the GaN NW-based piezo-generator. The lateral bending of the generator induces a thinning and thus a compression of the active layer as illustrated in (b). (c) Output voltages generated by the piezo-generator as a function of its deformation for a bending cycle rate of 3 Hz (top) and 6 Hz (bottom). (d) Band diagram of the interface between the top of GaNNWs and the Pt electrode. The red dashed lines illustrate the modulation of the Schottky barrier height due to the induced piezo-potential.

Acknowledgments

This work was supported by the by EU Horizon 2020 ERC project "NanoHarvest" (Grant 639052), the French National Research Agency though the GANEX program (ANR-11-LABX-0014) and the Region Ile-de-France in the framework of DIM Nano-K and ESSEN project. The authors are also grateful to Hezhi Zhang for fruitful discussions.

measured power density defines the new state of the art and offers promising prospects for the development of high-efficiency and ultra-compact piezo-generators based on GaN NWs.


Hezhi Zhang
email: hezhi.zhang@u-psud.fr

Agnès Messanvi

Christophe Durand

Joël Eymery

Pierre Lavenus

Andrey Babichev

François H Julien

Maria Tchernycheva

InGaN/GaN core/shell nanowires for visible to ultraviolet range photodetection

Keywords: core/shell nanowire, InGaN/GaN, photodetector, MOVPE

We report on the fabrication and characterization of single nitride nanowire visible-to-ultraviolet p-n photodetectors. Nitride nanowires containing 30 InGaN/GaN radial quantum wells with 18% indium fraction were grown by catalyst-free metal-organic vapour phase epitaxy. Single nanowires were contacted using optical lithography. As expected for a radial p-n junction, the current-voltage (I-V) curves of single wire detectors show a rectifying behavior in the dark and a photocurrent under illumination. The detectors present a response in the visible to UV spectral range starting from 2.8 eV. The peak responsivity is 0.17 A/W at 3.36 eV. The on-off switching time under square light pulses is found to be below 0.1 sec.

1 Introduction InGaN/GaN two-dimensional quantum well (QW) superlattices have been intensively studied as materials for visible and ultraviolet light emission and detection [1][2][3] . They are also investigated in view of photovoltaic conversion 4,5 . However, the performance of InGaN devices is negatively affected by the presence of a large number of threading dislocations 6,[START_REF] Jain | Overstraeten[END_REF] and by a strong internal polarization field in c-oriented thin films 8,9 . Nanowires (NWs) can solve these two major problems. The dislocation density is drastically reduced due to the free lateral surface, which allows efficient relaxation of the misfit strain. For the core/shell QWs grown on the non-polar mplane facets the potential profile is not distorted by the internal field leading to a good overlap between the electron and hole wavefunctions. In addition, the NW design presents promising advantages for photodetectors 10,11 because of independent optimization of light absorption and carrier collection, efficient light trapping and thus a possibility to reduce the active material volume without loosing efficiency. Previously, NW-based core/shell n-i-n InGaN/GaN detectors have been investigated 12 . Thanks to the high photoconductive gain, these devices exhibit a very high sensi-tivity (in the 10 4 A/W range), however their operation speed is low (return to equilibrium time constants are in the range of tens to hundreds of seconds). Core-shell p-n GaN single nanowires containing a thick (50 nm) InGaN layer have been also used as photodetectors in a photonic platform 10 . The replacement of the photoconductive n-i-n device by a p-n junction photodiode allowed to improve the operation speed (reported characteristic times were shorter than 0.1 sec). However, due to the strain accumulation, the structural quality of the InGaN radial layers is quite low leading to important non-radiative losses. The material quality can be improved by replacing a thick InGaN layer with a multi-quantum well heterostructure. Up to date, there have been no reports on core/shell p-n InGaN/GaN NW photodetectors using QWs for the light absorption.

In this work, we report for the first time the fabrication of a p-n junction NW photodiode containing radial InGaN/GaN QWs and we investigate the optoelectronic properties of these devices for their application in visible to UV range photodetection. The NWs were grown by catalyst-free Metal-organic vapour phase epitaxy (MOVPE) on sapphire substrates. The photoresponse was studied on a model case of a single NW contacted by means of optical lithography. The NW detectors present a response in the visible to UV spectral range starting from 2.8 eV. The peak responsivity is 0.17 A/W at 3.36 eV. This value corresponds to a more than two times enhancement with respect to the previous report for core/shell p-n photodiodes based on a thick InGaN radial layer 10 . The on-off switching time under square light pulses is found to be below 0.1 sec, which is much shorter than the typical response time of photoconductive detectors suffering from a slow current decay.

Experimental

The N-polar GaN wires were grown by MOVPE on c-sapphire substrates using in-situ SiNx thin film pre-deposition 13,14 . The bottom wire part (~10 µm long) was heavily n + -doped. Then a non-intentionally doped GaN segment (~10-15 µm long, residual n-doping ~10 18 cm -3 ) was deposited. The upper part of the wires was coated with 30 InGaN/GaN radial QWs (6 nm/24 nm) along nonpolar m-plane facets and then overgrown with a p-doped GaN shell layer as reported in Refs 13,15 . The targeted indium composition in the QWs is ~18%. The NW morphology is illustrated in figure 1. The average wire length is 20-25 µm and the diameter is 1-2 µm.

To fabricate single wire photodetectors, the NWs were detached from their native substrate by ultra-sonic cutting and dispersed on a SiO 2 /Si substrate. The NWs were then encapsulated in a hydrogen silsesquioxane polymer (HSQ) transformed into SiOx after thermal baking for 1 hour at 400 °C. The excess of HSQ was etched in diluted HF solution (HF:H 2 O = 1:200 volume) to expose the lateral NW facet for contacting. The first contact was defined on the p-GaN shell by optical lithography. The Ni /Au (10 nm / 150 nm) was deposited followed by an annealing at 500 °C in ambient atmosphere for 5 min to favor the formation of ohmic contact on p-type GaN. The second electrode to n-type GaN core was also defined by optical lithography and Ti/Al/Ti/Au (10/20/10/200 nm) metallization. A scanning electron microscopy (SEM) image of a typical device is shown in figure 2 (a) and a schematic picture illustrating the device structure is presented in figure 2 (b).

Electrical and optical characterization 3.1 Current-voltage characterization

Figure 3 (a) presents the I-V curve (in logarithmic scale) of a single NW photodetector in the dark and under illumination. In the dark, the I-V curve shows a typical rectifying behavior without significant current leakage (dark current is below 10 -11 A at -1 V bias). The non-zero dark current at zero bias (below 1 pA) and the current accidents in the -0.75 to 0 V range are due to the measurement uncertainty of our electrical detection system. Under illumination with λ=370 nm, the photodetector exhibits a strong photocurrent. The induced current under zero bias reaches ~200 pA. As expected for a p-n junction photodiode, the current signal does not significantly change with the reverse bias (for biases from 0 to -1.5V the current remains between 200 and 240 pA). Under forward bias, the current under illumination first slightly decreases and then increases following the dark current behavior of a p-n junction under forward bias. The I-V curve for higher forward current levels (up to 50 nA) is shown in figure 3 (b) (in linear scale) under illumination and in the dark. It is seen that for forward bias above 1.5 V the slope of the I-V curve under illumination is slightly steeper that the slope of the dark I-V curve. This indicates the variation of the access resistance under illumination (possibly, the resistance of the p-GaN shell). It should be noted that NW photodetectors due to the surface Fermi level pinning are often characterized by strong photoconductive effects leading to an elevated photocurrent gain 16 . NW photoconductors can even present persistent photoconductivity. However, as seen in figure 3 (b), in the presently-investigated case of a radial p-n junction these negative effects are minimized thanks to the core-shell geometry. Indeed, the dark and illuminated I-V curves under the forward bias are almost identical, whereas a strong current increase is expected for a NW photoconductor 16 .

Time dependent photocurrent transient measurement

In order to investigate the operation speed of the photodetector, the time dependent photocurrent transient measurements were performed under zero bias. The NW detector was illuminated with square light pulses with different wavelengths using a wavelength tunable light source (Xe lamp coupled with a monochromator). The photocurrent signal has been normalized with respect to the incident light intensity at different wavelengths. As shown in figure 4 (a), the photodetector presents a response to wavelengths starting from 430 nm (photocurrent = 4.7 pA) corresponding to the absorption in the InGaN/GaN radial QWs. The response increases for shorter wavelength. The rise and the decay switching times for all illumination wavelengths are below the time resolution of the measurement system (~0.1 sec). We note that no persistent photoconductivity effects that would lead to slow signal transients were observed. To characterize the switching time with a better precision, the illumination was mechanically chopped at different frequencies and the detector signal was recorded. The -3dB cutoff frequency under zero bias and λ=370 nm is found to be 30 Hz.

Photocurrent spectroscopy

To better analyze the spectral dependence of the detector reponse, photocurrent spectra of a single NW device were measured at room temperature using a wavelength tunable Xenon lamp light coupled with a Jobin Yvon Triax 180 spectrometer. The photocurrent spectrum under zero bias of a typical device in the visible-to-UV spectral range is reported in figure 5 (in a logarithmic scale). The calibration of the light source output intensity was done using a commercial calibrated photodiode sensor and the photocurrent spectrum was normalized accordingly. The detector responsivity is defined as :

! = ! !! ! !"#
, where I ph is the photocurrent and P opt is the incident optical power. As seen in figure 5, the photoresponse appears below the GaN bandgap at around 2.8 eV (442 nm). This response is attributed to the absorption in the QWs and the subsequent extraction of the electron-hole pairs from the quantum wells by the electric field of the radial p-n junction. However, the responsivity in the visible range is rather weak (~0.015 A/W at λ=442 nm). Between 2.8 eV and 3.36 eV (370 nm), the photocurrent rapidly increases and reaches its maximum of 0.17 A/W at 3.36 eV (~370 nm). This increase can be understood as a consequence of the inhomogeneous In compositions in the QWs 17 . The regions with a lower In content contribute to the photocurrent at shorter wavelengths. In addition, the carrier extraction from the shallow In poor QW parts is more efficient than from the deep In-rich QW parts. The signal remains almost constant until 3.54 eV (350 nm) and then decreases by 35-40 % toward the UV range (300 nm). This decrease for wavelengths shorter than the GaN bandgap is due to the loss of photoexcited carriers absorbed in the p-GaN shell region. Indeed, these carriers can recombine on surface states before diffusing to the p-n junction field region and thus they do not contribute to the photocurrent. With respect to the previously reported NW core/shell p-n photodiodes based on a thick radial InGaN layer 10 , the peak responsivity of the multi-QW photodiode is increased from 0.075 A/W to 0.17 A/W. We attribute this enhancement to the improvement of the structural properties of the multi-QW nanowires. Indeed, thick InGaN layers exhibit plastic relaxation due to the lattice mismatch with the GaN core. Therefore, thick InGaN layers are expected to have a large number of non-radiative defects leading to carrier losses. The multi-QW design allows to improve the structural quality leading to a higher responsivity.

Conclusion

In summary, we fabricated and characterized single NW photodetectors operating in the visible-to-UV spectral range. Self-assembled NWs containing 30 InGaN/GaN core/shell QWs ( 6 nm/24 nm ) in the middle of a radial pn junction were synthesized by MOVPE. The electrical properties of the photodetector as well as the time dependent current transits and the photocurrent spectra were analyzed. Contrary to the NW photoconductors, the detector presents no persistent photoconductivity and the response time is below 0.1 sec, which is limited by experimental setup timporal resolution. The detector response extends from 2.8 eV (442 nm) to the deep UV range. The peak responsivity is 0.17 A/W at 3.36 eV (370 nm).

Figure 2 (Figure ( 1 )

 21 Figure 2 (a) SEM picture of the processed single nanowire visible-UV photodetector and (b) the schematic picture of a single nanowire photodetector

Figure 4

 4 Figure 4 Time dependent photocurrent for on/off illumination cycles at different wavelength from 450 nm to 370 nm. Inset shows a close-up on one illumination cycle with λ = 430 nm corresponding to the QW absorption.

Figure 3 (

 3 Figure 3 (a) I-V curve of a single nanowire photodetector in the dark and under illumination. (b) I-V curve for larger forward biases showing only a weak current modification under illumination.

Figure 5

 5 Figure 5 Room temperature photocurrent spectrum of a single nanowire photodetector.

Acknowledgements This work has been partially financially supported by Laboratory of Excellency 'GaNeX' (ANR-11-LABX-2014) and 'NanoSaclay' (ANR-10-LABX-0035), by ANR-14-CE26-0020-01 project 'PLATOFIL', by FP7 Marie Curie projects 'Funprob' and by EU ERC project "NanoHarvest" (grant no. 639052). A. Babichev acknowledges the support of RFBR (Project number 14-02-31485, 15-02-08282), Russian Science Foundation (Project number 15-12-00027) for photoconductivity measurements support, and of the Scholarship of the President of the Russian Federation (Grant number SP-4716.2015.1). The device processing has been performed at CTU-IEF-Minerve technological platform, member of the Renatech RTB network.