Romain Joly
email: romain.joly@ujf-grenoble.fr

How opening a hole affects the sound of a flute A one-dimensional mathematical model for a tube with a small hole pierced on its side

Keywords: thin domains, convergence of operators, resonance, mathematics for music and acoustic. AMS subject classification: 35P15, 35Q99

In this paper, we consider an open tube of diameter ε > 0, on the side of which a small hole of size ε 2 is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one the open parts of the tube. We show that this spectrum converges when ε goes to 0 to the spectrum of an explicit onedimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open.

Introduction and main result

In this paper, we obtain a one-dimensional model for the resonances of a tube with a small hole pierced on its side. Our 1: Three different kind of tubes without holes and their resonances. [START_REF] Casado-Díaz | The diffusion equation in a notched beam[END_REF]. We show that this kind of techniques applies to the mathematical modelling of music instruments.

Basic facts on wind instruments.

The acoustic of flutes is a large subject of research for acousticians. Basically, a flute is the combination of an exciter which creates a periodic motion (a fipple, a reed etc.) and a tube, whose first mode of resonance selects the note produced. Studying the acoustic of a flute combines a lot of problems as the influence of the shape of the tube, the study of the creation of oscillations by blowing in the fipple. . . see [START_REF] Rossing | The Science of Sound[END_REF], [START_REF] Fletcher | The Physics of Musical Instruments[END_REF], [START_REF] Dickens | Flute acoustics: measurements, modelling and design[END_REF], [25] and [5] for nice introductions. In this paper, we will not consider the creation of the periodic excitation, we rather want to study mathematically the resonances of the tube of the flute and how an open hole affects it. Therefore, we simplify the problem by making the following usual assumptions: -the pressure of the air in the tube follows the wave equation and therefore the resonances of the tube are the squareroots of the eigenvalues of the corresponding Laplacian operator.

-on the inner surface of the tube, the pressure satisfies homogeneous Neumann boundary condition.

-where the tube is open to the exterior, we assume that the pressure is equal to the exterior pressure which may be assumed to be zero without loss of generality.

We can roughly classify the tube of the wind instruments in three different categories, depending on which end of the tube is open. See Figure 1. It is known since a long time that the resonances of the tubes of Figure 1 can be approximated by the spectrum of the one-dimensional Laplacian operator on (0, L) with either Dirichlet or Neumann boundary conditions, depending on whether the corresponding end is open or not (see for example [START_REF] Courant | Methods of mathematical physics[END_REF]). Notice that this rough approximation can explain simple facts: a tube with a closed end sounds an octave lower than an open tube of the same length (enabling for example to make shorter organ pipes for low notes) and moreover it produces only the odd harmonics (explaining the particular sounds of reed instruments).

In this article, we study how the one-dimensional limit is affected by opening one of the holes of the flute, say a hole at position a ∈ (0, L). At first sight, one may think that it is equivalent to cutting the tube at the place of the open hole. In other words, the note is the same as the one produced by a tube of length a. This is roughly true for flutes with large holes as the modern transverse flute, except that one must add a small correction and the length ã of the equivalent tube is slightly larger than a. This length ã is called the effective length. This kind of approximation seems to be the most used one by acousticians. It states that the resonances of the tube with an open hole are: a fundamental frequency1 π/ã and harmonics kπ/ã, k ≥ 2. However, the approximation of the resonances by the ones of a tube of length ã is too rough for flutes with small holes as the baroque flute or the recorder. In particular, the approximation by effective length fails to explain the following observations, for which we refer e.g. to [START_REF] Benade | On the Mathematical Theory of Woodwind Finger Holes[END_REF] and [START_REF] Wolfe | Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes[END_REF]:

-the effective length depends on the frequency of the waves in the tube. In other words, the harmonics are not exact multiples of the fundamental frequency.

-closing or opening one of the holes placed after the first open hole of the tube changes the note of the flute. This enables to obtain some notes by fork fingering, as it is common in baroque flute or recorder. We also enhance that some effects of the baroque flute or of the recorder are produced by half-holing, that is that by half opening a hole (some flutes have even holes consisting in two small close holes to make half-holing easier). In these cases, the effective length ã is not only related to the position a of the first open hole, which makes the method of approximation by effective length less relevant.

The purpose of this article is to obtain an explicit one-dimensional mathematical model for the flute with a open hole, which could be more relevant in the case of small holes than the approximation by effective length. The models used by the acousticians are based on the notion of impedance. The model introduced here rather uses the framework of differential operators.

The thin domains techniques. The fact that the behaviour of thin three dimensional objects as a rope or a plate can be approximated by one-or two-dimensional equations has been known since a long time, see [START_REF] Hadamard | La théorie des plaques élastiques planes[END_REF] and [START_REF] Courant | Methods of mathematical physics[END_REF] for example. In general, a thin domain problem consists in a partial differential equation (E ε ) defined in a domain Ω ε of dimension n, which has k dimensions of negligible size with respect to the other n-k dimensions. The aim is then to obtain an approximation of the problem by an equation (E) defined in a domain Ω of dimension n -k. It seems that the first modern rigorous studies of such approximations mostly date back to the late 80's: [START_REF] Lobo | Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites[END_REF], [START_REF] Anné | Spectre du laplacien et écrasement d'anses[END_REF], [START_REF] Arrieta | Eigenvalue problems for non-smoothly perturbed domains[END_REF], [START_REF] Hale | Reaction-diffusion equation on thin domains[END_REF], [START_REF] Schatzman | On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions[END_REF]... There exists an enormous quantity of papers dealing with thin domain problems of many different types. We refer to [START_REF] Raugel | Dynamics of partial differential equations on thin domains[END_REF] for a presentation of the subject and some references.

In this paper, the domain Ω ε is the thin tube of the flute and we hope to model the behaviour of the internal air pressure by a one-dimensional equation. It is well known that the wave equation in a simple tube can be approximated by the one-dimensional wave equation. Even the case of a far more complicated domain squeezed along some dimension is well understood, see [START_REF] Prizzi | The effect of domain squeezing upon the dynamics of reaction-diffusion equations[END_REF] and the references therein. We will assume in this paper that the open parts of the tube yield a Dirichlet boundary condition for the pressure in the tube. In fact, we could study the whole system of a thin tube connected to a large room and show that, at a first order of approximation, the effect of the connection with a large domain is the same as a the one of a Dirichlet boundary condition, see [START_REF] Beale | Scattering frequencies of resonators[END_REF], [START_REF] Arrieta | Eigenvalue problems for non-smoothly perturbed domains[END_REF], [START_REF] Jimbo | Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels[END_REF] and the other works related to the "dumbbell shape" model. The main difficulty of our problem comes from the different scales: the open hole on the side of the tube is of size ε 2 , whereas the diameter of the tube is of size ε. Thin domains involving different order of thickness have been studied in [START_REF] Ciuperca | Reaction-diffusion equations on thin domains with varying order of thinness[END_REF], [START_REF] Casado-Díaz | The diffusion equation in a notched beam[END_REF], [START_REF] Casado-Díaz | Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths[END_REF], [START_REF] Casado-Díaz | Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities[END_REF] and the related works. The methods used in this paper are mainly based on these last articles of J. Casado-Díaz, M. Luna-Laynez and F. Murat.

Notations and main result.

For ε > 0, we consider the domain

Ω ε = (0, 1) × (-ε, 0) × (-ε/2, ε/2) .
We split any x ∈ Ω ε as x = (x 1 , x 2 , x 3 ) = (x 1 , x). Let a ∈ (0, 1) and δ > 0. We denote by ∆ ε the positive Laplacian operator with the following boundary conditions:

       Dirichlet B.C. u = 0 on (0, ε) × {0} × (-ε/2, ε/2) ∪ {1} × (-ε, 0) × (-ε/2, ε/2) ∪ (a -δε 2 /2, a + δε 2 /2) × {0} × (-δε 2 /2, δε 2 /2) Neumann B.C. ∂ ν u = 0 elsewhere.
We denote by H 1 0 (Ω ε ) the Sobolev space corresponding to the above Dirichlet boundary conditions. The domain Ω ε is represented in Figure 2.

In this paper, we show that, when ε goes to 0, the spectrum of the operator ∆ ε converges to the one of the one-dimensional operator A, defined by where

A : D(A) -→ L 2 (0, 1) u -→ -u ′′ x 1 = 0 x 1 = 1 (a, 0, 0) δε 2 ε ε ε
D(A) = {u ∈ H 2 ((0, a) ∪ (a, 1)) ∩ H 1 0 (0, 1) | u ′ (a + ) -u ′ (a -) = αδu(a)
} and where α is the positive constant given by

α = K |∇ζ| 2 , (1.1)
with ζ being the auxiliary function introduced in Proposition 3.2 below. Notice that both ∆ ε and A are positive definite self-adjoint operators and that

∀u, v ∈ D(∆ ε ) , ∆ ε u|v L 2 (Ωε) = Ωε ∇u(x)∇v(x)dx , (1.2) ∀u, v ∈ D(A) , Au|v L 2 (0,1) = 1 0 u ′ (x)v ′ (x)dx + αδu(a)v(a) . (1.3) Let 0 < λ 1 ε < λ 2 ε ≤ λ 3 ε ≤ .
. . be the eigenvalues of ∆ ε and let 0 < λ 1 < λ 2 ≤ λ 3 ≤ . . . be the ones of A. The purpose of this paper is to prove the following result.

Theorem 1.1. When ε goes to 0, the spectrum of ∆ ε converges to the one of A in the sense that

∀k ∈ N * , λ k ε -------→ ε-→0 λ k .
Theorem 1.1 yields a new model for the flute, which is discussed in Section 2. The proof of Theorem 1.1 consists in showing lower-and upper-semicontinuity of the spectrum, which is done is Sections 4 and 5 respectively. We use scaling techniques consisting in focusing to the hole at the place (a, 0, 0). These techniques follow the ideas of [START_REF] Casado-Díaz | The diffusion equation in a notched beam[END_REF] (see also [START_REF] Casado-Díaz | Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths[END_REF] and [START_REF] Casado-Díaz | Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities[END_REF]). The corresponding technical background is introduced in Section 3.

Acknowledgements: the interest of the author for the mathematical models of flutes started with a question of Brigitte Bidégaray and he discovered the work of J. Casado-Díaz, M. Luna-Laynez and F. Murat following a discussion with Eric Dumas. The author also thanks the referee for having reviewed this paper so carefully and so quickly. which are ±i λ k ε . Theorem 1.1 shows that the frequencies λ k ε are asymptotically equal to the frequencies µ > 0 such that µ 2 is an eigenvalue of A. A straightforward computation shows that µ 2 is an eigenvalue of A, with corresponding eigenfunction u, if and only if

u(x) = C sin(µx) x ∈ (0, a) C sin(µa) sin(µ(1-a)) sin(µ(1 -x)) x ∈ (a, 1)
with some C = 0 and with µ > 0 solving

αδ = -µ sin µ sin(µa) sin(µ(1 -a)) := f a (µ) , (2.1)
see Figure 3.

Using the above computations, we can do several remarks about the resonances of the flute with a small open hole, as predicted by our model.

• The eigenfunctions of A corresponds to the expected profile of the pressure in the flute with an open hole, see Figure 3 and the ones of [5], [START_REF] Coltman | Acoustical analysis of the Boehm flute[END_REF] and [START_REF] Wolfe | Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes[END_REF].

• The note of the flute corresponds to the fundamental frequency µ = √ λ 1 . To obtain a given note, one can adjust both δ (the size of the hole) and a (the place of the hole). This enables to place smartly the different holes to obtain some notes by combining the opening of several holes (fork fingering). We can also compute the change of frequency produced by only half opening the hole (half-holing). Notice that changing the shape of the hole affects the coefficient α.

• The overtones of the flute correspond to the other frequencies µ = √ λ k with k ≥ 2. We can see in Figure 3 that they are not exactly harmonic, i.e. they are not multiples of the fundamental frequency. This explains why the sound of flutes, which have only a small hole opened, is uneven and not as pure as the sound produced by a simple tube. In other words, our model directly explains the observation that the effective length approximation depends on the considered frequency. Moreover, when µ increases, the slope of f a becomes steeper due to the factor µ in (2.1) and the solutions of (2.1) are closer to µ = kπ. This is consistant with the observation that high frequencies are less affected by the presence of the hole than low frequencies, see [START_REF] Wolfe | Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes[END_REF] or [25]. However, notice that this is only roughly true since for example one can see on Figure 3 that the second overtone is almost equal to 3π, whereas the fourth one is less close to 5π. This comes from the fact that a = 0.7 is almost a node of the mode sin(3πx).

• Of course, when δ = 0, we recover the equation sin µ = 0 corresponding to the eigenvalue of the open tube without hole. When δ → +∞, i.e. when the hole is very large, we recover the equations sin(µa) = 0 or sin(µ(1 -a)) = 0, which correspond to two separated tubes of lengths a and 1 -a (in fact the part (a, 1) is not important because this is not the part of the tube which is excited by the fipple). When the hole is of intermediate size, the fundamental frequency corresponds to a tube of intermediate length ã ∈ (a, 1), but the overtones are not the same as the ones of the tube of length ã.

• The thin domain techniques used here are general and do not depend on the fact that the section of the tube Ω ε is a square and not a disk. If the surface g(x) of the section of the tube is not constant (think at the end of a clarinet), then the operator ∂ 2 xx in the definition of A must be replaced by 1 g(x) ∂ x (g(x)∂ x .), see [START_REF] Hale | Reaction-diffusion equation on thin domains[END_REF]. Of course, if there are several open holes, then other terms of the type αδu(a)v(a) appear in (1.3).

To conclude, we obtain in this article a mathematical model for the flute with a small open hole, which consists in a one-dimensional operator different from a simple Laplacian operator. It yields simple explanation of some observations as the fact that the overtones are not harmonic.

1

x 2 = 0 1/δε ∂K(hole) ∂K(up) Box K ε Box B ε Figure 4: The cube K ε , part of the half-space K = {x ∈ R 3 , x 2 < 0}
, and the corresponding boundaries. When ε goes to 0, the cube K ε converges to K, whereas the hole ∂K(hole) remains unchanged.

Focusing on the hole: the rescaled problem

When ε goes to zero, if one rescales the domain Ω ε with a ratio 1/(δε 2 ) to focus on the hole, then one sees the rescaling domain Ω ε converging to the half-space x 2 < 0 (see Figure 4). The purpose of this section is to introduce the technical background to be able study our problem in this rescaled frame. For the reader interested in more details about the Poisson problem in unbounded domain, we refer to [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF].

The space Ḣ1 (K)

Let K be the half-space {x ∈ R 3 , x 2 < 0}. For any ε > 0, we introduce the cube

K ε = -1 2δε , 1 2δε × -1 δε , 0 × -1 2δε , 1 2δε
as shown in Figure 4. We denote by ∂K ε (hole) the part of the boundary (-1/2, 1/2) × {0} × (-1/2, 1/2) corresponding to the hole. We denote ∂K ε (up) the remaining part of the upper face. We also denote by ∂K(hole) and ∂K(up) the corresponding parts of the boundary of the half-space K. See Figure 4.

We introduce the space Ḣ1 (K) defined by

Ḣ1 (K) = {v ∈ H 1 loc (K) , ∇v ∈ L 2 (K) and v = 0 on ∂K(hole)} (3.1)
and we equip it with the scalar product

ϕ|ψ Ḣ1 (K) = K ∇ϕ.∇ψ . (3.2)
We also introduce the space Ḣ1 0 (K) which is the completion of

C ∞ 0 (K) = {ϕ ∈ C ∞ (K) / supp(ϕ) is compact and ϕ ≡ 0 on ∂K(hole)} (3.3)
with respect to the Ḣ1 scalar product defined in (3.2). Let χ ∈ C ∞ (K) be such that χ ≡ 1 outside a compact set, χ ≡ 0 on ∂K(hole) and ∂ ν χ ≡ 0 on ∂K(up). Following [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF], we get the following results.

Theorem 3.1. The spaces Ḣ1 (K) and Ḣ1 0 (K) equipped with the scalar product (3.2) are Hilbert spaces and

Ḣ1 (K) = Ḣ1 0 (K) ⊕ Rχ , (3.4)
this sum being a direct sum of closed subspaces. Moreover, a function u ∈ Ḣ1 (K) belongs to Ḣ1 0 (K) if and only if it belongs to L 6 (K). As a consequence, the splitting of u ∈ Ḣ1 (K) given by (3.4) is uniquely determined by u = u + uχ, where

u = lim ε→0 1 |K ε | Kε u(x)dx
is the average of u, which is well defined.

Proof : direct sum (3.4) is a particular case of Theorem 2.15 of [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF]. The equivalence between u ∈ Ḣ1 0 (K) and u ∈ L 6 (K) is given by Theorem 2.8 of [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF]. Let u = u + cχ with u ∈ Ḣ1 0 (K) and c ∈ R. Since u ∈ L 6 (K), we have Kε | u| ≤ C|K ε | 5/6 and thus the average of u is well defined and equal to 0. Since the average of χ is well defined and equal to 1, the average of u is also well defined and it is equal to c.

The function ζ

We now introduce the function ζ, which is used to define the coefficient α in (1.1).

Proposition 3.2. There is a unique weak solution ζ of

       ∆ζ = 0 on K ζ = 0 on ∂K(hole) ∂ ν ζ = 0 on ∂K(up) ζ = 1 (3.
∀ φ ∈ Ḣ1 0 (K) , K ∇ ζ∇ φ = - K ∆χ. φ .
See [START_REF] Simader | The Dirichlet problem for the Laplacian in bounded and unbounded domains[END_REF] for a discussion on this kind of variational problems.

The function ζ yields a different way to write the scalar product in Ḣ1 (K).

Proposition 3.3. The function ζ is the orthogonal projection of χ on the orthogonal space of Ḣ1 0 (K) in Ḣ1 (K). Thus, for all u and v in Ḣ1 (K), there exist two unique functions u and v in Ḣ1 0 (K) such that u = u + uζ and v = v + vζ. Moreover,

u|v Ḣ1 (K) = K ∇ u(x).∇ v(x) dx + αu.v ,
where α is defined by (1.1).

Weak K ε -convergence

As one can see in Figure 4, if (u ε ) is a sequence of functions defined in Ω ε , then the rescaled functions w ε are only defined in the box K ε and not in the whole space K. Hence, we have to introduce a suitable notion of weak convergence. Proposition 3.4. Let (w ε ) ε>0 be a sequence of functions of H 1 (K ε ) vanishing on ∂K ε (hole). Assume that ∃C > 0 , ∀ε > 0 ,

Kε |∇w ε | 2 ≤ C .
Then, there exists a subsequence (w εn ) n∈N , with ε n → 0, which converges weakly to a function w 0 ∈ Ḣ1 (K) in the sense that

∀ϕ ∈ Ḣ1 (K) , Kε n ∇w εn ∇ϕ ------→ εn-→0 K ∇w 0 ∇ϕ .
Moreover, the average of w 0 is given by

w 0 = lim εn→0 1 |K εn | Kε n w ε . (3.6)
Before starting to prove Proposition 3.4, we recall Poincaré-Wirtinger inequality.

Lemma 3.5. (Poincaré-Wirtinger inequality)

There exists a constant C > 0 such that, for any ε > 0 and any function ϕ

∈ H 1 (K ε ), Kε ϕ(x) - 1 |K ε | Kε ϕ(s)ds 6 dx ≤ C Kε |∇ϕ(x)| 2 dx 3 . ( 3

.7)

Proof : First, let us set ε = 1. The classical Poincaré inequality (see [START_REF] Evans | Partial differential equations[END_REF] for example) states that

K 1 ϕ(x) - 1 |K 1 | K 1 ϕ(s)ds 2 dx ≤ C K 1 |∇ϕ(x)| 2 dx .
Thus, the right-hand side controls the H 1 -norm of ϕ -ϕ. Then, the Sobolev inequalities shows that (3.7) holds for ε = 1. Now, the crucial point is to notice that the constant C in (3.7) is independent of the size of the cube K ε since both sides of the inequality behave similarly with respect to scaling.

Proof of Proposition 3.4 : First notice that Ḣ1 0 (K) is separable due to the density of C ∞ 0 -functions. Hence, Ḣ1 (K) is also separable and by a diagonal extraction argument, we can extract a subsequence ε n → 0 such that for all ϕ ∈ Ḣ1 (K), Kε n ∇w εn ∇ϕ converges to a limit L(ϕ) with L(ϕ) ≤ C ϕ Ḣ1 . By Riesz representation Theorem, there exists w 0 ∈ Ḣ1 (K) such that L(ϕ) = w 0 |ϕ .

To prove (3.6), we follow the arguments of [START_REF] Casado-Díaz | The diffusion equation in a notched beam[END_REF]. We set w ε = 1 |Kε| Kε w ε . Let p ∈ N. Lemma 3.5 and the fact that Kε |∇w ε | 2 is bounded, show that there exists a constant C independent of ε such that Kε |w ε (x) -w ε | 6 dx ≤ C. Thus,

∀ε < 1 p , K 1/p |w ε (x) -w ε | 6 dx ≤ C . (3.8)
By Sobolev inequality, we know that w ε is bounded in L 6 (K 1/p ) (remember that w ε vanishes on ∂K 1/p (hole)). Thus w ε is bounded and up to extracting another subsequence, we can assume that w εn converges to some limit β ∈ R. By a diagonal extraction argument, we can also assume that w εn converges to w 0 weakly in L 6 (K 1/p ), for any p ∈ N. As a consequence, (3.8) implies that

K 1/p |w 0 (x) -β| 6 dx ≤ lim sup ε-→0 K 1/p |w ε (x) -w ε | 6 dx ≤ C .
Since the previous estimate is uniform with respect to p ∈ N and since K 1/p grows to K when p goes to +∞, we obtain that w 0 -β belongs to L 6 (K) and thus w 0 -βχ ∈ L 6 (K). Theorem 3.1 shows that w 0 -βχ belongs to Ḣ1 0 (K) i.e. β = w 0 .

Lower-semicontinuity of the spectrum

This section is devoted to the following result.

Proposition 4.1. ∀k ∈ N * , 0 ≤ lim sup ε→0 λ k ε ≤ λ k .
Proof : Let (u k ) be a sequence of eigenfunctions of A corresponding to the eigenvalues λ k . Since A is symmetric, we may assume that u k |u j L 2 (0,1) = 0 for k = j. The main idea of the proof of Proposition 4.1 is to construct an embedding I ε : H 1 0 (0, 1) → H 1 0 (Ω ε ) such that the functions I ε u k are almost L 2 -orthogonal and such that

Ωε |∇I ε u k | 2 Ωε |I ε u k | 2 ------→ ε-→0 λ k . (4.1)
The definition of the embedding I ε : H 1 0 (0, 1) → H 1 0 (Ω ε ) is as follows. Far from the hole: we split the functions u k into two parts u k |(0,a) and u k |(a,1) , we slightly rescale them so that they are defined in (ε, a -ε/2) and (a + ε/2, 1) respectively, and we embed both parts in L 2 (Ω ε ) by setting

ϕ k ε (x) = u k a a -3ε/2 (x 1 -ε) and ψ k ε (x) = u k a + 1 -a 1 -a -ε/2 (x 1 -a -ε/2) .
Near the hole: let ζ ∈ Ḣ1 (K) be as in Proposition 3.2 and let ζ = ζ + χ be the splitting given by Theorem 3.1 (where we use that ζ = 1 by definition). By the definition of Ḣ1 0 (K), there exists a sequence of functions ( ζε ) ∈ C ∞ 0 (K) converging to ζ in Ḣ1 0 (K). Therefore, there exists a sequence ζ ε = ζε + χ ∈ C ∞ (K) ∩ Ḣ1 (K) such that ζ ε ≡ 1 outside a compact set and (ζ ε ) converges strongly to ζ when ε goes to zero. Notice that we may assume that ζ ε ≡ 1 outside a compact set of the cube K ε defined in Section 3. We set ζε (x) = ζ ε ((x -(a, 0, 0))/(δε 2 )) and I ε u k = u k (a) ζε in the cube B ε = (a, 0, 0) + δε 2 K ε . Summarizing: the whole embedding I ε is described by Figure 5. Calculing the scalar products: by change of variables, we have

Bε | ζε | 2 = δ 3 ε 6 Kε |ζ ε | 2 ≤ δ 3 ε 6 Kε | ζε | 2 1/2 + Kε |χ| 2 1/2 2 .
Since χ is a bounded C ∞ function and since the volume of K ε is of order 1/ε 3 , we have

Kε |χ| 2 = O(1/ε 3 ). Due to Theorem 3.1, ζε converges to ζ in L 6 (K) and thus Kε | ζε | 2 ≤ Kε | ζε | 6 1/3 Kε 1 2/3 = O(1/ε 2 ) . ε ε ε u k ε = ψ k ε u k ε = ϕ k ε Box B ε u k ε ≡ 0 u k ε ≡ u k (a) u k ε = u k (a) ζε
= I ε u k of u k ∈ H 1 0 (0, 1) into H 1 0 (Ω ε ) (lateral view).
Therefore, we get that

Bε | ζε | 2 = O(ε 3 ). Thus, the L 2 -norm of u k ε = I ε u k is mostly due to the L 2 -norms of ϕ k
ε and ψ k ε and so, for any k and j,

u k ε |u j ε L 2 (Ωε) = ε 2 u k |u j L 2 (0,1) + o(ε 2 ) . (4.2)
On the other hand, we have

Bε |∇ ζε | 2 = δε 2 Kε |∇ζ ε | 2 . Since (ζ ε ) converges to ζ in Ḣ1 (K)
and due to the definition (1.1) of α, Kε |∇ζ ε | 2 converges to α. Therefore, for any k and j,

Ωε ∇u k ε ∇u j ε = Ωε ∇ϕ k ε ∇ϕ j ε + Ωε ∇ψ k ε ∇ψ j ε + u k (a)u j (a) Bε |∇ ζε | 2 = ε 2 a 0 ∂ x u k (x)∂ x u j (x)dx + 1 a ∂ x u k (x)∂ x u j (x)dx + δu k (a)u j (a) Kε |∇ζ ε | 2 + o(ε 2 ) = ε 2 1 0 ∂ x u k (x)∂ x u j (x)dx + αδu k (a)u j (a) + o(ε 2 ) = ε 2 Au k |u j L 2 + o(ε 2 ) . (4.3)
Hence the previous estimates yield the limit (4.1).

Applying the Min-Max formula: for ε small enough, (4.2) implies that the functions u k ε are linearly independent. Due to the Min-Max Principle (see [START_REF] Reed | Methods of Modern Mathematical Physics IV: Analysis of Operators[END_REF] for example), we know that

λ k ε ≤ min p 1 <p 2 <...<p k max c∈R k * Ωε | k i=1 c i ∇I ε u p i | 2 Ωε | k i=1 c i I ε u p i | 2 . ( 4.4)
The above estimates (4.2) and (4.3) show that, for any c ∈ R k * , we have

Ωε | k i=1 c i ∇I ε u p i | 2 Ωε | k i=1 c i I ε u p i | 2 = A k i=1 c i u p i | k i=1 c i u p i L 2 k i=1 c i u p i 2 L 2 + o(1) ,
where the remainder o( 1) is uniform with respect to c when ε goes to zero. Using the Min-Max Principle another time, we get min

p 1 <p 2 <...<p k max c∈R k * Ωε | k i=1 c i ∇I ε u p i | 2 Ωε | k i=1 c i I ε u p i | 2 = λ k + o(1) .
This finishes the proof of Proposition 4.1.

5 Upper-semicontinuity of the spectrum Let ε > 0 and let (u k ε ) be a sequence of eigenfunctions of ∆ ε corresponding to the eigenvalues (λ k ε ). We can assume that the functions u k ε are orthogonal in L 2 (Ω ε ) and that u k ε L 2 = ε. To work on a fixed domain, we set Ω = (0, 1) 3 and we introduce the functions

v k ε = Ju k ε where J is the canonical embedding of H 1 (Ω ε ) into H 1 (Ω), that is that Ju k ε (y) = v k ε (y) = v k ε (y 1 , ỹ) = u k ε (y 1 , εỹ) .
We have

-∂ 2 y 1 y 1 + 1 ε 2 ∂ 2 ỹ ỹ v k ε = λ k ε v k ε and v k ε L 2 = 1 .
By multiplying the previous equation by v k ε and integrating, we get

Ω |∂ y 1 v k ε | 2 + 1 ε 2 |∂ ỹv k ε | 2 = λ k ε .
(5.1) Proposition 4.1 shows that (λ k ε ) ε>0 is bounded. Therefore, up to extracting a subsequence, we may assume that (λ k ε ) converges to λ k 0 = lim inf ε→0 λ k ε when ε goes to 0 and that (v k ε ) converges to a function v k 0 ∈ H 1 (Ω), strongly in H 3/4 (Ω) and weakly in H 1 (Ω). Moreover, (5.1) shows that v k 0 depends only on y 1 . In the following, we will abusively denote by v k 0 , either the function in H 1 (Ω) or the one-dimensional function in H 1 (0, 1).

The purpose of this section is to use the methods of [START_REF] Casado-Díaz | The diffusion equation in a notched beam[END_REF] (see also [START_REF] Casado-Díaz | Asymptotic behavior of diffusion problems in a domain made of two cylinders of different diameters and lengths[END_REF] and [START_REF] Casado-Díaz | Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities[END_REF]) to prove the following result. Proof : We recall that the functions v k ε are orthonormalised in L 2 (Ω) and converge strongly in L 2 (Ω) to v k 0 . Thus, the functions v k 0 are also orthonormalised. Since λ k 0 = lim inf ε→0 λ k ε , we know that λ 1 0 ≤ λ 2 0 ≤ . . . ≤ λ k 0 . Then, Proposition 5.1 shows that λ 1 0 , . . ., λ k 0 are k eigenvalues of A with linearly independent eigenfunctions, and thus that the largest one λ k 0 is larger than λ k .

The proof of Proposition 5.1 splits into several lemmas. To simplify the notations, we will omit the exponent k in the remaining part of this section and we will write u ε for u k ε , v ε for v k ε etc.

Lemma 5.3. Let B ε ⊂ Ω ε be any cube of size ε and let Γ ε be one of its faces. Then,

1 ε 3 Bε u ε (x)dx = 1 ε 2 Γε u ε (x)dx + o(1) . (5.2)
As a consequence, v 0 satisfies both Dirichlet boundary conditions v 0 (0) = v 0 (1) = 0.

Proof : We split the cube in slices B ε = ∪ s∈[0,ε] Γ ε (s) with Γ ε = Γ ε (0) and we set x = (s, x) with x ∈ Γ ε (s). For each s, we have

Γε(s) u ε (s, x)dx - Γε(0) u ε (0, x)dx ≤ Γε(ξ) s 0 |∇u ε (ξ, x)| dξdx ≤ ε √ s Γε(ξ) s 0 |∇u ε (ξ, x)| 2 dξdx .
To show (5.2), we integrate the above inequality from s = 0 to s = ε and we notice that

∇u ε L 2 = λ ε u ε L 2 = λ ε ε = O(ε).
The fact that v 0 (1) = 0 follows from v ε (1, x) = 0 and the strong convergence of v ε to v 0 in H 3/4 (Ω). To obtain the other Dirichlet boundary condition, we apply (5.2) to the cube

B ε = [0, ε] × [-ε, 0] × [-ε/2, ε/2]
at the left-end of Ω ε . Since u ε vanishes on the upper face of B ε , the average of u ε goes to zero in B ε . Applying (5.2) again, the average of u ε goes to zero on the left face of B ε . Thus, the average of v ε goes to zero on the left face Γ = {0} × [-1, 0] × [-1/2, 1/2] of Ω and hence Γ v 0 (0, ỹ)dỹ = 0 because v ε converges to v 0 in H 3/4 (Ω). Since v 0 does not depend on ỹ, this yields v 0 (0) = 0.

We now focus on what happens close to the hole at (a, 0, 0). To this end, we use the notations of Section 3 and we introduce the functions

w ε ∈ H 1 (K ε ) defined by ∀x ∈ K ε , w ε (x) = u ε ((a, 0, 0) + δε 2 x) .
The functions w ε will be useful to study the behaviour of u ε in the cube

B ε = (a -ε/2, a + ε/2) × (-ε, 0) × (-ε/2, ε/2) .
We show that they weakly converges to v 0 (a)ζ in Ḣ1 (K) in the following sense.

Lemma 5.4. For all ϕ ∈ Ḣ1 (K),

Kε ∇w ε ∇ϕ -----→ ε-→0 v 0 (a) K ∇ζ∇ϕ . Proof : We have Kε |∇w ε | 2 = 1 δε 2 Bε |∇u ε | 2 ≤ 1 δε 2 Ωε |∇u ε | 2 = 1 δε 2 λ ε Ωε |u ε | 2 = λ ε δ .
Moreover, the average of w ε in K ε is equal to the one of u ε in B ε , which converges to v 0 (a) due to Lemma 5.3 and the convergence of v ε to v 0 in H 3/4 (Ω). Applying Proposition 3.4, we obtain the weak convergence of a subsequence of w ε to a limit w 0 , whose average is w 0 = v 0 (a). To prove Lemma 5.4, it remains to show that w 0 = v 0 (a)ζ, which does not depend on the chosen subsequence (ε n ). Let ϕ ∈ C ∞ 0 (K) and assume that ε is small enough such that supp(ϕ) ⊂ K ε . We set Thus, w 0 is orthogonal to C ∞ 0 (K) and hence to Ḣ1 0 (K) and Proposition 3.3 implies that w 0 = w 0 ζ. Since we already know that w 0 = v 0 (a), Lemma 5.4 is proved.

Proof of Proposition 5.1 : We have shown in Lemma 5.3 that v 0 satisfies Dirichlet boundary condition at x 1 = 0 and x 1 = 1. Let φ ∈ H 1 0 (0, 1) be a test function. We also denote by φ the canonical embedding of φ into H 1 (Ω). We embed φ into Ω ε by setting φ ε = I ε φ, where I ε is the embedding introduced in the proof of Proposition 4.1. Using the notations of Figure 5 The limits of the different terms are as follows. First, notice that

Ωε ∇u ε ∇φ ε = λ ε Ωε u ε φ ε = ε 2 λ ε Ω v ε Jφ ε
where Jφ ε is the canonical embedding of φ ε in H 1 (Ω). Obviously, Jφ ε converges to Jφ in L 2 (Ω) and we know that v ε converges to v 0 in L 2 (Ω). Thus,

Ωε ∇u ε ∇φ ε = ε 2 λ 0 Ω v 0 φ + o(ε 2 ) = ε 2 λ 0 1 0 v 0 φ + o(ε 2 ) .
In the parts x 1 < a -ε/2 and x 1 > a + ε/2, we know that v ε converges to v 0 weakly in H 1 (Ω) and obviously Jϕ ε and Jψ ε converge to φ strongly in H 1 . Moreover, notice that Jϕ ε and Jψ ε only depends on x 1 . Hence,

x 1 <a-ε/2 ∇u ε ∇ϕ ε + x 1 >a+ε/2 ∇u ε ∇ψ ε = ε 2 x 1 <a-ε/2 ∂ x 1 v ε ∂ x 1 (Jϕ ε ) + x 1 >a+ε/2 ∂ x 1 v ε ∂ x 1 (Jψ ε ) = ε 2 a 0 ∂ x 1 v 0 ∂ x 1 φ + 1 a ∂ x 1 v 0 ∂ x 1 φ + o(ε 2 ) . = ε 2 1 0 ∂ x 1 v 0 ∂ x 1 φ + o(ε 2 ) .
The term of (5. In conclusion, when ε goes to 0, Equality (5.3) shows that

λ 0 1 0 v 0 φ = 1 0 ∂ x 1 v 0 ∂ x 1 φ + αδv 0 (a)φ(a) .
Since this holds for all φ ∈ H 1 0 (0, 1), going back to the variational form of A given in (1.3), this shows that v 0 is an eigenfunction of A for the eigenvalue λ 0 (remember that v 0 L 2 = 1 and so v 0 is not zero).

Figure 2 :

 2 Figure 2: The domain Ω ε . The grey parts correspond to Dirichlet boundary conditions, the other ones to Neumann boundary conditions.

Figure 3 :

 3 Figure 3: Left, the first eigenfunction of A, i.e. the fundamental mode of resonance of the flute with an open hole. Right, the graphic of the function f a and the intersections with the line y = αδ giving the frequencies of the flute. The references values are a = 0.7 and αδ = 5.

5 )

 5 in the sense that ζ ∈ Ḣ1 (K), ζ = 1 and ∀ϕ ∈ C ∞ 0 (K) , K ∇ζ.∇ϕ = 0 . Proof : Theorem 3.1 shows that ζ = χ + ζ with ζ ∈ Ḣ1 0 (K). Then, Proposition 3.2 is a direct application of Lax-Milgram Theorem to the variational equation

Figure 5 :

 5 Figure 5: The embedding u k ε= I ε u k of u k ∈ H 1 0 (0, 1) into H 1 0 (Ω ε ) (lateral view).

Proposition 5 . 1 .

 51 For all k ∈ N * , the function v k 0 is an eigenfunction of A for the eigenvalue λ k 0 . Proposition 5.1 finishes the proof of Theorem 1.1 since we immediately get the uppersemicontinuity of the spectrum. Corollary 5.2. ∀k ∈ N * , lim inf ε→0 λ k ε ≥ λ k .

  ∀x ∈ B ε , φε (x) = ϕ x -(a, 0, 0) δε 2and we extend φε by zero in Ω ε . Sinceu ε L 2 = ε and φε L 2 (Ωε) = δ 3/2 ε 3 ϕ L 2 (K) ,

  , we haveΩε∇u ε ∇φ ε = x 1 <a-ε/2 ∇u ε ∇ϕ ε + φ(a) Bε ∇u ε ∇ ζε + x 1 >a+ε/2 ∇u ε ∇ψ ε (5.3)

3 )

 3 in the box B ε is more delicate, but all the work has already been done in Lemma 5.4. Indeed we have Bε ∇u ε ∇ ζε = δε 2 Kε ∇w ε ∇ζ ε . By definition ζ ε converges to ζ strongly in Ḣ1 (K). Thus, Lemma 5.4 implies that Bε ∇u ε ∇ ζε = δε 2 v 0 (a) K ∇ζ∇ζ + o(ε 2 ) = αδv 0 (a)ε 2 + o(ε 2 ) .

  arguments are based on recent thin domain techniques of

		Sketch of the tube (the open parts are in grey)	main resonance mode frequencies
	flute, recorder,	π/L,
	open	organ	2π/L,
	pipe. . .		3π/L . . .
	closed	organ	π/2L,
	pipe,	pan-	3π/2L,
	pipes. . .		5π/2L . . .
	reed	instru-	π/2L,
	ments (clarinet,	3π/2L,
	oboe. . . )		5π/2L . . .
	Figure

We use in this article the mathematical habit to identify the frequencies to the eigenvalues of the wave operator. To obtain the real frequencies corresponding to the sound of the flute, one has to divide them by

2π


J Delahaye

J Honoré

T Grenet

Slow conductance relaxation in insulating granular Al: evidence for screening effects

Keywords: numbers: 72.80.Ng, 61.20.Lc, 73.23.Hk

It is shown that the conductance relaxations observed in electrical field effect measurements on granular Al films are the sum of two contributions. One is sensitive to gate voltage changes and gives the already reported anomalous electrical field effect. The other one is independent of the gate voltage history and starts when the films are cooled down to low temperature. Their relative amplitude is strongly thickness dependent which demonstrates the existence of a finite screening length in our insulating films and allows its quantitative estimate (about 10nm at 4K). This metallic-like screening should be taken into account in the electron glass models of disordered insulators.

During the last 20 years, slow and glassy conductance relaxations were found in several disordered insulating systems [1][2][3]. These relaxations have been revealed and often studied by means of field effect measurements. In such experiments, disordered insulating films are used as (weakly) conducting channels of MOSFET devices which allow us to measure their conductance response to gate voltage (V g ) changes. After a quench at e.g. 4K, a slow and endless decrease of the conductance is found as long as V g is kept constant, and any V g change triggers a new conductance relaxation. Moreover, the system keeps some memory of its V g history: Any stay under a fixed V g value remains printed for some time in G(V g ) sweeps as a conductance dip centered on this value. We have also shown recently in granular Al thin films that these V g induced relaxations display aging, i.e. the dynamics of the system depends on the time spent at low temperature, a characteristic property of glassy systems [4].

Several experimental findings suggest that these slow conductance relaxations could reflect the properties of the "electron glass" [5,6], a glassy state theoretically predicted in the 1980s for disordered insulators [7,8]. According to theoretical and numerical studies, a system of electrons with ill-screened interactions and disorder will need an infinite time to reach its equilibrium state at low temperature, the relaxation towards equilibrium being characterized by a conductance decrease of the system [9]. Up to now, the electron glass problem has received considerable theoretical developments but only few experimental illustrations, which explain the interest in the electrical field effect results. Moreover, most of the experiments have focused on the electrical conductance and its V g -induced relaxations but less is known concerning the dielectric properties and their possible time evolution at low T .

Since even after a long stay under a fixed V g the conductance decrease shows no sign of saturation, it is not possible to define the conductance relaxation relative to the equilibrium value, which is unknown. Instead, short excursions to V g values never explored before are often used to define a V g history-free reference conductance G ref [2][3][4][5]. This history-free conductance was sometimes called the "off-equilibrium" conductance since the system has never been allowed to equilibrate at these V g s. In the present Letter, we show that G ref is not constant in time after a cool down to 4.2K in granular Al films thicker than 10nm. We also show that this feature demonstrates the existence of a screening length in our insulating granular films and allows its quantitative estimate.

Our granular Al films were prepared by e-gun evaporation of Al under a partial pressure of O 2 , as described elsewhere [3]. By changing the O 2 pressure, we can tune the resistance of the films from metallic to insulating. For insulating samples, x-rays and TEM studies have revealed an assembly of crystalline Al grains with a typical size of a few nanometers. They are believed to be separated by thin insulating Al 2 0 3-x layers. MOSFET devices are made by deposition of granular Al films on top of heavily doped Si wafers (the gate) covered by a 100nm thick thermally grown SiO 2 layer (the gate insulator).

The film conductance G was measured by using a two terminal ac technique, employing a FEMTO current amplifier DLPCA 200 and a lock in amplifier SR 7265. Source-drain voltage was such that G stays in the Ohmic regime. All the electronic equipments sensitive to room temperature drifts were placed in a thermalized chamber with a T stability better than 0.1K. This last point was crucial in order to follow G variations with a precision of 10 -3 % during weeks of measurements.

For 4.2K measurements, the MOSFET devices are mounted in a box filled with He exchange gas and plunged into a 100l liquid He dewar. The sample temperature follows the mK variations of the liquid He bath around 4.2K and a carbon glass thermometer close by was used to correct these temperature variations. The time required to cool down the sample from room temperature to 4.2K is about 10min.

For Fig. 1a, a MOSFET device with a 20nm thick granular Al channel was cooled down to 4.2K and main- tained at this temperature under V geq = 0V . Fast V g sweeps from -15 to +15V (250s long) were taken every 6000s after the cool down. All the G(V g ) curves display a conductance dip centered on V geq which reflects the memory of the relaxation associated with the stay under V geq . Far enough from V geq (here for |V g | > 5V ), G is roughly constant and this baseline reflects the "off-equilibrium" conductance G ref previously mentioned. Looking at the time evolution of G(V g ) curves, two features are salient. First, the amplitude of the dip increases as a function of time. This result is well known from previous studies [2][3][4][5]: the longer the stay under V geq , the more pronounced the dip is. Second, and this is the new feature we aim to discuss here, the baseline conductance also decreases as a function of time. As highlighted in Fig. 1b, the G decrease is well described at any V g s by a ln t dependence, more pronounced in the dip region because of the superposed baseline relaxation and dip growth.

To be more quantitative, we can define a parameter whose physical meaning will become clear later, the slope ratio SR of the G relaxation slope in Fig. 1b at V geq to the G ref relaxation slope (at -10V for example). For the sample of Fig. 1, SR ≃ 2.9. We have measured 4 different 20nm thick films with R 2 values from 10M Ω to 10GΩ. A baseline relaxation was always observed, with SR at 4K between 2 and 3. For one film, the gate and the gate insulator were respectively Al and alumina and consistent results were obtained which excludes any role of a specific gate insulator material. Detailed investigation of the vertical electrical homogeneity of the films has also shown that the baseline relaxation is not related to a specific layer close to the film-substrate interface [START_REF]of the vertical electrical homogeneity of the granular Al films[END_REF].

We checked that this baseline relaxation was independent of the V g history by using the "two dip" protocol [5]. After some time under V geq1 , V geq was changed to V geq2 for the rest of the experiment. We observed the formation of a new dip at V geq2 and the erasure of the old one at V geq1 (as already known [3,5]), but the baseline relaxation in any V g range far enough from V geq1 and V geq2 continue as if no V geq change was imposed.

One may think that the baseline relaxation is induced by the V g themselves. Indeed, when measuring G(V g ) curves, a small dip starts to form at each measured point. The baseline thus reflects a short time relaxed G value (10s being typical for our sweep parameters). We know from the "two dip" protocol results that a dip formed during 10s will be erased roughly in about 100s [2,3]. In our case, we wait for 6000s between two V g sweeps, thus no memory of the previous sweep is expected out of the dip region. In one experiment, we checked that the baseline relaxation was unaffected by the suspension of V g sweeps during 30h.

An experimental artifact that may explain the baseline relaxation is the existence of a T drift after the cool down. Because of the rapid divergence of the films resistance at low T , the observed baseline variations correspond typically to drifts of few mK (≃ 2.5mK for the whole relaxation observed at 4.2K in Fig. 1). We have performed different tests (thermometer stability, comparison with and without He exchange gas) which indicate that the T drifts are much smaller than the baseline relaxation amplitude. The effect of the film thickness discussed below will definitely ruled out any interpretation in terms of a T drift.

If the baseline relaxation is a property of the granular film itself, how can we explain it? A simple hypothesis is to state that the granular Al film, although electrically insulating, has a metallic-like screening length L sc that is smaller than its thickness T h (T h = 20nm in Fig. 1). Then, only the layer of the film located at a distance smaller than L sc from the gate insulator is sensitive to V g changes. The conductance dip ∆G dip reflects the relaxation of this layer, whereas the conductance baseline G ref reflects the relaxation of the rest of the film. The conductance relaxation measured at V geq is then the sum of the two contributions: ∆G(V geq , t) = ∆G dip (t) + ∆G ref (t). Since the relaxations at any V g are well described by a ln t dependence, we can write ∆G ref (t) = -A ref ln t and ∆G dip (t) = -A dip ln t. The slope ratio SR is by defini-

tion equal (A

dip + A ref )/A ref .
If the film is homogeneous, it is natural to suggest that A dip and A ref are respectively proportional to L sc (the V g sensitive layer thickness) and T h -L sc (the V g insensitive layer thickness). Then, SR = T h /(T h -L sc ) as long as T h > L sc . The results for the 20nm films (SR between 2 and 3) give estimates for the screening length L sc between 10 and 13nm.

If our simple model is correct, then the baseline relaxation relative to that measured at V geq must change with the thickness of the films. This is illustrated in Fig. 2, where the experiment of Fig. 1 was reproduced for two granular Al films 10nm and 100nm thick. A clear thickness dependence is visible. Qualitatively, the baseline relaxation is almost absent for the 10nm thick film (Fig. 1a, SR = 20) while it is almost equal to that measured at V geq for the 100nm thick film (Fig. 1b, SR = 1.1). The agreement with the above simple picture is also quantitative. These SR values gives L sc estimates of about 10nm, close to the results obtained from 20nm thick films (10 -13nm). We stress that the SR variations observed for different thicknesses are much more important than those observed for a given thickness and different R 2 s.

Our simple model also predicts a thickness dependence for the amplitude of the conductance dip ∆G

/G = [G ref -G(V geq )]/G ref .
∆G/G data, measured for 10nm, 20nm and 100nm thick films are shown in Fig. 3 as a function of their percolation critical resistance R C [START_REF] Shklovskii | Electronic Properties of Doped Semiconductors[END_REF]. In 2D films (T h less than the percolation radius L 0 ), R C ≃ R 2 , but for thicker ones, R C ≃ (T h /L 0 )R 2 . Taking L 0 ≃ 20nm [START_REF] Delahaye | [END_REF], we estimate R C as R 2 for 10 and 20nm films and 5R 2 for 100nm films. All the samples were measured after being kept under V eq = 0V for t = 20h after a cool down at 4.2K. As can be seen and was already reported [1,3], ∆G/G increases significantly with R C . Besides, a clear reduction of ∆G/G is visible for a given R C as T h increases, in qualitative agreement with the fact that the conductance dip originates from a layer of a fixed thickness. The agreement is also quantitative. As long as T h < L sc , ∆G/G(t) does not depend on T h . Let us note its value

C(t) (of course C(t) de- pends on R C ). When T h > L sc , one has ∆G/G(t) = A dip ln t/(G 0 -A ref ln t) ≃ A dip ln t/G 0 = C(t)L sc /T h (∆G/G(t) ≪ 1)
. Then, plotting (T h /L sc ) × (∆G/G) for all the samples should give one single curve C(t, R C ). This is confirmed in the inset in Fig. 3 for which we have assumed a screening length of 10nm for all the films. Finally, our model provides a natural explanation for the increase of the ∆G/G data scatter with the thickness, as observed in Fig. 3. Small variations of the O 2 pressure or of the Al evaporation rate are always present during the deposition of granular Al films. Thus in thick samples, the measured R 2 (and the estimated R C ) does not necessarily reflect the resistance of the unscreened 10nm thick layer deposited on top of the gate insulator which determines the amplitude of the dip. The quality of the scaling in the inset in Fig. 3 is indeed an indirect test of the vertical homogeneity of the films, as discussed in details in Ref. [START_REF]of the vertical electrical homogeneity of the granular Al films[END_REF].

We now come to a discussion of our results. First, note that the thickness dependence of SR clearly demonstrates that the conductance relaxation exists in the "bulk"" of the film and is a property of the granular Al itself (and not of its interface with the gate or the surface oxide layer). The "bulk" relaxation after a cool down at 4.2K is well described by a ln t dependence without any sign of saturation over weeks of measurement. Such a ln t dependence was also observed for the time increase of the conductance dip amplitude after a cool down [2,3] (more complex laws are observed when V geq is fixed a time t e after the cool down, see [4]). It is a natural relaxation law for a system having a 1/τ distribution of relaxation times, as expected for an electron glass [13].

In our granular Al thin films, the electronic transport at finite T is believed to result from electron tunneling between metallic Al grains. A lower bound for L sc is then the typical grain size, which was found in recent TEM measurements to be 2 -4nm. A second important microscopic length scale is the percolation radius L 0 of the critical resistance network. Since in our films R C 100GΩ, the diffusion of charge carriers through the critical resistance network is expected to be faster than the time scale of the measurements (the intergrain capacitance and the capacitance of the granular Al channel to the gate are respectively ≃ 10 -19 F and ≃ 10 -11 F ). Thus, L 0 should be an upper bound for L sc as indeed observed: L 0 estimates from conductance fluctuations measurements are between 20nm and 40nm [START_REF] Delahaye | [END_REF], a few times our L sc value.

We note that no baseline relaxation was reported on 20nm thick indium oxide films which may indicate that L sc (4K) > 20nm in this system. Interestingly enough, L 0 was found to be ≃ 300nm in "crystalline" films [14], i.e. ≃ 10 times our estimate for granular Al films, reflecting a less dense percolating network in indium oxide consistent with a larger screening length.

Our observations address the important question of screening in a disordered insulator, a subject which has been little explored both theoretically and experimentally. Electron glass models are generally developed in the limit of strongly localized electrons [7][8][9] (Ref. [15,16] are exceptions). But in real systems, there is a mobility of the charge carriers at finite T which will give rise to a metallic-like screening. Numerical studies on disordered insulators have indeed found a transition between a metallic screening at high T where most of the electrons are diffusive and a slow dielectric response at low T where most of the electrons remain located in finite size clusters [17][18][19]. Such a transition was observed around 1K in capacitive measurements on a doped semiconductor [20]. According to the usual formula, L 2 sc = ǫǫ 0 /e 2 dN/dµ(E F ), where dN/dµ is the thermodynamic DOS. In a glassy phase and for a finite time t, the system cannot explore all the configurations and the thermodynamic DOS has to be replaced by a pseudo-equilibrium DOS dN/dµ(t) for which only relaxations faster than t are allowed. The first theoretical attempts to do so [7,21] have found that the short time (≃ Maxwell time) screening length diverges as T goes to 0 and decreases as a function of time [21].

In summary, we have described in insulating granular Al films the existence of a V g insensitive conductance relaxation. Its thickness dependence demonstrates the existence of a metallic screening length of about 10nm at 4K. Our results provide a new way to study the screening length and its relaxation in disordered insulators where an anomalous electrical field effect has been found. They also point to the need for more theoretical studies: how are the classical electron glass models predictions affected by the existence of a metallic screening length ?

This research has been partly supported by the French-National Research Agency ANR (contract N o 05-JC05-44044).

FIG. 1 :

 1 FIG. 1: (a) G(Vg) curves measured at different times t after a cool down to 4.2K. Vg = Vgeq = 0V between Vg sweeps. (b) Corresponding G(t) curves for negative Vgs in and out of the conductance dip region. The sample was 20nm thick with R2 = 550M Ω at 4.2K. See the text for the details.

FIG. 2 :

 2 FIG. 2: G(t) measurements after a cool down at 4.2K for: (a) a 10nm thick film, R2 = 6GΩ; (b) a 100nm thick film, R2 = 2M Ω. The G(t) relaxations are plotted at 0V (Vgeq) and at -10V (baseline relaxation G ref ).

FIG. 3 :

 3 FIG.3: ∆G/G values measured 20h after a cool down at 4.2K as a function of RC (see the text for the details). The films are 10nm (squares), 20nm (circles) and 100nm (triangles) thick. The lines are guides for the eye. In the inset, (∆G/G) × (T h /Lsc) versus RC . Typical error bars on each point are about 10%.