Itaï Ben

Adriane Kaïchouh

RECONSTRUCTION OF SEPARABLY CATEGORICAL METRIC STRUCTURES

We extend Ahlbrandt and Ziegler's reconstruction results ([AZ]) to the metric setting: we show that separably categorical structures are determined, up to bi-interpretability, by their automorphism groups.

Introduction

Categoricity offers an ideal setting for reconstruction: a lot of information on a categorical structure can be recovered from the action of its automorphism group. Indeed, the Ryll-Nardzewski theorem asserts that a classical countable structure is ℵ 0 -categorical if and only if its automorphism group acts oligomorphically on it. From this, definability in countably categorical structures translates as invariance under the action of the automorphism group.

An analogous phenomenon occurs in the continuous setting: a metric structure is separably categorical if and only if the action of its automorphism group is approximately oligomorphic ( [START_REF] Ben Yaacov | Model theory for metric structures, Model theory with applications to algebra and analysis[END_REF]theorem 12.10]). This continuous Ryll-Nardzewski theorem again implies that definability boils down to invariance by automorphisms (see section 1).

In this paper, we focus on a reconstruction result due to Ahlbrandt and Ziegler ( [AZ]) which states that countably categorical structures are determined, up to bi-interpretability, by their automorphism groups (regarded as topological groups). We extend Ahlbrandt and Ziegler's result to the continuous setting. More precisely, we introduce the notion of an interpretation between metric structures and prove that two separably categorical structures are bi-interpretable if and only if their automorphism groups are topologically isomorphic.

This guarantees that every model-theoretic property of separably categorical structures will translate into a topological property of their automorphism groups. Tsankov and the first author ( [BT]), and then Ibarlucía ( [I]), are precisely studying model-theoretic properties directly on groups.

Although our result encompasses its classical counterpart, the proof we give is fundamentally metric and is quite different from the original one. Indeed, we apply a construction of Melleray ([M, theorem 6]) that provides a canonical way to make a metric structure out of any Polish group (we will call this the hat structure associated to the group; see subsection 3.3 for a definition). The heart of the reconstruction consists in showing that every separably categorical metric structure is in fact bi-interpretable with the hat structure of its automorphism group.

Definability

In this section, we prove the aforementioned fact that in separably categorical structures, definability amounts to invariance under the action of the automorphism group. First, we introduce the following item of notation.

Notation. If ρ is a bounded pseudometric on a structure M, then (M, ρ) will denote the quotient metric space induced by ρ. For such a ρ, let ρ ω be the pseudometric on M ω defined by

ρ ω (a, a ′ ) = n<ω 1 2 n ρ(a n , a ′ n ).
When ρ is a metric, so is ρ ω , which then induces the product topology on M ω .

Proposition 1. Let M be a separably categorical metric structure and G its automorphism group.

Let P : S n (T ) coincide. This implies that Φ is continuous for the logic topology as well. Thus, by theorem 9.9 of [BBHU], the predicate P is definable.

M n → [0, 1] a continuous predicate on M. Then P is definable in M if and only if P is G-invariant. Proof. ⇒] If P is definable, there is a sequence (ϕ k ) k 1 of formulas which converges uniformly to P . Now G preserves (interpretations of) formulas so P is also G-invariant. ⇐] Suppose that P is G-invariant.
Remark 2. The same holds for predicates in an infinite number of variables. In fact, if M ω is endowed with d ω , then the Ryll-Nardzewski theorem can be reformulated as follows: a metric structure M is separably categorical if and only if the space (S ω (M), d ω ) of types in infinitely many variables is compact. Thus, the proof above readily adapts to an infinite number of variables.

Reconstruction up to interdefinability

We begin by reconstructing separably categorical structures up to interdefinability, mirroring

Ahlbrandt and Ziegler's theorem 1.1 (in [AZ]). The proof is exactly the same as in the discrete setting.

Definition 3. Let M and N be two structures on the same universe. We say that M and N are interdefinable if they have the same definable relations.

Proposition 4. Let M and N be two separably categorical metric structures on the same universe, in languages L M and L N respectively. Then M and N are interdefinable if and only if their automorphism groups are equal.

Proof. ⇒] Assume that M and N are interdefinable and let R be a relation in L N . Since it is definable in N, it is definable in M as well, so proposition 1 implies that it is Aut(M)-invariant.

Thus Aut(M) preserves every relation in L N so Aut(M) ⊆ Aut(N). Similarly, we obtain that Aut(N) ⊆ Aut (M).

⇐] Conversely, assume that Aut(M) = Aut(N) and let R be definable in M. Then, by proposition 1, it is Aut(M)-invariant hence Aut(N)-invariant by assumption. Thus, R is definable in N and the two structures have the same definable relations.

Reconstruction up to bi-interpretability

3.1. Interpretations. In the classical setting, an interpretation of a structure M in an other structure N is an embedding of M into a definable quotient of a finite power of N, that is, into the imaginaries of N. As Usvyatsov and the first author pointed out in [START_REF] Yaacov | Continuous first order logic and local stability[END_REF], the right definition of imaginaries in metric structures should allow classes of infinite tuples and this is also true for interpretations.

Definition 5. Let M and N be two metric structures in languages L M and L N respectively. An interpretation of M in N consists of the following data:

• a definable pseudometric ρ on N ω and

• an isometric map ϕ : (M, d M ) → (N ω , ρ) such that • the predicate P : N ω → [0, 1] defined by P (x) = ρ(x, ϕ(M)) is definable in N and
• for every formula F in L M , the formula

P F : ϕ(M) r → [0, 1] defined by P F (x) = F (ϕ -1 (x))
is definable in N.

To Then the equivalence relation on N n induced by restriction of ρ is well-defined and definable, and it yields an interpretation of M in N.

If M, N and K are metric structures, ϕ :

(M, d M ) → (N ω , ρ N ) is an interpretation of M in N and ψ : (N, d N ) → (K ω , ρ K
) is an interpretation of N in K, then we can compose the interpretations ψ and ϕ as follows.

Taking the product of ψ, we get an isometric map

ψ ω : (N ω , d ω N ) → (K ω×ω , ρ ω K ) = (K ω , ρ ω K ). Now, since ψ(N) is definable in (K ω , ρ K ), the image ψ ω (N ω ) is definable in (K ω , ρ ω K ) too.
Besides, ρ N is a definable pseudometric, so its pushforward by ψ ω also is. Then, by [B1, proposition 3.6], it extends to a definable pseudometric ρ on (K ω , ρ ω K ). Thus, the isometric map

ψ ω • ϕ : (M, d M ) → (K ω , ρ) is an interpretation of M in K.
Remark 7. If N is separably categorical and ρ is a definable pseudometric on N ω , then the structure (N ω , ρ) is separably categorical too (and thus definability corresponds to invariance by the automorphism group). Indeed, the automorphism group of N acts approximately oligomorphically on N ω and thus on (N ω , ρ) too. This implies that the whole automorphism group of (N ω , ρ) is approximately oligomorphic so, by the Ryll-Nardzewski theorem, that (N ω , ρ) is separably categorical.

In particular, any structure that is interpretable in a separably categorical one is itself separably categorical. That is the reason why it is necessary to impose an oligomorphicity restriction in theorems 14 and 16. Definition 8. Let M and N be two metric structures. We say that M and N are bi-interpretable if there exist interpretations ϕ of M in N and ψ of N in M such that ψ • ϕ and ϕ • ψ are definable.

In the rest of this section, we argue that interpretations between separably categorical structures correspond to continuous homomorphisms between their automorphism groups.

3.2. From interpretations to group homomorphisms. The first side of this correspondence is not too surprising, for it amounts to saying one can get information on the automorphism group from the structure. The process is however nicely functorial.

Proposition 9. Let M and N be two metric structures and ϕ an interpretation of M in N. Then ϕ induces a homomorphism of topological groups Aut(ϕ) from Aut(N) to Aut (M).

Proof. Let g be an automorphism of N. Then g leaves ϕ(M) and the predicates P R invariant so it induces an automorphism of (ϕ(M), (P R )) and thus of M. More formally, if a is an element of M, we put Aut(ϕ)(g)(a) = ϕ -1 (g(ϕ(a))). Then Aut(ϕ) is the conjugation by ϕ so it is a group homomorphism.

And since ρ is continuous, it is easy to see that Aut(ϕ) is continuous.

The map ϕ → Aut(ϕ) is functorial: it respects composition.

Lemma 10. Let M, N and K be metric structures, ϕ an interpretation of M in N and ψ an interpretation of N in K. Then Aut(ψ

• ϕ) = Aut(ψ) • Aut(ϕ).
Lemma 11. Let M be a separably categorical metric structure and ϕ an interpretation of M in itself. Then ϕ is definable in M if and only if Aut(ϕ) = id Aut(M ) .

Proof. ⇒] If ϕ is definable, then ϕ is Aut(M)-invariant. Then, if g ∈ Aut(M) and a ∈ M, we have Aut(ϕ)(g)(a) = ϕ -1 (g(ϕ(a))) = ϕ -1 (ϕ(g(a))) = g(a) and thus Aut(ϕ) is the identity. ⇐] If Aut(ϕ) is the identity, the same computation shows that ϕ is Aut(M)-invariant. Since ϕ is continuous (it is isometric), this implies that ϕ is definable (by proposition 1). This lemma will yield the first direction of theorem 17.

3.3.

A special structure: a group with a hat. We now proceed to the second part of the correspondence: the actual reconstruction. To this aim, we come down to a canonical structure built from the automorphism group and with which the structure is bi-interpretable.The following construction is due to Melleray ([M,theorem 6]).

Let M be a metric structure and G be its automorphism group. Whenever we endow G with a compatible left-invariant metric d L , we can consider the structure G whose universe is the left completion G L of (G, d L ) and whose relations are all those maps of the form R C (x) = d(x, C), for some orbit closure C of ( G L ) n under the diagonal action of G. Then the automorphism group of

G is G.
Now fix a dense sequence ξ ∈ M ω . Then, the metric on G given by d ξ (g, h) = d(gξ, hξ) is a compatible left-invariant metric.

Proposition 12. The structure G (obtained from this particular metric d ξ ) is interpretable in M.

Proof. Consider the map ψ :

g → gξ from (G, d ξ ) to G • ξ ⊆ M ω .
It is isometric so it extends to the left completion of G. Then ψ is an interpretation of G in M.

Indeed, the predicate P :

x → d(x, ψ( G)) = d(x, G • ξ) on M ω is G-invariant so it is definable in M by proposition 1. Moreover, if C is an orbit closure and R = R C is the associated predicate in G, we have P R (gx) = R(ψ -1 (gx)) = R(gψ -1 (x)) = R(ψ -1 (x)) because R is invariant by the automorphism group = P R (x),
so P R is definable, which completes the proof.

Remark 13. In fact, since the image of ψ is dense, G • ξ is exactly the left completion of G and from now on, we identify G with G • ξ.

The above proposition, along with remark 7, implies that if M is separably categorical, then so is G. And in that case, if d L is any other compatible left-invariant metric, then the associated hat structure is bi-interpretable with G: the two metrics generate the same topology so they are continuous with respect to each other, and their left-invariance implies, by proposition 1, that they are definable from each other. All the hat structures obtained from G are bi-interpretable and we will therefore identify them.

Moreover, if M is separably categorical, then the structure M is also interpretable in G. In fact,

we have the following more general result which will be the key ingredient in the proof of theorem 16.

Theorem 14. Let N be a metric structure and let H be a subgroup of Aut(N) which acts approximately oligomorphically on N. Then N is interpretable in H.

Proof. Let ζ be a dense sequence in N. Then H = H • ζ. Now the assumption ensures that the space N H of orbit closures of N by H is compact.

The intuition for the proof is to say that N is not far from being the product H × N H and moreover that compact spaces should be interpreted in every structure. As a matter of fact, we will build a particular system of representatives of N H that H will interpret.

We begin by building a tree T representing this compact quotient N H. For this, we will choose 2 ) in N with a finite number of balls of radius 1 4 centered in elements of ζ so that the second level of our tree consists of the indices of those centers, and so on.

The construction ensures that for every infinite branch of T , the sequence (ζ σ(i) ) converges in N. Moreover, every orbit closure corresponds to an infinite branch of T (maybe even several): for every a in N, there exists an infinite branch σ of T such that the limit of the sequence (ζ σ(i) ) is in the closure of the orbit of a. Let [T ] be the set of infinite branches of T .

We now embed N isometrically into (the completion of) a quotient of H • ζ × [T ], which we identify with H × [T ]. This will give the base map for our interpretation.

Endow the set H • ζ × [T ] with the following pseudometric ρ((x, σ), (y, τ )) = lim i→∞ d(x σ(i) , y τ (i) ).
Since for every branch σ in [T ], the sequence (ζ σ(i) ) converges, this is also true of every (x σ(i) ) with

x in H • ζ, so ρ is well-defined.

We now define a map ϕ :

(H • ζ × [T ], ρ) → N by ϕ(x, σ) = lim i→∞ x σ(i)
. By definition of ρ, the map ϕ is isometric. In addition, the image of ϕ is dense in N. Indeed, let a be an element of N and ǫ > 0. There exists a branch σ in [T ] such that (ζ σ(i) ) converges to some a ′ in N which is in the same H-orbit closure as a, that is, there exists h ∈ H such that d(h(a ′ ), a) < ǫ, so d(ϕ(hζ σ(i) ), a) < ǫ, hence the density.

Thus, the isometric map ϕ can be extended to an isometry from the completion of (H

• ζ ×[T ], ρ)
onto N. Then its inverse, call it φ, is the desired isometric map between N and the completion of ( H × [T ], ρ). This was the first step in our intuition.

In order to see φ as an interpretation of N in H, it remains to interpret [T ] in H, in other words, to code the branches of T in a power of H (that is H • ζ via the identification of the previous remark). The map φ will then induce a map N → H × H ω , which will be the desired interpretation.

A branch can be coded by a sequence of zeroes and ones 1 . Then we code 2 each bit by a pair of

elements of H • ζ. Consider the pseudometric on H • ζ × H • ζ defined by δ((x, x ′ ), (y, y ′ )) = |d(x 0 , x ′ 0 ) -d(y 0 , y ′ 0 )|,
which compares the differences between the first coordinates of the two sequences of the pair. This is a definable pseudometric and we code the bit 0 by the δ-class of (ζ, ζ) and the bit 1 by the δ-class of (ζ, h 0 ζ) where h 0 is some element of H that does not fix ζ 0 . Note that the code is invariant under the action of H.

Finally, we identify branches of T with their codes in ( H 2 , δ) ω and we transfer the pseudometric ρ on H × [T ] to a definable pseudometric ρ on H × ( H 2 , δ) ω . Note that the elements of ( H 2 , δ) ω that code a branch of [T ] may not cover the whole of ( H 2 , δ) ω , but [B1]'s proposition 3.6 allows us to extend ρ to ( H 2 , δ) ω all the same.

So we can now rewrite the map φ as a map from N to the completion of ( H × ( H 2 , δ) ω , ρ). The oligomorphicity of the action of H on N implies that the structure H = H • ζ, whose automorphism group is H, is separably categorical. Since ρ is invariant under the action of H, proposition 1 then yields that the pseudometric ρ is definable in H. Therefore, this new map φ is an interpretation of N in H.

Corollary 15. If M is separably categorical, then the structures M and G are bi-interpretable.

Proof. The proposition implies in particular that M is interpretable in G. Thus, it suffices to show that the compositions of the interpretations constructed in the previous propositions are definable. Both interpretations respected the actions of the automorphism groups so proposition 1 and remark 7 allow us to conclude.

3.4. Reconstruction. We are now ready to complete the reconstruction.

1 There are many ways of doing so; we pick one. For instance, we may say that given a branch of T , we follow the levels of T one by one, and we put a 1 in our sequence when we hit an element of our branch and a 0 otherwise.

2

There are also many ways of coding zeroes and ones in a power of

d ′ L (g 1 , g 2 ) = inf{d L (g 1 k 1 , g 2 k 2 ) : k 1 , k 2 ∈ Ker(f )}.
Since Ker(f ) is normal, this indeed defines a pseudometric, which induces a compatible metric on H. Then (H, d ′ L ), which we identify with H (see subsection 3.3), is the quotient 3 of G by the definable pseudometric d ′ L and is thus interpretable in G.

Theorem 17. Let M and N be separably categorical metric structures. Then M and N are bi-interpretable if and only if their automorphism groups are isomorphic as topological groups.

Proof. ⇒] Assume that ϕ and ψ are interpretations that witness the bi-interpretability of M and N. Then lemma 11 implies that Aut(ϕ • ψ) = id Aut(N ) and Aut(ψ • ϕ) = id Aut(M ) . But Aut(ϕ • ψ) = Aut(ϕ) • Aut(ψ) so Aut(ϕ) = Aut(ψ) -1 and Aut(ψ) is an isomorphism of topological groups between Aut(M) and Aut(N). Note that for this direction, we do not need the categoricity of the structures.

⇐] By corollary 15, M is bi-interpretable with Aut(M) and N with Aut(N). Now if the two groups are isomorphic as topological groups, then their associated hat structures are biinterpretable (by the discussion following remark 13).

Example 18. In [B2], it is shown, by an explicit computation, that the probability algebra M of the unit interval is bi-interpretable with the space N = L 1 ([0, 1], [0, 1]) of [0, 1]-valued random variables, identified up to equality almost everywhere. Our reconstruction theorem allows us to recover this result in a more elegant way. Indeed, the probability algebra of [0, 1] is separably categorical, thus its automorphism group G = Aut(µ) is Roelcke-precompact ([R, theorem 5.2]).

Moreover, G is also the automorphism group of N. The space of orbit closures of N under the action of G can be identified with the space of probability measures on [0, 1]. Indeed, given a measurable map in N, multiply it by G to make it non-decreasing. The resulting map is then the characteristic function of some probability measure on I.

Thus, the space of orbit closures of N is compact. This suffices, by [START_REF] Yaacov | Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups[END_REF]theorem 2.4], to get that the action of G on N is approximately oligomorphic, hence that the structure N is also separably categorical. Theorem 17 then applies, proving that M and N are bi-interpretable.

  If a and b have the same type in M n , then, since M is approximately homogeneous ([BBHU, corollary 12.11]) and P is continuous, the G-invariance of P gives that P (a) = P (b). Thus, P induces a metrically continuous map Φ : S n (T ) → [0, 1] on types, defined by Φ(p) = P (a) for a ∈ M n of type p. Since every type is realized in M (by the Ryll-Nardzewski theorem [BU1, fact 1.14]), the map Φ is well-defined. Now, by the Ryll-Nardzewski theorem again, the logic topology and the d-metric topology on

  verify the last condition, it suffices to check it on relation symbols in L M and on predicates of the form (x, y) → d(x, f (y)), where f is a function symbol in L M . Remark 6. If M and N are classical structures, they can be made into discrete metric structures. Then every interpretation of M in N (in the metric sense, as defined above) induces a classical interpretation of M in N. To see this, given a metric interpretation ϕ of M in N, use the continuity of the associated pseudometric to choose a big enough n such that the elements in the image ϕ(M) (which is discrete) are determined by their restriction to the first n coordinates.

  representatives, within ζ, of a dense sequence of orbit closures that witnesses the compactness of this quotient, and T will be the tree of their indices in ζ. More precisely, we build the tree by induction: first, there exist ζ n 1 , ..., ζ n k in ζ such that the balls of radius 1 2 centered in the closures of the orbits of ζ n 1 , ..., ζ n k cover all of the quotient. The indices of those elements constitute the first level of our tree. For the next step, we cover each of the balls B(ζ n i , 1

  Theorem 16. Let M and N be two metric structures, with M separably categorical. Let f : Aut(M) → Aut(N) be a continuous group homomorphism whose image acts approximately oligomorphically on N. Then N is interpretable in M.Proof. Set G = Aut(M) and H = f (G). Since H acts approximately oligomorphically on N, theorem 14 implies that N is interpretable in H. And by proposition 12, the structure G is interpretable in M. It then suffices to show that H is interpretable in G. Now H is the quotient of G by the closed normal subgroup Ker(f ). If d L is a left-invariant metric on G, then we can endow G with the following left-invariant pseudometric

H • ζ. Here we go for a method which compares two sequences of a pair in a very simple way.

Acknowledgements. The authors wish to thank Tomás Ibarlucía for his careful proofreading and the ensuing conversations. Part of this work was carried out during the Universality and homogeneity trimester program at the Hausdorff Institute for Mathematics in Bonn; we are thankful for the extended hospitality and excellent working conditions.


Member, IEEE Olivier Marcelot
email: olivier.marcelot@isae.fr

Member, IEEE Magali Estribeau
email: magali.estribeau@isae.fr

Member, IEEE Vincent Goiffon
email: vincent.goiffon@isae.fr

Member, IEEE Philippe Martin-Gonthier
email: philippe.martin_gonthier@isae.fr

Franck Corbière
email: franck.corbiere@isae.fr

Romain Molina
email: romain.molina@isae.fr

Sébastien Rolando
email: sebastien.rolando@isae.fr

Member, IEEE Pierre Magnan
email: pierre.magnan@isae.fr

Study of CCD Transport on CMOS Imaging Technology: Comparison Between SCCD and BCCD, and Ramp Effect on the CTI

Keywords: Charge, charge transfer, charge-coupled devices, CMOS image sensors (CIS), deep submicrometer process, transfer inefficiency, trapped charge

This paper presents measurements performed on charge-coupled device (CCD) structures manufactured on a deep micrometer CMOS imaging technology, in surface channel CCD and in buried channel CCD mode. The charge transfer inefficiency is evaluated for both CCD modes with regard to the injected charge, and the influence of the rising and falling time effect is explored. Controlling the ramp and especially reducing its abruptness allows to get much lower charge transfer inefficiency in buried CCD mode. On the contrary, we did not observe any effect of the ramp on surface channel CCD mode because of the presence of interface traps at the silicon-oxide interface.

I. INTRODUCTION

C ONVENTIONAL charge-coupled devices (CCDs) are systems allowing the charge transfer between devices on a semiconductor material. They have been extensively used for imaging application, like pixel CCD array or TDI sensors [START_REF] H O L L A N D | Fully-depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon[END_REF]- [START_REF] Lepage | Time-delay-integration architectures in CMOS image sensors[END_REF]. CCDs use the dedicated processes that allow overlapping of polysilicon gates to achieve very low charge transfer inefficiency (CTI). On the contrary, CMOS processes do not use overlapping, and the gates have to be separated by a minimum gap; therefore, a high-quality CCD structure is challenging to obtain. However, with the scaling down of the CMOS technology, it is now possible to realize very narrow polysilicon gap, and to come close to a true CCD device.

We propose in this paper to realize CCD structures on a deep micrometer CMOS imaging technology, using a gap of 130 nm between adjacent gates. The CTI is studied with regard to the injected charge in surface channel charge-coupled device (SCCD) mode and in buried channel charge-coupled device (BCCD) mode. On one hand we will show the benefit of the buried channel structure, and on the other hand the effect of controlling the ramp of the gate signals. The following section presents the tested devices and the experimental conditions. Section III details the CTI and conversion gain factor (CVF) evaluation method used. Section IV gives the experimental results and analyses, for both SCCD and BCCD devices.

II. EXPERIMENTAL DETAILS

CCD test structures are manufactured on a 5 µm epitaxy layer of p-silicon doped at 1.0 × 10 15 At/cm 3 . The process used is a deep micrometer imaging CMOS process, including pinned photodiode devices. A special option allows the user to draw narrow polysilicon gap until 130 nm, using bottom antireflective coating (BARC) material etching. The gap length between adjacent gates was checked by a FEG-SEM observat i o ni nc r o s ss e c t i o np r e p a r e db yaF o c u s e dI o nB e a m( F I B ) (Fig. 1).

The CCD test structures are of two types: one has six transfer gates and the other one has 60 transfer gates. Gates are drawn 0.8-µm wide and 1.0-µm long with a gap of 130 nm. Fig. 2 shows simplified cross sections of a CCD test structure.

The CCD test structures are realized with two options. In the first one, no implant is executed below the poly gates; the charge transport is therefore operated in a surface CCD mode. In the second one, a special implant is processed under the gates in order to get a buried CCD mode. This special implant was developed with the help of Synopsys TCAD tool in a way to get a buried phosphorus profile which can be depleted [Fig. 3(a)]. Considering these conditions, the buried phosphorus implant simulated has a concentration of 6 × 10 16 P. c m -3 at 70 nm under the surface.

These CCD test structures are operated in a three-phase mode (Fig. 4), which means that three different signals control all gates. Charge are injected via an injection drain and an injection gate, using the fill-and-spill technique [START_REF] Prigozhin | CCD charge injection structure at very s m a l ls i g n a ll e v e l s[END_REF], [START_REF] Tompsett | Surface potential equilibration method of setting charge in charge-coupled devices[END_REF]. The gate control signal is swung between 0 and 3.3 V relative to the substrate, according to the CMOS technology. Under these conditions, the surface is not inverted at the low transport voltage. Fig. 3(b) shows a simulation of the potential profile 10 nm under the gate in the SCCD mode.

All measurements were done with an overlapping of the clock signal of 500 ns between two phases, including ramps (Fig. 4).

The clock signal is generated by a Tektronix DG2020A data generator, and the transfer gates are controlled either by the Tektronix, or by Arbitrary Waveform Generator AWG Keithley 3390 elements. The floating diffusion (FD) of the CCD structure is read by a standard readout chain, as can be found in CMOS pixel array [START_REF] Theuwissen | CMOS image sensors: State-of-the-art and future perspectives[END_REF]. The output signal is monitored on a digital oscilloscope.

III. CTI AND CVF EVA L UAT I O N METHOD

A. CTI Evaluation Method

The CTI is defined as the fraction of charge lost from one phase gate to the adjacent phase gate in a charge transfer device. It characterizes the transfer quality of the device, and must be as small as possible. While CTI less than 10 -5 [START_REF] Hoople | Characteristics of submicrometer gaps in buried-channel CCD structures[END_REF], [START_REF] Lee | A 360 000 pixel charge-coupled color-image sensor for imaging photographic negative[END_REF] could be obtained in real CCD device, it is much more tricky to get very good CTI on CMOS technology. The reason is the trapping of carriers at interface states and the gap between adjacent gates leading to energetic barrier [START_REF] Mutoh | Driving voltage reduction of shift registers in IT-CCD image sensors[END_REF].

The CTI parameter is estimated considering the number of injected charge Q inj compared with the number of transferred charge Q tr , for a given number of n transfers

CTI = 1 n × Q inj -Q tr Q inj . ( 1
)
CTI may be evaluated calculating the injected charge and measuring the transferred charge. Injected charge are estimated using straightforward formula, knowing the gate dimensions and the applied voltage [START_REF] Tompsett | Surface potential equilibration method of setting charge in charge-coupled devices[END_REF]. Transferred charge are estimated from the potential shift of the floating node V out ,a n d knowing the charge to voltage conversion gain (CVF) of the read out chain

Q tr (e-) = V out CVF . (2)
More accurate methods have been developed, like extended pixel edge response (EPER), first pixel response, or with an X-ray source, and are commonly employed to characterize CCD architectures [START_REF] Waczynski | A comparison of charge transfer efficiency measurement techniques on proton damaged n-channel CCDs for the hubble space telescope wide-field camera 3[END_REF]. These methods require a high number of transfer gates to measure the CTI.

In our case, we have used a different way to evaluate the CTI. We made the assumption that the CTI is constant over the entire CCD. One CCD structure is made of six transfer gates and is used as a reference of transferred charge

Q tr -6G = V out -6G CVF . (3)
A second CCD structure has 60 gates and gives a second transferred charge number, related to a higher number of transfer gates

Q tr -60G = V out -60G CVF . ( 4
)
CTI is then calculated from the value of transferred charge of these two structures, via

CTI = V out -6G -V out -60G V out -6G 1 (n 60G -n 6G ) (5)
where n 6G and n 60G are, respectively, the transfer number in the CCD test structure containing six transfer gates and 60 transfer gates. We assume that gate dimensions of structures with 6 and 60 gates are physically identical in a way that it does not affect the CTI calculation. This method takes advantage to avoid the calculation of injected charge and the measurement of CVF, which both introduce artifacts on the final result. In addition, the extraction of V out is averaged on more than 200 samples, and the standard deviation σ given by the oscilloscope is used to calculate the error made on CTI measurement by means of the following equation:

Err CTI = σ 6G + σ 60G V out -6G (n 60G -n 6G ) . (6)
In all measurements we did, the CTI is evaluated for different charge injections via the injection gate control.

To confirm and compare results with other works, the CTI was also evaluated using the EPER method [START_REF] Waczynski | A comparison of charge transfer efficiency measurement techniques on proton damaged n-channel CCDs for the hubble space telescope wide-field camera 3[END_REF], [START_REF] Janesick | Scientific Charge-Coupled Devices[END_REF], although our longer CDD structures have only 60 gates. The EPER gives a CTI based on the number of deferred charge, while our method gives a CTI depending both on deferred and lost charge. Consequently, one should expect a higher CTI in our case compare with the EPER one.

B. CVF Evaluation Method

Using (5) allows obtaining a value of the CTI without the need to know the value of the CVF. However, measurements of the CVF have been performed to be able to get the variation of the CTI with regards to the number of injected charge. To do so, all transfer gates are off in order to isolate the CCD part to the readout part. In addition, the CCD structure is uniformly illuminated which creates photoelectrons in the FD. When RS gate is activated (Fig. 2), a current flows in the RS transistor drain. It represents the current necessary to reset the floating node and is related to the voltage drop due to illumination by (Fig. 5)

I = q CVF V T RS ( 7
)
where q is the elementary charge, CVF is the conversion gain, V is the voltage drop, and T RS is the time during when RS transistor gate is activated. Actually, the average current I ave is measured during the signal period T ,w h i c h is equivalent to the measurement of the current I during the reset time T RS .

Using different illumination levels yields to different voltage drops and different drain currents, allowing to get a measurement of the CVF more accurate (not influenced by leakage currents), as shown in Fig. 6 [START_REF] Hopkinson | Use and Calibration of Detector Array Equipment[END_REF].

The CVF is therefore extracted from the slope of the curve V (I ave ) using ( 7). Voltage drop V against the average current I ave for different illumination levels. The CVF is deduced from the slope. Fig. 7. CTI against charge injection in the SCCD structures. A reset procedure is applied on the structure before injecting and transferring charge (with RST) or is not applied (without RST). The CTI is evaluated using our method and the EPER method.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

A. SCCD Transport 1) Use of Digital Command for the Gate Control: The Tektronix data generator controls all gates. Rising/falling edges applied are less than 10 ns according to oscilloscope monitoring, and the overlap between two transfer gate signals is 500 ns including the rising/falling edges. The structures are evaluated using two methods; in one case a reset procedure is applied to the CCD structure before charge injection and transfer, and in the other case the reset procedure is not applied. The reset procedure is to empty the entire CCD structure, by keeping "on" the reset transistor (RS) during a complete CCD cycle. The CTI is measured with regard to the injected charge and is shown in Fig. 7.

As it can be seen on the figure, the lower is the number of injected charge, the higher is the CTI. The increase of CTI measured by our method is even higher when less than 5000 electrons are transferred. This trend is known in CCD [START_REF] Waczynski | A comparison of charge transfer efficiency measurement techniques on proton damaged n-channel CCDs for the hubble space telescope wide-field camera 3[END_REF], [START_REF] Hardy | Charge transfer efficiency in proton damaged CCD's[END_REF]- [START_REF] Brodersen | Experimental characterization of transfer efficiency in charge-coupled devices[END_REF], and is mostly attributed to interface traps. Indeed, smaller amount of charge packets have a lower charge density, and therefore the smaller packets interact with more traps per electron of signal, leading to an increase of the CTI. This is also shown by a TCAD simulation in Fig. 8, where the structure is simulated in 2-D, with Fig. 8. TCAD simulation of the CTI against injected charge for a six transfer gates structure with an injection gate and an injection drain. CTI is measured by comparing the injected charge with the stored charge under the last gate. In the case of the interface trap model at the Si/SiO 2 interface, the acceptor concentration is 5×10 9 cm -2 and the capture cross section is 1×10 -15 cm 2 .

1.0-µm-long transfer gate. Without trap model, CTI is ideally null. Using a Si/SiO 2 interface trap model CTI is showing a more realistic trend, which means that it increases with the diminution of injected charge.

However, this behavior might be also attributed to other factors [START_REF] Tompsett | The quantitative effects of interface states on the performance of charge-coupled devices[END_REF]- [START_REF] Banghart | A model for charge transfer in buried-channel chargecoupled devices at low temperature[END_REF], like:

1) silicon bulk defects; 2) weak fringing fields;

3) potential barrier between gates due to the polygap.

As our device is realized on an epitaxy, one may suppose that the crystalline quality is not degraded by a significant presence of defects and does not affect the transfer efficiency. The fringing field drift is coming from the potential difference between adjacent gates, and it generally dominates the end of the transfer. An approximate expression for the fraction of charge remaining under the gate after the transfer time due to fringing field drift is given by [START_REF] Banghart | A model for charge transfer in buried-channel chargecoupled devices at low temperature[END_REF]

ε (t) = exp - t τ ff ,τ ff = L µ n E ff ( 8
)
where t is the transfer time, τ ff is the fringing field induced drift lifetime, E ff is the fringing field, µ n is the average electron mobility in the channel, and L is the gate length. Applied here, it yields a fringing field induced drift lifetime of 2 ps in SCCD mode, which gives a negligible contribution to the CTI. Therefore, the main contributors to the CTI increase at low charge injection level are most likely the interface traps and the energetic barrier between gates.

If we compare the case with and without reset procedure before charge injection, we see that without reset the CTI is slightly reduced. This trend is enhanced when only a few electrons are injected, because when the reset is not applied before the charge injection, some interface traps are already occupied and therefore do not capture signal electrons. This result shows that interface states play a major role in the increase of CTI with regards to the decrease of charge injection.

Fig. 7 also shows the CTI evaluated by means of the EPER method. This method is showing the deferred charge Fig. 9.

CTI against charge injection in the SCCD structures, with a reset procedure before charge injection. Transfer gates are controlled by the Tektronix data generator or by Keithley AWG with three different rising or falling edges. The CTI is evaluated using our method.

and not the lost charge and consequently gives better CTI. The comparison between the two methods giving the CTI shows that the transfer inefficiency is dominated by deferred charge at high injection level, and by lost charge (interface states) at low injection level.

An average CTI based on our measurement method is extracted for more than 5000 electrons and gives CTI = 2, 2 × 10 -3 with reset and CTI = 1, 8 × 10 -3 without reset. These CTI values are comparable with other devices found in the literature, like in the papers by Borg et al [START_REF] Borg | Evaluation of a surface-channel CCD manufactured in a pinned active-pixel-sensor CMOS process[END_REF]. (5.7 × 10 -4 < CTI < 7 × 10 -3 ) and Fife et al. [START_REF] Fife | A 0.5µm pixel frametransfer CCD image sensor in 110nm CMOS[END_REF] (CTI = 1 × 10 -3 ).

2) Use of AWG for the Gate Control: To investigate the effect of the rising or falling edges on the CTI, the same experiment was done with AWG Keithley controlling the transfer gates. Rising or falling edges are set and monitored on oscilloscope from 10 to 140 ns, and the overlap between two transfer gates signal is kept at 500 ns (Fig. 4). Fig. 9 shows the obtained results.

Only a small effect is visible for a high number of transferred charge. CTI is very slightly reduced when the rising or falling edges are 60 and 140 ns. Interface traps still have a strong effect on the CTI, and the impact of longer ramp is only visible on large amount of charge.

B. BCCD Transport 1) Use of Digital Command for the Gate Control:

The BCCD structures are identical to the SCCD structures, except the buried channel implant below the transfer gates from the injection drain to the floating diffusion. As in the SCCD test structures, the devices are evaluated using two methods; in one case a reset procedure is applied to the CCD structure before charge injection and transfer, and in the other case the reset procedure is not applied (Fig. 10).

Contrary to the SCCD structures, the BCCD structures show a slight decrease of the CTI with the diminution of injected charge for more than 5000 electrons, using our CTI measurement method. This shows the buried channel benefit, which means that charge are carried away from the silicon-oxide interface and much less charge are captured by Fig. 10. CTI against charge injection in BCCD structures. A reset procedure is applied on the structure before injecting and transferring electrons (with RST) or is not applied (without RST). The CTI is evaluated using our method and the EPER method. interface traps. The higher is the amount of injected charge; the closer is the SCCD CTI to the BCCD one. The reason is the increase of the charge packets, which leads to a volume extension and a rapprochement between the surface and the charge packets. However, as it can be seen for less than 5000 injected electrons, an important increase of the CTI with the decrease of injected charge takes place. The root cause of this trend might be the interface traps on the STI oxide along the channel, and probably the energetic barrier between gates. Indeed, in this particular design, STI edges are not passivated and are in contact with the buried channel (Fig. 2, crosssection BB'), which create interface traps. This is also confirmed by the curves showing the CTI measured by the EPER method. In this case, the CTI measurement does not consider the trapped charge, and one cannot see any strong increase of the CTI for less than 5000 injected electrons. A limitation due to weak fringing fields is excluded as we found a fringing field drift lifetime of 30 ps in BCCD mode, yielding again to a CTI contribution because of negligible fringing field.

An average CTI based on our measurement method is extracted for more than 5000 electrons and gives CTI = 9, 8 × 10 -4 with reset and CTI = 5, 7 × 10 -4 without reset. The CTI is well decreased by the use of a buried channel.

2) Use of AWG for the Gate Control: The effect of the rising or falling edges on the transfer quality is investigated with AWG Keithley wave generators. The measurements performed on the buried channel structures are shown in Fig. 11.

In the case of the buried transport, the effect of the rising or falling edges on the transfer efficiency is strongly visible. For more than 5000 injected electrons, the use of an AWG and especially the slow ramp improve the transfer efficiency and low value of CTI may be obtained (CTI = 2 × 10 -4 ). Similar results are obtained with 60 and 140 ns ramps. However, the error made during the measurement might hide a small difference between the two ramps. The higher is the amount of charge, the higher is the CTI. However, for less than 5000 electrons interface traps strongly impact the transfer and the CTI increases until near 0.01. One can deduce here that slow rising or falling edges improve the buried transport and help to go through the barrier pocket between gates, but do not influence the trapping mechanism of interface traps. Average CTI are extracted for more than 5000 electrons and gives very good value: CTI = 7.1×10 -4 (ramp 10 ns), CTI = 4.1×10 -4 (ramp 60 ns), and CTI = 3.8 × 10 -4 (ramp 140 ns).

V. C ONCLUSION

We have studied in this paper CCD structures manufactured on a deep micrometer CMOS imaging technology with gaps of 130 nm between transfer gates. In surface channel mode, the CTI obtained are comparable with other works and usable for a CCD device. The transfer is strongly impacted by interface traps for a low level of injected electrons, and adjustment of AWG rising or falling edges do not have any impact on it. However, the use of a buried channel device improves significantly the transfer efficiency. We found that the CTI of BCCD structures were decreasing with the reduction of the amount of injected electrons until CTI = 2 × 10 -4 ,w h i c hi s a very good result. Moreover, we showed that slow rising or falling edges help to go through energetic barriers between gates in the BCCD mode. As with the SCCD structure, for small charge packets, the transfer is most likely impacted by interface traps and the CTI increases a lot. Further work is necessary, in particular in optimizing the design to limit the interface trap impact. For example, one could think about passivating the STI sides with Pwell implant, or shifting the STI away from the CCD canal with the restriction to draw poly contact on active area. He is currently an Associate Professor with the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse.

Philippe Martin-Gonthier (M'09) received the M.S. degree from ENSERB, Bordeaux, France, in 1998, and the Ph.D. degree from the University of Toulouse, Toulouse, France, in 2010, both in electrical engineering.

He joined the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse, in 1998, as a Microelectronic Designer, where he is currently a Scientist.

Franck Corbière received the master's degree in microelectronic from LIRMM Montpellier II University, Montpellier, France, in 1997.

He joined the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse, in 1998, as a Microelectronic Designer.

Romain Molina received the Electronic Engineering degree from ENSEIRB Bordeaux University, Bordeaux, France, in 2008.

He joined the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse, in 2010, as a Microelectronic Designer.

Sébastien Rolando received the M.S. and Ph.D. degrees in electrical engineering from the SUPAERO and University of Toulouse, Toulouse, France, in 2004 and 2008, respectively.

He is currently a Microelectronic Designer with the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse.

Pierre Magnan (M'99) received the Degree in electrical engineering from the University of Paris, Paris, France, in 1980.

He is currently a Full Professor and the Head of the Image Sensor Research Group.

Fig. 1 .

 1 Fig. 1. SEM cross sectional view of a CCD test structure centered on three transfer gates.

Fig. 2 .

 2 Fig. 2. Cross sectional views of a CCD test structure and corresponding designed plan view.

Fig. 3 .

 3 Fig. 3. (a) TCAD simulation of the potential distribution in BCCD below one gate polarized at 0 V, 3.3 V depleted or 3.3 V with stored electrons. (b) TCAD simulation of the potential profile 10 nm under the oxide, in the SCCD mode. V -TG1 = V -TG3 = 0V .

Fig. 5 .

 5 Fig. 5. Principle of the CVF measurement; measured elements are the voltage drop V and the average current I ave knowing the signal period T .

Fig. 6 .

 6 Fig.6.Voltage drop V against the average current I ave for different illumination levels. The CVF is deduced from the slope.

Fig. 11 .

 11 Fig. 11.CTI against charge injection in the BCCD structures with a reset procedure before charge injection. Transfer gates are controlled by the Tektronix data generator or by Keithley AWG with three different rising or falling edges. The CTI is evaluated using our method.

Olivier

  Marcelot (M'12) received the Ph.D. degree in boron diffusion and activation control in silicon from the University Paul Sabatier of Toulouse, Toulouse, France, in 2007. He is a Research Scientist of physics of photodetector with Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse. Magali Estribeau (M'11) received the M.S. and Ph.D. degrees in electrical engineering from SUPAERO, Toulouse, France, in 2000 and 2004, respectively. She is currently a Research Scientist with the Image Sensor Research Team, Institut Supérieur de l'Aéronautique et de l'Espace, Toulouse. Vincent Goiffon (S'08-M'09) received the M.S. and Ph.D. degrees in electrical engineering from SUPAERO and University of Toulouse, Toulouse, France, in 2005 and 2008, respectively.

ACKNOWLEDGMENT

The authors would like to thank P. Salles and G. Benassayag from the CEMES-CNRS (Toulouse), who did the FIB preparation and the SEM observation on our samples.


Pierre Étoré

Ester Mariucci
email: ester.mariucci@imag.fr.

L 1 -DISTANCE FOR ADDITIVE PROCESSES WITH TIME-HOMOGENEOUS LÉVY MEASURES

We give an explicit bound for the L 1 -distance between two additive processes of local characteristics (f j (•), σ 2 (•), ν j ), j = 1, 2. The cases σ = 0 and σ(•) > 0 are both treated. We allow ν 1 and ν 2 to be time-homogeneous Lévy measures, possibly with infinite variation. Some examples of possible applications are discussed.

Introduction and main result

In this note we give an upper bound for the L 1 -distance between the laws induced on the Skorokhod space by two additive processes observed until time T > 0. By the L 1distance between two σ-finite measures µ 1 and µ 2 on (E, E ) such that µ 1 is absolutely continuous with respect to µ 2 we mean

L 1 (µ 1 , µ 2 ) = 2 sup A∈E µ 1 (A) -µ 2 (A) = E dµ 1 dµ 2 -1 dµ 2 .
Note that, with our definitions, the L 1 -distance is twice the so called total variation distance.

Giving bounds for the L 1 -distance is a classical problem, which, in the last decades, has been reinterpreted in more modern terms via Stein's method (see, e.g., [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF][START_REF] Ross | Fundamentals of Stein's method[END_REF][START_REF] Peccati | The chen-stein method for poisson functionals[END_REF]). This kind of problems arises in several fields such us Bayesian statistics, convergence rates of Markov chains or Monte Carlo algorithms (see [START_REF] Gibbs | On choosing and bounding probability metrics[END_REF], Section 4 and the references therein). However, to the best of our knowledge, results bounding the L 1 -distance between laws on the Skorokhod space are much less abundant. In this setting other kinds of distances have been privileged such as the Wasserstein-Kantorovich-Rubinstein metric (see [START_REF] Gairing | Coupling distances between lévy measures and applications to noise sensitivity of sde[END_REF]). More relevant to our purposes is a result due to Memin and Shiryayev [START_REF] Mémin | Distance de Hellinger-Kakutani des lois correspondant à deux processus à accroissements indépendants[END_REF] computing the Hellinger distance between the laws of any two processes with independent increments. In particular this gives a bound for the L 1 -distance between additive processes. In order to state their result let us fix some notation.

Let {x t } be the canonical process on the Skorokhod space (D, D) and denote by P (f,σ 2 ,ν) the law induced on (D, D) by an additive process having local characteristics (f (•), σ 2 (•), ν). We will denote such a process by {x t }, P (f,σ 2 ,ν) and we will write

P (f,σ 2 ,ν) T
for the restriction of P (f,σ 2 ,ν) to the σ-algebra generated by {x s : 0 ≤ s ≤ T } (see Section 2 for the precise definitions). Our purpose is to bound

L 1 P (f1,σ 2 1 ,ν1) T , P (f2,σ 2 2 ,ν2 T .
From now on we will assume that σ

2 1 (•) = σ 2 2 (•) = σ 2 (•), otherwise this distance is 2 (see,
e.g., [START_REF] Charles | The inner product of path space measures corresponding to random processes with independent increments[END_REF][START_REF] Jacod | of Grundlehren der Mathematischen Wissenschaften[END_REF]). We also need to define the following quantities:

γ νj = |y|≤1 yν j (dy), j = 1, 2; ξ 2 = T 0 (f 2 (r) -f 1 (r) -(γ ν2 -γ ν1 )) 2 σ 2 (r) dr.
Theorem 1.1 (Memin and Shiryayev). Let {x t }, P (f1,σ 2 ,ν1) and {x t }, P (f2,σ 2 ,ν2) be two additive processes with ν 1 and ν 2 Lévy measures such that ν 1 is absolutely continuous with respect to ν 2 and satisfying:

(1)

H 2 (ν 1 , ν 2 ) := R dν 1 dν 2 (y) -1 2 ν 2 (dy) < ∞.
The following upper bounds hold, for any 0 < T < ∞: If σ 2 > 0 then

L 1 P (f1,σ 2 ,ν1) T , P (f2,σ 2 ,ν2) T ≤ 8 1 -exp - ξ 2 8 - T 2 H 2 (ν 1 , ν 2 ) .
If σ 2 = 0 and f 1f 2 ≡ γ ν1γ ν2 , then

L 1 P (f1,0,ν1) T , P (f2,0,ν2) T ≤ 8 1 -exp - T 2 H 2 (ν 1 , ν 2 ) .
Observe that (1) implies γ νj < ∞, j = 1, 2. When σ 2 = 0 it follows from Theorem 1.2 that, for example, L 1 P

(γ ν 1 ,0,ν1) T , P (γ ν 2 ,0,ν2) T ≤ 2 T L 1 (ν 1 , ν 2 ).
The proof of Theorem 1.1, however, makes heavy use of general theory of semimartingales. This note originated from the research for a proof based only on classical results for Lévy processes, Esscher-type transformations and the Cameron-Martin formula. It turned out that this method, when applicable, gives sharper bound on the L 1 -distance. More precisely, our main result is as follows.

Theorem 1.2. Let {x t }, P (f1,σ 2 ,ν1) and {x t }, P (f2,σ 2 ,ν2) be two additive processes with ν 1 and ν 2 Lévy measures such that ν 1 is absolutely continuous with respect to ν 2 and satisfying:

L 1 (ν 1 , ν 2 ) < ∞. Then, the following upper bounds hold, for any 0 < T < ∞.

If σ 2 > 0 then

L 1 P (f1,σ 2 ,ν1) T , P (f2,σ 2 ,ν2) T ≤ 2 sinh T L 1 (ν 1 , ν 2 ) + 2 1 -2φ - ξ 2 .
If σ 2 = 0 and f 1f 2 ≡ γ ν1γ ν2 , then

L 1 P (f1,0,ν1) T , P (f2,0,ν2) T ≤ 2 sinh T L 1 (ν 1 , ν 2 ) .
Remark that, in the case ν 1 = ν 2 = 0, i.e. where there are no jumps, the upper bound in Theorem 1.2 is achieved. Indeed, an explicit formula for the L 1 -distance between Gaussian processes is well known. Denoting by φ the cumulative distribution function of a normal random variable N (0, 1), we have, for any 0 < T < ∞:

L 1 P (f1,σ 2 ,0) T , P (f2,σ 2 ,0) T = 2 1 -2φ - 1 2 T 0 (f 1 (t) -f 2 (t)) 2 σ 2 1 (t)
dt whenever the right-hand side term makes sense (see, e.g., [START_REF] Brown | Asymptotic equivalence of nonparametric regression and white noise[END_REF]).

The reason for our interest in the L 1 -distance lies in the fact that it is a fundamental tool in the Le Cam theory of comparison of statistical models ( [START_REF] Le | Asymptotic methods in statistical decision theory[END_REF][START_REF] Le | Asymptotics in statistics[END_REF]). More precisely, the presented result will be needed in a forthcoming paper by the second author, establishing an equivalence result, in the Le Cam sense, for additive processes. Similar estimations appear in many other results concerning the Le Cam ∆-distance. See for example [START_REF] Brown | Asymptotic equivalence of nonparametric regression and white noise[END_REF][START_REF] Nussbaum | Asymptotic equivalence of density estimation and Gaussian white noise[END_REF][START_REF] Carter | Deficiency distance between multinomial and multivariate normal experiments[END_REF], where the L 1 -distance between Gaussian processes is computed or [START_REF] Milstein | Diffusion approximation for nonparametric autoregression[END_REF][START_REF] Dalalyan | Asymptotic statistical equivalence for scalar ergodic diffusions[END_REF][START_REF] Genon | Asymptotic equivalence of nonparametric diffusion and euler scheme experiments[END_REF] concerning diffusion processes without jumps. In recent years, however, there is a growing interest in models with jumps due to their numerous applications in econometrics, insurance theory or financial modelling. Because of that, it is useful to dispose of simple formulas for estimating distances between such processes.

Theorem 1.2 is proved in Section 3. In Section 2 we collect some preliminary results about additive processes that will play a role in the proof. Before that, we give some examples of situations where our result can be applied. The choice of these examples are inspired by the models exhibited in [START_REF] Cont | Financial modelling with jump processes[END_REF].

Example 1.3. (L 1 -distance between compound Poisson processes)

Let {X 1 t } and {X 2 t } be two compound Poisson processes on [0, T ] with intensities λ j > 0, j = 1, 2 and jump size distributions G j ; i.e. {X j t } is a Lévy process of characteristic triplet λ j |y|≤1 yG j (dy), 0, λ j G j . Furthermore, let A be a subset of R and suppose that G j is equivalent to the Lebesgue measure restricted to A. Denote by g j the density dGj dLeb |A ; then, an application of Theorem 1.2 yields:

L 1 X 1 , X 2 ≤ 2 sinh T A |λ 1 g 1 (y) -λ 2 g 2 (y)|dy . Example 1.4. (L 1 -

distance between additive processes of jump-diffusion type)

An additive process of jump-diffusion type on [0, T ] has the following form:

X t = t 0 f (r)dr + t 0 σ(r)dW r + Nt i=1 Y i , t ∈ [0, T ],
where {W t } is a standard Brownian motion, {N t } is the Poisson process counting the jumps of {X t }, and Y i are jumps sizes (i.i.d. random variables). Consider now the additive processes of jump-diffusion type {X j t } having local characteristics (f j (•) + λ j |y|≤1 yG j (dy), σ 2 (•), λ j G j ), j = 1, 2 and suppose again that G j is equivalent to the Lebesgue measure restricted to some A ⊆ R. Letting g j denote the density of G j as above, we have:

L 1 X 1 , X 2 ≤ 2 sinh T A |λ 1 g 1 (y)-λ 2 g 2 (y)|dy +2 1-2φ - T 0 (f 1 (t) -f 2 (t)) 2 4σ 2 (t) dt .
Example 1.5. (L 1 -distance between tempered stable processes) Let {X 1 t } and {X 2 t } be two tempered stable processes, i.e. Lévy processes on R with no gaussian component and such that their Lévy measures ν j have densities of the form

dν j dLeb (y) = C - |y| 1+α e -λ j -|y| I y<0 + C + y 1+α e -λ j + y I y>0 , j = 1, 2,
for some parameters C ± > 0, λ j ± > 0 and α < 2. Then the hypothesis (1) is satisfied and Theorem 1.2 bounds the L 1 -distance by:

2 sinh T C + ∞ 0 e -λ 1 + y -e -λ 2 + y y 1+α dy + C - 0 -∞ e -λ 1 -|y| -e -λ 2 -|y|
|y| 1+α dy .

Preliminary results

2.1. Additive processes.

Definition 2.1. A stochastic process {X t } = {X t : t ≥ 0} on R defined on a probability space (Ω, A, P) is an additive process if the following conditions are satisfied.

(1) X 0 = 0 P-a.s.

(2) For any choice of n ≥ 1 and 0 ≤ t 0 < t 1 < . . . < t n , random variables X t0 , X t1 -X t0 , . . . , X tn -X tn-1 are independent. (3) There is Ω 0 ∈ A with P(Ω 0 ) = 1 such that, for every ω ∈ Ω 0 , X t (ω) is rightcontinuous in t ≥ 0 and has left limits in t > 0. (4) It is stochastically continuous.

Thanks to the Lévy-Khintchine formula, the characteristic function of any additive process {X t } can be expressed, for all u in R, as:

(2) E e iuXt = exp iu t 0 f (r)dr - u 2 2 t 0 σ 2 (r)dr -t R (1 -e iuy + iuyI |y|≤1 )ν(dy) ,
where f (•), σ 2 (•) are functions on L 1 [0, T ] and ν is a measure on R satisfying

ν({0}) = 0 and R (|y| 2 ∧ 1)ν(dy) < ∞.
In the sequel we shall refer to (f (•), σ 2 (•), ν) as the local characteristics of the process {X t } and ν will be called Lévy measure. This data characterises uniquely the law of the process {X t }. In the case in which f (•) and σ(•) are constant functions, a process {X t } satisfying ( 2) is said a Lévy process of characteristic triplet (f, σ 2 , ν).

Let D = D([0, ∞), R) be the space of mappings ω from [0, ∞) into R that are rightcontinuous with left limits. Define the canonical process x :

D → D by ∀ω ∈ D, x t (ω) = ω t , ∀t ≥ 0.
Let D t and D be the σ-algebras generated by {x s : 0 ≤ s ≤ t} and {x s : 0 ≤ s < ∞}, respectively (here, we use the same notations as in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]).

Let {X t } be an additive process defined on (Ω, A, P) having local characteristics (f (•), σ 2 (•), ν). It is well known that it induces a probability measure P (f,σ 2 ,ν) on (D,D) such that {x t }, P (f,σ 2 ,ν) is an additive process identical in law with ({X t }, P) (that is the local characteristics of {x t } under P (f,σ 2 ,ν) is (f (•), σ 2 (•), ν)). For all t > 0 we will denote P (f,σ 2 ,ν) t for the restriction of P (f,σ 2 ,ν) to D t . In the case where |y|≤1 |y|ν(dy) < ∞, we set γ ν := |y|≤1 yν(dy). Note that, if ν is a finite Lévy measure, then the process {x t }, P (γ ν ,0,ν) is a compound Poisson process.

Here and in the sequel we will denote by ∆x r the jump of process {x t } at the time r:

∆x r = x r -lim s↑r x s .
Definition 2.2. Consider {x t }, P (f,σ 2 ,ν) and define the jump part of {x t } as We now recall the Lévy-Itô decomposition, i.e. the decomposition in continuous and discontinuous parts of an additive process.

Theorem 2.3 (See [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], Theorem 19.3). Consider {x t }, P (f,σ 2 ,ν) and define {x d,ν t } and {x c,ν t } as in 3 and 4, respectively. Then the following hold. (i) There is D 1 ∈ D with P (f,σ 2 ,ν) (D 1 ) = 1 such that, for any ω ∈ D 1 , x d,ν t (ω) is defined for all t ∈ [0, T ] and the convergence is uniform in t on any bounded interval, P (f,σ 2 ,ν) -a.s. The process {x d,ν t } is a Lévy process on R with characteristic triplet (0, 0, ν).

(ii) There is D 2 ∈ D with P (f,σ 2 ,ν) (D 2 ) = 1 such that, for any ω ∈ D 2 , x c,ν t (ω) is continuous in t. The process {x c,ν t } is an additive process on R with local characteristics (f (•), σ 2 (•), 0). (iii) The two processes {x d,ν t } and {x c,ν t } are independent. 2.2. Change of measure for additive processes. For the proof of Theorem 1.2 we also need some results on the equivalence of measures for additive processes. By the notation ≪ we will mean "is absolutely continuous with respect to".

19).

Let {x t }, P (0,0,ν) and {x t }, P (η,0,ν) be two Lévy processes on R, where

(5) η = |y|≤1 y(ν -ν)(dy)
is supposed to be finite. Then P (η,0,ν) t ≪ P (0,0,ν) t for all t ≥ 0 if and only if ν ≪ ν and the density dν dν satisfies

(6) dν dν (y) -1 2 ν(dy) < ∞.
Remark that the finiteness in [START_REF] Genon | Asymptotic equivalence of nonparametric diffusion and euler scheme experiments[END_REF] implies that in [START_REF] Gairing | Coupling distances between lévy measures and applications to noise sensitivity of sde[END_REF]. When P

(η,0,ν) t ≪ P (0,0,ν) t
, the density is The convergence in [START_REF] Gibbs | On choosing and bounding probability metrics[END_REF] is uniform in t on any bounded interval, P (0,0,ν) -a.s. Besides, {U t (x)} defined by ( 7) is a Lévy process satisfying E P (0,0,ν ) [e Ut(x) ] = 1, ∀t ∈ [0, T ].

dP (η,0,ν) t dP (0,0,ν) t (x) = exp(U t (x)),
2.2.2. Case σ 2 > 0. Lemma 2.5. Let ν 1 ≪ ν 2 be Lévy measures such that R dν 1 dν 2 (y) -1 2 ν 2 (dy) < ∞. (8) Define η = |y|≤1 y(ν 1 -ν 2 )(dy), (9)
which is finite thanks to (8), and consider real functions f 1 , f 2 and σ > 0 such that

(10) T 0 f 1 (r) -f 2 (r) -η σ(r) 2 dr < ∞, T ≥ 0.
Then, under P (f2,σ 2 ,ν2) ,

M t (x) = exp C t (x) + D t (x) (11)
is a (D t )-martingale for all t in [0, T ], where

C t (x) := t 0 f 1 (r) -f 2 (r) -η σ 2 (r) (dx c,ν2 r -f 2 (r)dr) - 1 2 t 0 f 1 (r) -f 2 (r) -η σ(r) 2 dr, (12)
D t (x) := lim ε→0 r≤t ln dν 1 dν 2 (∆x r )I |∆xr|>ε -t |y|>ε (ν 1 -ν 2 )(dy) .
The convergence in [START_REF] Milstein | Diffusion approximation for nonparametric autoregression[END_REF] is uniform in t on any bounded interval, P (f2,σ 2 ,ν2) -a.s.

Proof. The existence of the limit in ( 12) is guaranteed by (8) (see Theorem 2.4). Since

t 0 1 σ(r) (dx c,ν2 r -f 2 (r)dr
) is a standard Brownian motion under P (f2,σ 2 ,0) , we have that

t s f1(r)-f2(r)-η σ 2 (r) (dx c,ν2 r -f 2 (r)dr) has normal law N 0, t s f1(r)-f2(r)-η σ(r)
2 dr , hence

E P (f 2 ,σ 2 ,0) [exp((C t -C s )(x))] = 1. Theorem 2.3 entails that {x c,ν2
t } and {x d,ν2 t } are independent under P (f2,σ 2 ,ν2) . Moreover, the law of {C t (x)} (resp. {D t (x)}) is the same under P (f2,σ 2 ,ν2) or P (f2,σ 2 ,0) (resp. P (f2,0,ν2) or P (0,0,ν2) ). Further, using Theorem 2.4, we know that {D t (x)} is a Lévy process such that E P (0,0,ν 2 ) [exp(D t-s (x))] = 1 for all s < t. These facts together with the independence of the increments of ({x t }, P (f2,σ 2 ,ν2) ) and the stationarity of {D t (x)} imply:

E P (f 2 ,σ 2 ,ν 2 ) [M t (x)|D s ] = E P (f 2 ,σ 2 ,ν 2 ) M s (x) exp (C t -C s )(x) + (D t -D s )(x) D s = M s (x)E P (f 2 ,σ 2 ,ν 2 ) [exp((C t -C s )(x) + (D t -D s )(x))] = M s (x)E P (f 2 ,σ 2 ,0) [exp((C t -C s )(x))]E P (0,0,ν 2 ) [exp((D t -D s )(x))] = M s (x)E P (0,0,ν 2 ) [exp(D t-s (x))] = M s (x).
Lemma 2.6. Suppose that the hypothesis (8) and (10) of Lemma 2.5 are satisfied.

Then, using the same notations as above, P (f1,σ 2 ,ν1) t ≪ P (f2,σ 2 ,ν2) t for all t and the density is given by: [START_REF] Charles | The inner product of path space measures corresponding to random processes with independent increments[END_REF] dP

(f1,σ 2 ,ν1) t dP (f2,σ 2 ,ν2) t (x) = M t (x).
Proof. For s < t, we prove that E P (f 2 ,σ 2 ,ν 2 ) exp(iu(x t -x s )) Mt Ms (x)|D s = E (f1,σ 2 ,ν1) P

[exp(iu(x tx s )]. To that aim remark that, thanks again to Theorem 2.3:

E P (f 2 ,σ 2 ,ν 2 ) e iu(xt-xs) M t (x) M s (x) D s = E P (f 2 ,σ 2 ,ν 2 ) e iu(x c,ν 2 t -x c,ν 2 s +x d,ν 2 t -x d,ν 2 s ) M t (x) M s (x) D s = E P (f 2 ,σ 2 ,0) e iu(xt-xs) e (Ct-Cs)(x) E P (0,0,ν 2 ) e iu(xt-xs) e (Dt-Ds)(x) . ( 14
)
Let us now compute the first factor of ( 14):

E P (f 2 ,σ 2 ,0) e iu(xt-xs) e (Ct-Cs)(x) = E P (f 1 -•η,σ 2 ,0) e iu(xt-xs) = exp iu t s (f 1 (r) -η)dr - u 2 2 t s σ 2 (r)dr .
In the first equality we used the Girsanov theorem, thanks to the fact that t 0 1 σ(r) (dx rf 2 (r)dr) is a Brownian motion under P (f2,σ 2 ,0) , while the second one follows from [START_REF] Carter | Deficiency distance between multinomial and multivariate normal experiments[END_REF]. We compute the second factor of ( 14) by means of Theorem 2.4 and another application of ( 2): E P (0,0,ν 2 ) e iu(xt-xs) e (Dt-Ds)(x) = E P (0,0,ν 2 ) e iuxt-s e Dt-s(x) = E P (η,0,ν 1 ) e iuxt-s = exp (ts) iuη -R (1e iuy + iuyI |y|≤1 )ν 1 (dy) .

Consequently:

(15) E P (f 2 ,σ 2 ,ν 2 ) e iu(xt-xs) M t (x) M s (x) D s = E P (f 1 ,σ 2 ,ν 1 ) [e iu(xt-xs) ] ∀0 ≤ s ≤ t.
Fix t and define a probability measure P t on D t by P

t (B) = E P (f 2 ,σ 2 ,ν 2 ) [M t I B ] for B ∈ D t .
As a consequence of Lemma 2.5 and the Bayes rule, the two processes given by {x s : 0 ≤ s ≤ t}, P (f1,σ 2 ,ν1) t

and {x s : 0 ≤ s ≤ t}, P t are identical. Indeed, by [START_REF] Nussbaum | Asymptotic equivalence of density estimation and Gaussian white noise[END_REF], both have independent increments and the prescribed characteristic function. Consequently, (13) holds.

Proof of Theorem 1.2

For the proof we will need the following three calculus lemmas. Lemma 3.1. Let X be a random variable with normal law N (m, σ 2 ). Then

E 1 -e X = 2 φ - m σ -φ - m σ -σ , where φ(x) = 1 √ 2π x -∞
e -y 2 2 dy.

Proof. By definition we have = E P (γ ν 2 ,0,ν 2 ) x) .

E 1 -e X = 1 √ 2πσ ∞ -∞ |1 -e x |e -(x-m) 2 2σ 2 dx = 1 √ 2πσ 0 -∞ (
T e A + (x) -e A -(
In order to compute the last quantity we apply Theorem 2.4 and the fact that both A + (x) and A -(x) have the same law under P

2. 2 . 1 .

 21 Case σ 2 = 0. Theorem 2.4 (See [18], Theorems 33.1-33.2 and [19] Corollary 3.18, Remark 3.

with ( 7 )

 7 U t (x) = lim ε→0 r≤t ln dν dν (∆x r )I |∆xr|>ε -|y|>ε t dν dν (y) -1 ν(dy) , P (0,0,ν) -a.s.

1 -.Lemma 3 . 2 .Lemma 3 . 3 .dν 1 dν 2 ≥1 and h -= dν1 dν2 I dν 1 dν 2 <1T 1 -T 1 +

 132332211 e x )e -(x-mTo conclude, just split the sums inside the integrals and use the change of variables y = x-m σσ , resp. y = x-m σ For all x, y in R we have:(16) |1e x+y | ≤ 1 + e x 2 |1e y | + 1 + e y 2 |1e x |.Proof. By symmetry we restrict to x ≥ 0.• x, y ≥ 0: In this case we have that |1 -e x+y | is exactly equal to 1+e x 2 |1e y | + 1+e y 2 |1e x |. • x ≥ 0, y ≤ 0, x + y ≥ 0:Then the member on the right hand side of (16) is equal to e xe y ≥ e x -1 ≥ e x+y -1. • x ≥ 0, y ≤ 0, x + y ≤ 0: In this case the member on the right of (16) is equal to e xe y ≥ 1e y ≥ 1e x+y . With the same notations as in Theorem 1.2 and Lemma 2.5, we have: (17)E P (0,0,ν 2 ) T 1-exp(D T (x)) = E P (γ ν 2 ,0,ν 2 ) T 1-exp(D T (x)) ≤ 2 sinh T R L 1 (ν 1 , ν 2 ) . Proof. Because of Theorem 2.3 it is clear that E P (0,0,ν 2 ) T |1-exp(D T (x))| = E P (γ ν 2 ,0,ν 2 ) T |1exp(D T (x))| .In order to simplify the notations let us writeA ± (x) := lim ε→0 r≤T ln h ± (∆x r )I |∆(xr)|>ε -T |y|>ε (h ∓ (y) -1)ν 2 (dy)with h + = dν1 dν2 I , so thatD T (x) = A + (x) + A -(x).Then, using Lemma 3.2 and the fact that A + (x) ≥ 0 and A -(x) ≤ 0 we get:E P (γ ν 2 ,0,ν 2 ) T |1 -D T (x)| = E P (γ ν 2 ,0,ν 2 ) exp(A + (x) + A -(x))≤ E P (γ ν 2 ,0,ν 2 ) e A + (x) 2 1e A -(x) + 1 + e A -(x) 2 1e A + (x)

(γ ν 2

 2 ,0,ν2) T and P (0,0,ν2) T :E P (γ ν 2 ,0,ν 2 ) T e A + (x)e A -(x) = exp T R (h + (y)h -(y))ν 2 (dy) exp T R (h -(y)h + (y))ν 2 (dy) = 2 sinh T R (h + (y)h -(y)ν 2 (dy) ν 2 (dy) .Proof of Theorem 1.2. Case σ 2 > 0: With the same notations as in Lemma 2.5 and by means of Lemma 2.6 one can write


R Renoud

C Borderon

H W Gundel

Measurement and modelisation of dielectric properties of ferroelectrics thin layers

Keywords:

In this study, the real and imaginary parts of the complex permittivity of PZT ferroelectric thin films are studied in the frequency range of 100 Hz -100 MHz. The permittivity is well fitted by the Cole-Cole model. The variation of the relaxation time with temperature is described by the Arrhenius law and an activation energy of 0.38 eV is found. Due to its nonlinear character, the dielectric response of the ferroelectric sample depends on the amplitude of the applied ac electric field. The permittivity is composed of three different contributions: the first is due to intrinsic lattice, the second to domain wall vibrations and the third to domain wall jumps between pinning centers. The last one depends on the electric field so it is important to control the field amplitude to obtain the desired values of permittivity and tunability.

INTRODUCTION

More recently, functional ferroelectric thin films have shown a growing interest for telecommunication applications. The tunability of the dielectric constant under an applied electric field allows realizing of intelligent devices. However, the use of these materials is at present limiting because they present relatively high dielectric losses and a too low tunability. Thus a better understanding of the ferroelectric dielectric characteristics is still required.

In practical devices, the ferroelectric thin films are often used in a metal -insulator -metal geometry (MIM structure). Hence we have chosen chemical solution deposition (CSD) for an elaboration of the films on stainless steel substrates. Dielectric measurements were performed with an Agilent 4294A impedance analyzer. The relative permittivity and the losses were obtained in the 100 Hz -100 MHz frequency range at temperatures varying from ambient temperature to 250°C. The real and the imaginary part of the permittivity were deduced and modeled using a Cole-Cole approach. The effects of temperature and of the amplitude of the ac electric field on the permittivity are studied. This allows us determination of the parameters which have a main influence on the tunability of the ferroelectric material.

II. EXPERIMENTAL PROCEDURE

The Pb(Zr 0.43 Ti 0.57 )O 3 thin films (PZT 43/57) were realized by a modified sol-gel process based on the use of an alkoxide precursor. The lead acetate Pb(OOCCH 3 ) 2 is dissolved in acetic acid CH 3 COOH and the solution is heated at 100°C until complete dissolution. The solution is then cooled down at room temperature before the addition of the zirconium and the titanium n-propoxides which are mixed with a molar ratio of 43/57. Ethylene glycol HO-CH 2 -CH 2 -OH is added in order to retain the mobility of the atoms and significantly reduces the appearance of cracks in the film [START_REF] Giridharan | Structural, morphology and electrical studies on ferroelectric bismuth titanate thin films prepared by sol-gel technique[END_REF]. Moreover, addition of ethylene glycol stabilizes the solution and prevents from precipitation [START_REF] Tian | Synthesis and dielectric characteristic of Ba1-xSrxTiO3 thin films-based strontium-barium alkoxides derivatives[END_REF]. Solution is then filtrated and deposited by spin coating at 4000 rpm during 20 sec on stainless steel substrates (AISI 304) and the samples are annealed during 2 min in a pre-heated open air furnace at 650°C. Multiple spincoating was used when thicker PZT layers are desired.

The samples were characterized (thickness and surface of the films) by scanning electron microscopy (SEM) (Jeol 6400) and the crystallinity is determined by X-ray diffraction (Siemens D5000). The samples that crystallized in the tetragonal phase, present a polycristalline structure with grains having an average diameter of 90 nm. The measured thickness e is 1 µm. Square gold electrodes of 0.5 mm larger (surface S = 0.25 mm 2 ) were evaporated through a shadow mask on the films in order to form metal-ferroelectric-metal capacitors in a parallel-plate structure.

The P-E hysteresis loops (figure 1) were measured using a classic Sawyer-Tower circuit to verify the polarization state of the sample. The remanente polarization P r is equal to 0.25 µC•cm -2 while the value of the coercitive field is We also measure the ε'-E loop in order to determine the tunability of the material (figure 2). For that, we use the classic definition [START_REF] Al-Shareef | Tunability and calculation of the dielectric constant of capacitor structures with interdigital electrodes[END_REF] (0)

E c = 95 kV•cm -1 .
( ) % 100 (0) bias E tunability ε ε ε ′ ′ - = × ′ (1)
where ε'(0) and ε'(E bias ) are the permittivity without and under the bias electric field E bias . The samples present here a tunability of 42.1 %.

-

′ ′ ′ ′ = = ⋅ ( 2
)
ε 0 is the permittivity of free space.

III. RESULTS AND DISCUSSIONS

A first series of measurements allows us to obtain the real part ε' and imaginary part ε" of the permittivity as a function of the frequency at different temperatures. Results are reported on figure 3. A relaxation is observed on the frequency domain studied and is attributed to the domain wall movements [START_REF] Borderon | Développement et étude des matériaux ferroélectriques accordables en vue d'une application pour les antennes intelligentes[END_REF]. Under the influence of the electric field, domain walls can either vibrate around their equilibrium position (pinning centers) or jump from a center to another. The relaxation time τ is defined by τ = 1/ω 0 where ω 0 is the pulsation for which ε"(ω) is maximum (or for which ε'(ω) admits an inflection point). His relaxation time increases with temperature and the values of ε s and ε ∞ ( ( )

0 lim s ω ε ε ω → ′ = , ( ) lim ω ε ε ω ∞ →∞ ′ =
) also increase. At low frequencies, diffusion process appears for the highest values of T. Generally, one can write [START_REF] Bunget | Physics of solids dielectrics[END_REF] ( ) ( ) where

( ) ( ) 2 2 0 2 2 0 d 1 d 1 g g τ ε ω ε ε τ ω τ τ ωτ ε ω ε τ ω τ ∞ ∞ ∞ ′ = +Δ + ′′ = Δ + ∫ ∫ (3)
Δε = ε s -ε ∞ . g(τ)
is the distribution of the relaxation time which is specific to studied relaxation. Generally, it is impossible to obtain g(τ) only from the knowledge of ε'(ω) and ε"(ω). This can be also expressed by

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 2 1 1 1 sin 2 1 2 s i n 2 cos 2 . 1 2 s i n 2 p p p p p p β β β β β β β π ε ω ε ε β π β π ε ω ε β π - ∞ - - - - - + ′ = +Δ + + ′′ = Δ + + (5)
Parameters ε ∞ , Δε, τ and β are determined by fitting curves of figure 3 with equations (5). We found that ε ∞ follows a Curie-Weiss law (T C = 770 K) while Δε does not depend on the temperature (figure 5). Thus ε s also follows a Curie-Weiss law as awaited for ferroelectric materials. As shown on figure 6, the relaxation time τ obeys to an Arrhenius law which corresponds to a thermally activated process

0 exp a B E k T τ τ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ . ( 6
)
E a is the activation energy and equals here to 0.38 eV. We found also τ 0 = 12.6 ps and 1/τ 0 well corresponds to the phonon vibration frequency. k B is the Boltzmann constant. On the temperature domain studied here, β seems to have a linear dependence with temperature (figure 7), decreasing when T increases. However, it is difficult to interpret this result because β has not a direct physical signification.

As already mentioned, the movement of domain walls under the influence of a low ac electric field E exc can corresponds either to a vibration around the equilibrium position, or to a jump towards another equilibrium position. For the permittivity, that results in the hyperbolic law [START_REF] Borderon | Description of the low field nonlinear dielectric properties of ferroelectric and multiferroics materials[END_REF]

2 2 2 2 2 bulk rev irrev bulk rev exc E ε ε ε ε ε ε α = + + = + + . ( 7
)
ε bulk is the permittivity due to intrinsic lattice contributions, ε rev the dielectric constant related to domain walls vibrations which correspond to a reversible process [START_REF] Boser | Statistical theory of hysteresis in ferroelectric materials[END_REF]. The parameter α represents domain walls pinning and is related to an irreversible modification of the local polarization [START_REF] Gharb | Dielectric nonlinearity of Pb(Yb1/2Nb1/2)O3-PbTiO3 thin films with {100} and {111} crystallographic orientation[END_REF]. Both parameters depend on the crystal structure, but α also reflects the presence of impurities, dopants or defects [START_REF] Boser | Statistical theory of hysteresis in ferroelectric materials[END_REF]. This law is only valid if E exc < E C /2 (= 45 kV•cm -1 here) [START_REF] Borderon | Description of the low field nonlinear dielectric properties of ferroelectric and multiferroics materials[END_REF][START_REF] Boser | Statistical theory of hysteresis in ferroelectric materials[END_REF]. In a second series, the temperature is fixed at 300 K and ε' and ε" are measured for two values of E exc , 1 kV•cm -1 and 9 kV•cm -1 (figure 7). The field effect is only perceptible at low frequencies and in particular, fits show that ε ∞ is independent of E exc . Consequently, for ω >> 1/τ, domain walls do not jump any more.

We have also measured the variations of ε' and ε" as the function of field E exc at various frequencies (1, 10, 100 and 1000 kHz) and plot the Argand diagrams for the different contributions to the permittivity (bulk, reversible and irreversible). Results are reported on figure 8.

It is noticed that each contribution to the permittivity is described by a Cole-Cole law with a very different dispersion parameter β. On a Argand diagram, the relaxation time is obtained by the frequency which corresponds to the maximum of ε". The values of τ estimated by this way for each contribution are practically the same excepted for the irreversible contribution. In this last case, a wall jumping from a site to another is difficult to realize and the relaxation time is very large. In both other cases, there are always some dipole vibrations, either in the domain wall or in the lattice. Thus, the relaxation times of bulk and reversible contributions are similar. To finish, we plot the tunability of the sample as a function of E exc (figure 9). It seems interesting to work with the greatest amplitude of E exc . However, losses also increase with the field amplitude [START_REF] Borderon | Développement et étude des matériaux ferroélectriques accordables en vue d'une application pour les antennes intelligentes[END_REF]. We can then define a figure of merit by

max Tunability . . . tan F O M δ = . ( 8
)
On figure 9, we see that the F.O.M. decrease when the field amplitude E exc increases. The losses increase more quickly than the real permittivity. So, to have a better compromise between the tunability and the losses, it thus appears preferable to work with an electric field of low amplitude.

IV. CONCLUSIONS

The study of PZT 43/57 shows that the permittivity is correctly described by the Cole-Cole model. The relaxation time is done by the Arrhenius law corresponding to a thermally activated process with an activation energy equals to 0.38 eV. The hyperbolic law shows that permittivity has three different contributions. The contribution associated with the domain wall jumps depends on the amplitude of the applied ac electric field. Thus, for devices including ferroelectric thin films, it is important to control this parameter. The values of the permittivity and tenability are significantly influences by the electric field. Because of the dielectric losses which also grow with the field, it is advised to use a reconfigurable device with a field amplitude the weakest possible.

Figure 1 .

 1 Figure 1. P-E hysteresis loops of PZT 43/57 at 300 K.

1 )Figure 2 .

 12 Figure 2. Butterfly ε-E loops of PZT 43/57 at 300 K.The capacitance C and the dielectric loss factor (tan δ) were measured using an impedancemeter (Agilent 4294A) with a low ac field E exc on 100 Hz -100 MHz domain frequencies. The real and imaginary parts of the permittivity (ε' and ε" respectively) are calculated with

Figure 3 .

 3 Figure 3. Evolution of ε' (top) and of ε" (bottom) as a function of the angular frequency ω and of the temperature T.

Figure 4 theFigure 4 .

 44 Figure 4 the Argand diagram, one realizes that data are well approximated by a Cole-Cole arc. In the Cole-Cole theory, the complex permittivity is done by (p = ωτ) [6]( ) ( )* 1 1 ip β ε ε ω ε ∞ -Δ = + +(4)with β a dispersion parameter which accounts for the width of the distribution of the relaxation time (β = 0 correspond to a single relaxation time, i.e. Debye formalism).

Figure 5 .

 5 Figure 5. Variation of εs, ε∞ and Δε with temperature.

Figure 6 .

 6 Figure 6. Variation of τ as a function of temperature.

Figure 7 .

 7 Figure 7. Variation of ε' as a function of ω for two values of Eexc.

1 )Figure 8 .

 18 Figure 8. Argand diagram for the bulk (top), the reversible (middle) and irreversible (bottom) contributions.

Figure 9 .

 9 Figure 9. Tunability (Top) and Figure Of Merit (bottom) as a function of Eexc.