Virginie Bonnaillie-No Ël
email: virginie.bonnaillie@bretagne.ens-cachan.fr

Monique Dauge
email: monique.dauge@univ-rennes1.fr

Nicolas Popoff
email: nicolas.popoff@univ-rennes1.fr

Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions

We study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle θ with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit θ → 0 and we prove a full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.

Introduction and main results

The aim of this paper is to study the eigenpairs of a Schrödinger operator with a degenerate electric potential of the form (t cos θ -s sin θ) 2 on the half-plane t > 0. This problem is motivated by the analysis of the third critical field H C 3 in the theory of superconductivity (see for instance [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF]). More precisely, the linearization of the Ginzburg-Landau functional leads to investigate the asymptotics of the lowest eigenvalues of Schrödinger operators with magnetic fields (ih∇ + A) 2 and Neumann boundary conditions on smooth domains Ω in R 3 . Then, near the boundary of Ω, the magnetic field can be approximated by a constant field which makes an angle θ ∈ 0, π 2 with the boundary (approximated by the tangent plane). Thus, after a choice of gauge, we are led to investigate the operator with Neumann conditions on the half-space R 3 + = {(r, s, t) ∈ R 3 : t > 0}:

h 2 D 2 s + h 2 D 2 t + (hD r + t cos θ -s sin θ) 2 ,
where D x denotes -i∂ x for any variable x. The first step in the study of this operator is a Fourier transform in r. If θ = 0, we are led to the so-called de Gennes operator on an halfline (see [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF]). If θ = 0, after a translation in s we are reduced to a Schrödinger operator with an electric potential on the half-plane R 2 + = {(s, t) ∈ R 2 : t > 0}:

h 2 D 2 s + h 2 D 2 t + (t cos θ -s sin θ) 2 .
After a rescaling, we can reduce to the case h = 1.

Thus, this is a natural question to wonder how the eigenpairs of this operator behave when θ goes to 0 (the form domain does not depend continuously on θ).

In this paper, we investigate this question and study the exponential concentration of eigenvectors.

Discrete and essential spectrum of model operators

We denote by x = (s, t) the coordinates in R 2 and by Ω the half-plane:

Ω = R 2 + = {x = (s, t) ∈ R 2 , t > 0}.
We study the self-adjoint Neumann realization on the half-plane Ω of the Schrödinger operator L θ with potential V θ :

L θ = -∆ + V θ = D 2 s + D 2 t + V θ ,
where V θ is defined for any θ ∈ (0, π 2 ) by V θ : x = (s, t) ∈ Ω -→ (t cos θ -s sin θ) 2 .

We can notice that V θ reaches its minimum 0 all along the line t cos θ = s sin θ, which makes the angle θ with ∂Ω. We denote by D N (L θ ) the domain of L θ and we consider the associated quadratic form q θ defined by:

q θ (u) = Ω |∇u| 2 + V θ |u| 2 dx,
whose domain D(q θ ) is:

D(q θ ) = {u ∈ L 2 (Ω), ∇u ∈ L 2 (Ω), V θ u ∈ L 2 (Ω)}.
The operator L θ is positive. We now recall the min-max principle which links the n-th eigenvalue to Rayleigh quotients (see [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]Theorem XIII.1], [23, p. 75]): Proposition 1.1 (min-max principle) Let A be a self-adjoint operator that is bounded from below, q A its quadratic form and D(q A ) its form domain. Let us define

µ n = sup Ψ 1 ,...,Ψ n-1 ∈D(q A ) inf Ψ∈[Ψ 1 ,...,Ψ n-1 ] ⊥ Ψ∈D(q A ), Ψ =1 q A (Ψ) = inf Ψ 1 ,...,Ψn∈D(q A ) sup Ψ∈[Ψ 1 ,...,Ψn] Ψ =1
q A (Ψ) . (1.1)

Then, for each fixed n, we have the alternative "(a) or (b)":

(a) There are n eigenvalues (counted with multiplicity) below the bottom of the essential spectrum, and µ n is the n-th eigenvalue counted with multiplicity;

(b) µ n is the bottom of the essential spectrum, and in that case µ n = µ n+1 = ... and there are at most n -1 eigenvalues (counting multiplicity) below µ n .

Let σ n (θ) denote the n-th Rayleigh quotient of L θ defined by (1.1). Let sp dis (L θ ) and sp ess (L θ ) be its discrete and essential spectrum, respectively. Let us recall some fundamental spectral properties of L θ when θ ∈ 0, π 2 . It is proved in [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF] that sp ess (L θ ) = [1, +∞) and that θ → σ n (θ) is non decreasing. Moreover, the function (0, π 2 ) θ → σ 1 (θ) is increasing, and corresponds to a simple eigenvalue < 1 associated with a positive eigenfunction (see [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF]Lemma 3.6]). As a consequence θ → σ 1 (θ) is analytic (see for example [START_REF] Kato | Perturbation theory for linear operators[END_REF]Chapter 7]). Remark 1.2 By an even reflection through the boundary, our problem is equivalent to a problem set on the whole plane R 2 with a potential which reaches its minimum on the union of two half-lines: L θ has eigenvalues under its essential spectrum if and only if the half-lines are not colinear. It is interesting to note the analogy with quantum wave guides which have eigenvalues below their essential spectrum as soon as their middle fiber has a non zero curvature (see [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF][START_REF] Chenaud | Geometrically induced discrete spectrum in curved tubes[END_REF]). In all our investigations, of fundamental importance is the family of one-dimensional self-adjoint operators H ζ , ζ ∈ R, defined by:

H ζ = H ζ (t; D t ) = D 2 t + (t -ζ) 2 , (1.2)
on their common Neumann domain on the half-line:

{v ∈ H 2 (R + ), t 2 v ∈ L 2 (R + ), v (0) = 0}.
The spectral properties of this family of operators have been studied in [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF]. Let us recall some of these. We denote by µ(ζ) the lowest eigenvalue of H ζ , and by v ζ a normalized associated eigenfunction. We have the following limits (see [7, §3]):

lim ζ → +∞ µ(ζ) = 1 and lim ζ → -∞ µ(ζ) = +∞.
Let us also mention that

µ(ζ) = ζ → -∞ O(ζ 2 ). (1.3)
In addition, the function µ reaches (non degenerately) its minimum denoted by Θ 0 for a unique value ζ 0 (as proved in [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF]Theorem 4.3]). There holds (see [START_REF] Ël | Harmonic oscillators with Neumann condition on the halfline[END_REF] for refined numerical computations): ζ 2 0 = Θ 0 and Θ 0 0.590106125 .

Main results of the paper

Our results concern exponential decay estimates for eigenvectors of L θ and the asymptotic behavior of its eigenvalues in the small angle limit θ → 0. All along this paper, (σ(θ), u θ ) will denote an eigenpair of L θ with σ(θ) < 1. We prove the exponential decay estimates for u θ stated in the following two theorems, improving the results of [START_REF] Raymond | On the semi-classical 3D Neumann Laplacian with variable magnetic field[END_REF]. Our first result gives an isotropic exponential decay with a weight of the type e α|x| :

Theorem 1.3 Let (σ(θ), u θ ) be an eigenpair of L θ with σ(θ) < 1. We have:

∀α ∈ 0, 1 -σ(θ) , ∃C α,θ > 0, q θ (e α|x| u θ ) ≤ C α,θ u θ 2 L 2 (Ω) . (1.4)
Our second result is an anisotropic decay estimate in the orthogonal direction of the zero set of V θ : Theorem 1.4 Let 0 < β < 1 2 . Let (σ(θ), u θ ) be an eigenpair of L θ with σ(θ) < 1. Then there exists a constant K(β) such that

q θ (e βV θ u θ ) ≤ K(β) u θ 2 L 2 (Ω) .
(1.5)

Estimates (1.4) and (1.5) have different performances in different directions: For γ ∈ [0, π], let us consider the points x = r(cos γ, sin γ), r > 0, on the half-line of angle γ with ∂Ω. Then

|x| = r and V θ (x) = r 2 sin 2 (γ -θ).
Thus estimate (1.4) is stronger than (1.5) if γ = θ, but weaker as soon as γ = θ.

Then, we want to analyze the behavior of the eigenvalues below the essential spectrum when θ goes to zero. In a first step, we prove that the number of such eigenvalues tends to infinity: Theorem 1.5 We have the following upper bound for the n-th eigenvalue σ n (θ) of L θ :

σ n (θ) ≤ Θ 0 cos θ + (2n -1) sin θ, ∀n ≥ 1. (1.6)
Remark 1.6 If we denote by n(θ) the number of eigenvalues of L θ below the essential spectrum, we have:

n(θ) ≥ 1 -Θ 0 cos θ 2 sin θ + 1 2 . (1.7)
In [START_REF] Morame | Remarks on the spectrum of the Neumann problem with magnetic field in the half-space[END_REF]Theorem 2.1], it is proved that this number is finite for any chosen θ.

In a second step, we use semi-classical techniques to prove an expansion in powers of θ as θ → 0 for those eigenvalues: Theorem 1.7 For all n ≥ 1, there exists a sequence (β j,n ) j≥0 such that for all N ≥ 1 and J ≥ 1, there exist C N,J > 0 and θ N > 0 such that for all 1 ≤ n ≤ N and 0 < θ < θ N , σ n (θ) is an eigenvalue and

σ n (θ) - J j=0 β j,n θ j ≤ C N,J θ J+1 .
(1.8)

Moreover, β 0,n = Θ 0 and β 1,n = (2n -1) µ (ζ 0 ) 2 .
The proof of this theorem relies on the construction of quasimodes and on projection arguments which show that eigenvectors are close to quasimodes. In this way we prove at the same time a full expansion of eigenvectors associated with the lowest eigenvalues as θ → 0, see Section 4.3. A remarkable feature is that the eigenvector expansions do contain half-integer powers of θ, in contrast with the eigenvalue expansions.

Organization of the paper

After the present introduction, we prove in Section 2 the isotropic and anisotropic decay estimate of Theorems 1.3 and 1.4. In Section 3, we prove Theorem 1.5, which shows that the number of eigenvalues below 1 tends to infinity as the angle θ tends to 0. We also prove that eigenvalues densify on the whole interval [Θ 0 , 1] when θ → 0. Section 4 is devoted to the proof of Theorem 4.1 which immediately implies Theorem 1.7.

In Section 5 we present a series of computations of eigenpairs performed with the finite element library M ÉLINA [START_REF] Mélina | bibliothèque de calculs éléments finis[END_REF]. They illustrate the anisotropic exponential decay of eigenvectors, and also clearly display the four term asymptotic expansion for the n-th eigenvalue of L θ as θ → 0:

σ n (θ) = Θ 0 + (2n -1)a 1 θ -a 2,n θ 2 -a 3,n θ 3 + O(θ 3 ),
where a 1 0.7651882 and a 2,n , a 3,n are some positive coefficients. This expansion is coherent with (1.8), all the more since a 1 coincides with the 7-digit numerical approximation of µ (ζ 0 )/2 according to the 1D computations presented in [START_REF] Ël | Harmonic oscillators with Neumann condition on the halfline[END_REF]. In addition, for small angles θ, the eigenvectors show their resemblance with the quasimodes constructed in tensor product form, cf. Section 4.3 and Figures 8910.

Exponential decay of eigenvectors

The aim of this section is to prove Theorems 1.3 and 1.4. For that purpose, we need to recall some ingredients in order to implement the so-called "Agmon's estimates". These estimates are related to the Agmon distance the main properties of which can be found in [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF] (see also [10, §3.2]).

Preliminaries

Here we recall a few classical identities due to Agmon. There are consequences of the "IMS" formula and can be found in [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF] (see also [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF] and [START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2[END_REF] for the same kind of applications).

Lemma 2.1 Let u ∈ D N (L θ ) and Φ be a bounded and uniform Lipschitz function defined on Ω. Then we have

L θ u, e 2Φ u = q θ (e Φ u) -|∇Φ|e Φ u 2 L 2 (Ω) . (2.1)
Taking u = u θ in Lemma 2.1, we get the obvious corollary:

Corollary 2.2 Let (σ(θ), u θ ) be an eigenpair for L θ and Φ be a bounded and uniform Lipschitz function defined on Ω. We have the following identities:

Ω (σ(θ) + |∇Φ| 2 )e 2Φ |u θ | 2 = q θ (e Φ u θ ), (2.2
)

Ω |∇(e Φ u θ )| 2 + Ω (V θ -σ(θ) -|∇Φ| 2 )e 2Φ |u θ | 2 = 0. (2.3)
Let (Ω + , Ω -) be a partition of Ω: Ω = Ω + ∪ Ω -with Ω + ∩ Ω -= ∅, then we have

Ω + (V θ -σ(θ) -|∇Φ| 2 )e 2Φ |u θ | 2 ≤ sup Ω - V θ -σ(θ) -|∇Φ| 2 Ω - e 2Φ |u θ | 2 . (2.4)
In order to satisfy the hypotheses of this corollary, we will need to perform a partition of unity. This is the aim of the following two lemmas to explain how to deal with such a partition.

Lemma 2.3 Let χ ∈ C ∞ 0 (Ω) and u ∈ D(q θ ), then q θ (χu) = Ω |χ| 2 (|∇u| 2 + V θ |u| 2 ) + 1 2 Ω ∇|χ| 2 ∇|u| 2 + Ω |∇χ| 2 |u| 2 .
(2.5)

If we suppose moreover that u ∈ D N (L θ ), we have:

q θ (χu) = χ 2 L θ u, u + |∇χ|u 2 L 2 (Ω) . (2.6)
Lemma 2.4 Let (χ i ) i be a finite regular partition of unity with i χ 2 i = 1. Then for all u ∈ D(q θ ),

i q θ (χ i u) = q θ (u) + Ω i |∇χ i | 2 |u| 2 .
(2.7)

Isotropic decay of the eigenvectors

This subsection is devoted to the proof of Theorem 1.3. PRELIMINARIES. Let (χ 1 , χ 2 ) be a partition of unity on R + with χ 2 1 + χ 2 2 = 1 and:

0 ≤ χ 1 ≤ 1, χ 1 (r) = 1 if r ≤ 1, and 0 if r ≥ 2, 0 ≤ χ 2 ≤ 1, χ 2 (r) = 0 if r ≤ 1, and
1 if r ≥ 2.
We define

χ R 1 (x) = χ 1 ( |x| R ) and χ R 2 (x) = χ 2 ( |x| R ). (2.8)
We have ∇χ R j (x) = 1 R ∇χ j ( x R ). Thus we deduce:

∃C > 0, ∀j = 1, 2, ∀x ∈ Ω, |∇χ R j (x)| ≤ C R . (2.9)
Let us fix α > 0. As Agmon's distance, we choose the function:

Φ(s, t) = α √ s 2 + t 2 = α|x|.
(2.10)

It clearly satisfies |∇Φ| 2 = α 2 . We do not know yet that e Φ u θ ∈ D(q θ ). This is the reason why we use a cut-off function in order to use the Corollary 2.2. We define for k ∈ N:

   Φ k (x) = α|x| if |x| ≤ k, Φ k (x) = α(2k -|x|) if k ≤ |x| ≤ 2k, Φ k (x) = 0 if |x| ≥ 2k.
We have:

|∇Φ k | 2 = |∇Φ| 2 = α 2 if |x| ≤ 2k, |∇Φ k | 2 = 0 if |x| > 2k.
FIRST STEP. Using (2.2) and (2.7) we have:

Ω (σ(θ) + |∇Φ k | 2 )e 2Φ k |u θ | 2 = 2 j=1 q θ (χ R j e Φ k u θ ) - 2 j=1 Ω ∇χ R j 2 e 2Φ k |u θ | 2 . (2.11)
Let us choose ε ∈ (0, 1 -σ(θ)) and

α = 1 -ε -σ(θ).
(2.12) Thus, we have:

σ(θ) + |∇Φ| 2 = 1 -ε. (2.13)
It follows that:

Ω (σ(θ)+|∇Φ k | 2 )e 2Φ k |u θ | 2 = (1-ε) e Φ k u θ 2 L 2 (Ω) +(σ(θ)-1+ε) |x|>2k |u θ | 2 dx. (2.14) We choose R > 0 such that C 2 R 2 ≤ ε 4 , (2.15)
where C is the constant appearing in (2.9). Hence we get:

2 j=1 Ω ∇χ R j 2 |e Φ k u θ | 2 ≤ ε 2 e Φ k u θ 2 L 2 (Ω) . ( 2
ε 2 e Φ k u θ 2 L 2 (Ω) ≤ e Φ k u θ 2 L 2 (Ω) - 2 j=1 q θ (χ R j e Φ k u θ ) + (σ(θ) -1 + ε) |x|>2k |u θ | 2 dx ≤ e Φ k u θ 2 L 2 (Ω) -q θ (χ R 2 e Φ k u θ ). (2.17)
SECOND STEP. In order to bound from below the energy "far from the origin" q θ (χ R 2 e Φ k u θ ) we introduce a classical notation attached to Persson's lemma:

Σ(L θ , r) = inf q θ (u), u L 2 (Ω) = 1, u ∈ C ∞ 0 (Ω ∩ B r ) ,
where B r denotes the ball of radius r centered at 0, and B r its complement. It results from Persson's lemma (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF]) that the limit of Σ(L θ , r) as r → +∞ equals the bottom of the essential spectrum of L θ , thus 1:

lim r→+∞ Σ(L θ , r) = 1. (2.18)
We have:

q θ (χ R 2 e Φ k u θ ) χ R 2 e Φ k u θ 2 L 2 (Ω) ≥ Σ(L θ , R), (2.19)
and so:

q θ (χ R 2 e Φ k u θ ) ≥ Σ(L θ , R) |x|>2R e 2Φ k |u θ | 2 dx. (2.20)
Using (2.17), we get:

ε 2 e Φ k u θ 2 L 2 (Ω) ≤ |x|<2R e 2Φ k |u θ | 2 dx + (1 -Σ(L θ , R)) |x|>2R e 2Φ k |u θ | 2 dx.
Using (2.18), we can choose R large enough such that, besides (2.15):

1 -Σ(L θ , R) < ε 4 .
We deduce:

ε 4 Ω e 2Φ k |u θ | 2 ≤ |x|<2R e 2Φ k |u θ | 2 dx.
We finally get:

∀k ∈ N, e Φ k u θ 2 L 2 (Ω) ≤ 4 ε e 4αR u θ 2 L 2 (Ω) . (2.21) CONCLUSION. However, |e Φ k u θ | converges pointwise to |e Φ u θ | as k goes to infinity. It follows from Fatou's lemma that e Φ u θ ∈ L 2 (Ω).
The conclusion comes from:

∇(e Φ u θ ) 2 L 2 (Ω) + V θ e Φ u θ 2 L 2 (Ω) = (1 -ε) e Φ u θ 2 L 2 (Ω)
, as a direct consequence of (2.3) and (2.13).

Remark 2.5 This proof is the key point in order to prove that u θ is in the Schwartz's class, see [START_REF] Raymond | On the semi-classical 3D Neumann Laplacian with variable magnetic field[END_REF].

Examining the arguments of this proof, we can see that α and the constant C α,θ can be chosen uniformly in any closed interval

[θ 0 , θ 1 ] with θ 0 > 0 and θ 1 < π 2 . Since σ(θ) → 1 as θ → π
2 , it is impossible to obtain uniform estimates as θ → π 2 . When θ → 0, there is no valid uniform estimates in the tangential variable s. However, considering only a dependence with respect to t, we get a uniform control in θ: Proposition 2.6 Let η < 1. There exist C > 0 and γ > 0 such that for any eigenpair (σ(θ), u θ ) of L θ with σ(θ) ≤ η, there holds

Ω e 2γt |u θ | 2 dsdt ≤ C u θ 2 L 2 (Ω) .
(

2.22)

Proof: The proof is similar as for Theorem 1.3. We choose Φ = γt and instead (2.8) we use the partition of unity

(χ R j ) with respect to t: χ R 1 (x) = χ 1 ( t R ) and χ R 2 (x) = χ 2 ( t R ). (2.23)
The first step of the proof goes the same way with

ε ∈ (0, 1 -η),
and the key point of the second step is then the following lower bound which replaces (2.20):

q θ (χ R 2 e Φ u θ ) ≥ χ R 2 e Φ u θ 2 L 2 (Ω) . (2.24)
This inequality is a consequence of the fact that the support of χ R 2 is now far from the boundary of Ω and that the bottom of the spectrum of the self-adjoint realization on R 2 of

D 2 t + D 2 s + V θ is 1.
Thus, as ε is set and the size of R does not depend anymore on θ for this choice of Φ, we get that the upper bound in (2.21) is independent from θ.

Anisotropic decay from the minimum of the potential

In this section we prove the decay of u θ away from the minimum of V θ stated in Theorem 1.4. Let δ ∈ (0, 1). Following [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF], we introduce the function associated with Agmon's geodesics:

Φ(x) = (1 -δ) √ V θ (x) √ σ(θ) l 2 -σ(θ) + dl, (2.25)
where f + denotes the positive part of a function f . Let us notice that if we define the function

g(d) = d √ σ(θ) (l 2 -σ(θ)) + dl, we have Φ(x) = (1 -δ)g( V θ (x)).
It is an elementary computation to check that we have (uniformly in θ):

g(d) = d→+∞ d 2 2 + O(ln d) and g (d) = d→+∞ d + O(d -1 ). (2.26)
So Theorem 1.4 holds if and only if q θ (e Φ u θ ) is bounded uniformly in θ for all δ ∈ (0, 1).

Let us prove this. We choose δ ∈ (0, 1). By construction of Φ, we have:

|∇Φ| 2 = (1 -δ) 2 (V θ -σ(θ)) + .
(2.27)

Let η > 0, we define a partition of unity for Ω:

A + η = {(s, t) ∈ Ω, V θ (s, t) -σ(θ) > η} and A - η = {(s, t) ∈ Ω, V θ (s, t) -σ(θ) ≤ η}.
On A + η , we have:

V θ -σ(θ) -|∇Φ| 2 = (V θ -σ(θ))(2δ -δ 2 ) > η(2δ -δ 2 ).
(2.28)

Similarly, we have on A - η :

|V θ -σ(θ) -|∇Φ| 2 | = σ(θ) -V θ if V θ < σ(θ), (V θ -σ(θ))(2δ -δ 2 ) if not. Let us assume that 0 < η(2δ -δ 2 ) ≤ Θ 0 < σ(θ). (2.29)
Then, we have the following upper bound:

sup A - η V θ -σ(θ) -|∇Φ| 2 ≤ σ(θ).
(2.30)

We now combine (2.4), (2.28) and (2.30) in order to get:

η(2δ -δ 2 ) A + η e 2Φ |u θ | 2 ≤ A + η (V θ -σ(θ) -|∇Φ| 2 )e 2Φ |u θ | 2 ≤ σ(θ) A - η e 2Φ |u θ | 2 . (2.31)
Since u θ L 2 (Ω) = 1 and that Φ is maximal on the boundary of A - η , we get:

e Φ u θ L 2 (Ω) ≤ σ(θ) η(2δ -δ 2 ) + 1 exp √ σ(θ)+η √ σ(θ) (1 -δ) l 2 -σ(θ) dl. (2.32)
We denote by K(η, δ, σ(θ)) the right hand side of the last inequality. If we fix δ > 0, the function

R + × [Θ 0 , 1] → R (η, σ) → K(η, δ, σ)
is clearly positive and continuous. We notice that:

lim η→0 K(η, δ, σ) = +∞.
Recall that we assume the condition on η given by (2.29). We introduce the interval I(δ) = 0, Θ 0 2δ-δ 2 . This allows us to define the positive constant

K 0 (δ) = max σ∈[Θ 0 ,1] min η∈I(δ) K(η, δ, σ).
The minimum is achieved for a η 0 ∈ I(δ). Choosing this η 0 , we deduce from (2.32):

e Φ u θ L 2 (Ω) ≤ K 0 (δ). (2.33) If we define Φ(x) = 1 - δ 2 √ V θ (x) √ σ(θ) (l 2 -σ(θ)) + dl,
we have

e Φu θ L 2 (Ω) ≤ K 0 δ 2 .
Because of (2.26), we have easily

∃K 1 (δ) > 0, ∀d > 0, |d e -δ 2 g(d) | < K 1 (δ). (2.34) We notice that √ V θ e Φ-Φ = √ V θ e -δ 2 g( √ V θ )
and with (2.34), we deduce:

∃K 1 (δ) > 0, V θ e Φ-Φ L ∞ (Ω) ≤ K 1 (δ). (2.35)
Therefore, we have:

V θ e Φ u θ L 2 (Ω) ≤ V θ e Φ-Φ L ∞ (Ω) e Φu θ L 2 (Ω)
, and finally, with a new constant K 2 (δ):

V θ e Φ u θ L 2 (Ω) ≤ K 2 (δ).
(2.36)

With the definition of Φ, we also get:

|∇Φ| e Φ u θ L 2 (Ω) ≤ K 3 (δ).
(2.37) Using (2.1), we finally obtain

q θ (e Φ u θ ) = |∇Φ|e Φ u θ 2 L 2 (Ω) + σ(θ) e Φ u θ 2 L 2 (Ω) ≤ K(δ).
3 Densification of the spectrum for small angles

In this section, we investigate the behavior of the eigenvalues below 1.

An upper bound

In this section, we give the proof of Theorem 1.5. In order to get the announced upper bound, we will construct a family of quasimodes and use the min-max principle.

Let us introduce first some tools from spectral theory of self-adjoint operators (see for example [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]). We denote by b θ (u) the Rayleigh quotient associated with a function u for L θ :

∀u ∈ D(q θ ) \ {0}, b θ (u) = q θ (u) u 2 L 2 (Ω)
.

The bilinear form associated with q θ is defined on the form domain by:

a θ (u, v) = Ω D t u D t v + D s u D s v + V θ uv dx. Lemma 3.1 Let v ζ 0 be a normalized eigenvector associated with the first eigenvalue Θ 0 of the operator H ζ 0 (cf. (1.
2) and the properties recalled there), and let ψ n be the n-th Hermite function with the "physicists" convention. We recall that:

∀n ≥ 0, ∀x ∈ R, -ψ n (x) + x 2 ψ n (x) = (2n + 1)ψ n (x).
We define the normalized function u n,θ by:

u n,θ (s, t) = (cos θ sin θ) 1 4 v ζ 0 (t √ cos θ) ψ n s √ sin θ - ζ 0 √ tan θ . (3.1)
Then we have:

∀n ∈ N, ∀θ ∈ 0, π 2 , b θ ( u n,θ ) = Θ 0 cos θ + (2n + 1) sin θ. (3.2)
Proof: The function u n,θ is clearly in the form domain D(q θ ). We are going to estimate q θ ( u n,θ ). Let us make the following rescaling and translation:

   y = s √ sin θ - ζ 0 √ tan θ , z = t √ cos θ. (3.3)
Then

q θ ( u n,θ ) = Ω cos θ|v ζ 0 (z)ψ n (y)| 2 + sin θ|v ζ 0 (z)ψ n (y)| 2 + (z √ cos θ -y √ sin θ -ζ 0 √ cos θ) 2 |v ζ 0 (z)ψ n (y)| 2 dy dz = cos θ Ω |v ζ 0 (z)ψ n (y)| 2 + (z -ζ 0 ) 2 |v ζ 0 (z)ψ n (y)| 2 dy dz + sin θ Ω |v ζ 0 (z)ψ n (y)| 2 + y 2 |v ζ 0 (z)ψ n (y)| 2 dy dz -2 √ sin θ √ cos θ Ω y(z -ζ 0 )|v ζ 0 (z)ψ n (y)| 2 dy dz.
We have the following relations:

R + |v ζ 0 (z)| 2 + (z -ζ 0 ) 2 |v ζ 0 (z)| 2 dz = Θ 0 , (3.4)
R + (z -ζ 0 )|v ζ 0 (z)| 2 dz = 0, (3.5) R |ψ n (y)| 2 + y 2 |ψ n (y)| 2 dy = 2n + 1, (3.6)
where (3.5) is a direct consequence of the Feynman-Hellman formula (see [START_REF] Ismail | On the Hellmann-Feynman theorem and the variation of zeros of certain special functions[END_REF] and also Section 4.1). Thus we have

q θ ( u n,θ ) = Θ 0 cos θ ψ n 2 L 2 (R) + (2n + 1) sin θ v ζ 0 2 L 2 (R + ) . (3.7)
Since v ζ 0 and ψ n are normalized, and

u n,θ 2
L 2 (Ω) = 1, we deduce (3.2).
Lemma 3.2 The functions u n,θ , n ≥ 0, are orthogonal for the bilinear form a θ .

Proof: Let n = m be two integers. We recall that R ψ n ψ m = 0. As in the proof of Lemma 3.1, we have:

a θ ( u n,θ , u m,θ ) = Θ 0 cos θ R ψ n (y)ψ m (y) dy + sin θ R ψ n (y)ψ m (y) + y 2 ψ n (y)ψ m (y) dy -2 √ sin θ √ cos θ Ω y(z -ζ 0 )ψ n (y)ψ m (y)|v ζ 0 (z)| 2 dy dz.
For the second term, we make an integration by parts: Since the other terms are clearly equal to 0, we have a θ ( u n,θ , u m,θ ) = 0.

Combining Lemmas 3.1 and 3.2, we deduce Theorem 1.5: Indeed, we only have to apply the min-max principle with the functions (ũ n,θ ) n∈N which are orthogonal for the bilinear form associated with L θ .

We now show that the eigenvalues get dense in [Θ 0 , 1]. Then, there exist an eigenvalue λ of L θ and a constant C ζ > 0 such that:

Spectrum density

|µ(ζ) cos θ + (2n + 1) sin θ -λ| ≤ C ζ √ 2 cos θ sin θ √ n 2 + 1 . (3.8)
Proof: In the same way as previously, we define the functions:

u n,θ; ζ (s, t) = (cos θ sin θ) 1 4 v ζ (t √ cos θ)ψ n s √ sin θ - ζ √ tan θ . (3.9)
where v ζ is the normalized eigenfunction associated with the first eigenvalue µ(ζ) of H ζ (cf. § 1.1). These functions are clearly in the form domain of L θ . We have:

D 2 t u n,θ; ζ (s, t) = cos θ µ(ζ) -(t √ cos θ -ζ) 2 u n,θ; ζ (s, t), D 2 s u n,θ; ζ (s, t) = sin θ 2n + 1 -s √ sin θ - ζ √ tan θ 2 u n,θ; ζ (s, t).
We deduce

L θ u n,θ; ζ -(µ(ζ) cos θ + (2n + 1) sin θ) u n,θ; ζ = 2(cos θ sin θ) 1 2 ζ √ tan θ -s √ sin θ (t √ cos θ -ζ) u n,θ; ζ . (3.10)
Thus, noticing that u n,θ; ζ L 2 (Ω) = 1, we get:

L θ u n,θ; ζ -(µ(ζ) cos θ + (2n + 1) sin θ) u n,θ; ζ L 2 (Ω) = 2(cos θ sin θ) 1 2 (t -ζ)v ζ (t) L 2 (R + ) sψ n (s) L 2 (R) . (3.11) It is well known that R s 2 ψ 2 n (s) ds = n 2 + 1 2 ,
and if we define

C ζ = (t -ζ)v ζ L 2 (R + )
, we can conclude with the spectral theorem.

We can notice that the right part of (3.8) goes to infinity as n gets large, so the previous proposition is useless if θ is fixed and n goes to infinity. However we have the following proposition: Proposition 3. [START_REF] Ël | Harmonic oscillators with Neumann condition on the halfline[END_REF] We have the densification result:

∀λ 0 ∈ (Θ 0 , 1), ∀ε > 0, ∃θ * ∈ 0, π 2 , ∀θ ∈ (0, θ * ], dist sp dis (L θ ), λ 0 < ε. (3.12)
Proof: We only consider the case ε < 1. In the previous lemma we choose n = 0 and ζ such that µ(ζ) = λ 0 , which is possible since Θ 0 < λ 0 < 1 and µ(ζ) takes all values of [Θ 0 , 1) when ζ lays in R + . Thus we get (3.12).

In the next section, we improve the estimate of Theorem 1.5 for each fixed rank n when θ goes to zero.

Asymptotics of eigenvalues in the small angle limit

As it has been proved in Theorem 1.5, when θ goes to zero, the number of eigenvalues n(θ) below the essential spectrum tends to infinity. Thus, for any arbitrary integer N , we can find a value of θ small enough such that σ N (θ) < 1. In this section, we investigate the asymptotics of those eigenvalues and prove Theorem 1.7. We use again the scaling (3.3):

   y = s √ sin θ - ζ 0 √ tan θ , z = t √ cos θ.
In the new variables, the operator L θ rewrites

sin θD 2 y + cos θD 2 z + cos θ(z -ζ 0 -y √ tan θ) 2 = cos θ(L h + Θ 0 ),
where we have set h = tan θ and

L h = hD 2 y + D 2 z + (z -ζ 0 -yh 1/2 ) 2 -Θ 0 . (4.1)
We denote by s n (h) the n-th eigenvalue of L h . Due to the change of variables, we have

σ n (θ) = cos θ Θ 0 + s n (tan θ) .
Thus, Theorem 1.7 is clearly a consequence of the following asymptotics for s n (h) which we are going to establish:

Theorem 4.1 For all n ≥ 1, there exists a sequence (b j,n ) j≥0 such that for all N ≥ 1 and J ≥ 1, there exist C N,J > 0 and h 0 > 0 such that for all 1 ≤ n ≤ N and 0 < h < h 0 :

s n (h) - J j=0 b j,n h j ≤ C N,J h J+1 . Moreover b 0,n = 0 and b 1,n = (2n -1) µ (ζ 0 ) 2 .
Remark 4.2 It follows from Theorem 4.1 that for h small enough the eigenvalues s n (h), 1 ≤ n ≤ N , are simple.

The proof of Theorem 4.1 is organized in two main steps. Using the one-dimensional operators H ζ defined in (1.2), we can rewrite (4.1) as

L h = hD 2 y + H ζ 0 +y √ h (z; D z ) -Θ 0 .
In a first step we construct quasimodes by an expansion in powers of h 1/2 (natural power of h appearing in L h ), and using the spectral theorem, we get a family of approximate eigenvalues (constructed as asymptotic series in powers of h 1/2 and whose odd terms will be zero for some parity reason) and a rough upper bound for s n (h). In a second step, we establish a lower bound. The basic idea to get such a lower bound is to use a Born-Oppenheimer technique which consists of replacing H ζ 0 +y √ h by its ground energy µ(ζ 0 + y √ h) and to implement the standard harmonic approximation in the semi-classical limit for the one-dimensional operator L h,BO defined as:

L h,BO = hD 2 y + µ(ζ 0 + y √ h) -Θ 0 .
However L h,BO , seen as an operator acting on the domain of L h -i.e. as two-dimensional operator, has eigenvalues of infinite multiplicity, and we cannot use directly the min-max principle to compare its spectrum with the eigenvalues of L h . Thus, we have to justify, through Agmon estimates and a Grushin type argument, that the eigenvalues of L h are bounded from below by those of L h,BO seen as one-dimensional operator. Such a procedure was described in [START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF] for degenerate potentials in R n . Nevertheless, we cannot apply directly the techniques of [START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF] because the minimal line of the potential V θ goes to infinity and we work in a domain with boundary.

Construction of quasimodes

We can write L h as:

L h = P 0 + h 1/2 P 1 + hP 2 ,
with:

P 0 = D 2 z + (z -ζ 0 ) 2 -Θ 0 = H ζ 0 -Θ 0 , (4.2)
P 1 = -2(z -ζ 0 )y, (4.3)
P 2 = D 2 y + y 2 . (4.4)
We look for formal series solution of the equation L h u h = γ h u h in the form:

u h j≥0 ϕ j h j/2 and γ h = j≥0 γ j h j/2 .
We are led to the system:

h 0 : (P 0 -γ 0 )ϕ 0 = 0, (4.5)
h 1/2 : (P 0 -γ 0 )ϕ 1 = γ 1 ϕ 0 -P 1 ϕ 0 , (4.6)
h : (P 0 -γ 0 )ϕ 2 = γ 2 ϕ 0 + γ 1 ϕ 1 -P 2 ϕ 0 -P 1 ϕ 1 , (4.7)
h j/2 : (P 0 -γ 0 )ϕ j = j-1 k=0 γ j-k ϕ k -P 2 ϕ j-2 -P 1 ϕ j-1 . ( 4

.8)

Order h 0 . Considering (4.2), we choose γ 0 = 0 and the general solution of (4.5) is

ϕ 0 (y, z) = f 0 (y)v ζ 0 (z), (4.9)
for some f 0 to determine.

Order h 1/2 . To solve (4.6), a necessary and sufficient compatibility condition is:

γ 1 ϕ 0 (y, •) -P 1 ϕ 0 (y, •), v ζ 0 z = 0, ∀y ∈ R,
where •, • z denotes the standard L 2 scalar product on R + with respect to z. Using (4.3) and (4.9) this condition becomes

γ 1 f 0 (y) + 2yf 0 (y) (z -ζ 0 )v ζ 0 , v ζ 0 z = 0, ∀y ∈ R.
In order to evaluate the scalar product, we recall an easy computation. Let us take the derivative with respect to ζ of:

(H ζ -µ(ζ))v ζ = 0.
Choosing ζ = ζ 0 , we get:

(H ζ 0 -Θ 0 )(∂ ζ v ζ ) ζ=ζ 0 = 2(z -ζ 0 )v ζ 0 . (4.10)
We deduce:

R + (z -ζ 0 )v 2 ζ 0 (z) dz = 0. (4.11)
By (4.11), we get γ 1 = 0 and, thanks to (4.10), the general solution of (4.6) is given by:

ϕ 1 (y, z) = yf 0 (y)w ζ 0 (z) + f 1 (y)v ζ 0 (z), with w ζ 0 (z) := (∂ ζ v ζ ) ζ=ζ 0 , (4.12)
and where f 1 is to be determined.

Order h. Taking (4.9) and (4.12) into account we rewrite equation (4.7) in the form

(H ζ 0 -Θ 0 )ϕ 2 = γ 2 f 0 (y)v ζ 0 (z) -P 2 f 0 (y) v ζ 0 (z) -P 1 f 1 (y) v ζ 0 (z) -P 1 yf 0 (y) w ζ 0 (z)
From the previous calculations, we already know that a particular solution of the equation

(H ζ 0 -Θ 0 )ϕ = -P 1 f 1 (y)v ζ 0 (z) is yf 1 (y)w ζ 0 (z).
That is why we look for ϕ 2 in the general form

ϕ 2 (y, z) = ϕ ⊥ 2 (y, z) + yf 1 (y)w ζ 0 (z) + f 2 (y)v ζ 0 (z), (4.13)
where ϕ ⊥ 2 (y, z), v ζ 0 z = 0 for all y ∈ R. Note that as a consequence of the equality v ζ , v ζ z = 1 for all ζ, we have

w ζ 0 , v ζ 0 z = 0.
Thus ϕ ⊥ 2 has to solve

(H ζ 0 -Θ 0 )ϕ ⊥ 2 = γ 2 f 0 (y)v ζ 0 (z) -P 2 f 0 (y) v ζ 0 (z) -P 1 yf 0 (y) w ζ 0 (z).
The corresponding compatibility condition is:

γ 2 f 0 (y)v ζ 0 -P 2 f 0 (y) v ζ 0 -P 1 yf 0 (y) w ζ 0 , v ζ 0 z = 0, ∀y ∈ R, i.e. γ 2 f 0 (y) = P 2 f 0 (y) -2y 2 f 0 (y) (z -ζ 0 )w ζ 0 , v ζ 0 z , ∀y ∈ R, (4.14)
But we have the identity:

(H ζ 0 -Θ 0 )v (2)
ζ 0 = (µ (ζ 0 ) -2)v ζ 0 + 4(z -ζ 0 )w ζ 0 with v (2) ζ 0 (z) := (∂ 2 ζ v ζ ) ζ=ζ 0 .
Taking the scalar product •, v ζ 0 z , we find the well-known identity (see [3, p. 1283-1284] and also [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]):

µ (ζ 0 ) -2 = -4 (z -ζ 0 )w ζ 0 , v ζ 0 z , (4.15)
and the compatibility condition (4.14) becomes

H harm f 0 = γ 2 f 0 , with H harm = D 2 y + µ (ζ 0 ) 2 y 2 . (4.16)
Thus, for f 0 we take an eigenfunction f n, harm of H harm , and for γ 2 the associated eigenvalue

γ 2 = λ n, harm := µ (ζ 0 ) 2 (2n -1) (n ≥ 1). (4.17)
With this choice, ϕ ⊥ 2 exists and is unique.

Further terms Let us assume that the coefficients (γ k ) 0≤k≤j are determined. Let us also assume that, for 0 ≤ k ≤ j the functions ϕ k can be written in the form:

ϕ k (y, z) = ϕ ⊥ k (y, z) + yf k-1 (y)w ζ 0 (z) + f k (y)v ζ 0 (z),
where ϕ ⊥ k (y, •), v ζ 0 z = 0 for all y ∈ R and with the convention f -1 = 0. We assume that (ϕ ⊥ k ) 0≤k≤j and (f k ) 0≤k≤j-2 determined in S(R × R + ) and S(R), respectively, f j-1 and f j being still unknown.

This assumption is proven for j ≤ 2. Let us prove it for j + 1. For this, we write the equation of order j + 1:

(H ζ 0 -Θ 0 )ϕ j+1 = j k=0 γ j+1-k ϕ k -P 2 ϕ j-1 -P 1 ϕ j . (4.18)
We write ϕ j+1 in the form:

ϕ j+1 (y, z) = ϕ ⊥ j+1 (y, z) + yf j (y)w ζ 0 (z) + f j+1 (y)v ζ 0 (z).
Then equation (4.18) implies the following equation in ϕ

⊥ j+1 (H ζ 0 -Θ 0 )ϕ ⊥ j+1 = γ j+1 f 0 v ζ 0 + γ 2 f j-1 v ζ 0 -P 2 (f j-1 v ζ 0 ) -P 1 (yf j-1 w ζ 0 ) + R j , (4.19)
where

R j = j-2 k=1 γ j+1-k ϕ k + γ 2 ϕ ⊥ j-1 + γ 2 yf j-2 w ζ 0 -P 2 ϕ ⊥ j-1 -P 2 (yf j-2 w ζ 0 ) -P 1 ϕ ⊥ j
is known and belongs to S(R × R + ). The compatibility condition ensuring the solvability of (4. [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF]) is obtained by taking the scalar product with v ζ 0 . We calculate, cf (4.14)

γ j+1 f 0 (y)+γ 2 f j-1 (y) = (P 2 f j-1 )(y)-2y 2 f j-1 (y) (z -ζ 0 )w ζ 0 , v ζ 0 z -g j (y) , ∀y ∈ R,
where g j = R j , v ζ 0 z belongs to S(R). Thanks to (4.15) this equation in y can be put in the form:

(H harm -γ 2 )f j-1 = γ j+1 f 0 + g j . (4.20)
The compatibility condition ensuring the solvability of (4.20) is obtained by taking the scalar product with f 0 = f n, harm :

γ j+1 + g j , f 0 y = 0.
This determines γ j+1 , and then f j-1 (the unique solution orthogonal to f 0 ) thanks to the Fredholm alternative. The assertion is proven at the order j + 1.

Cancellation of odd terms

Let us now explain why, for j odd, we have γ j = 0. Let us first notice that either f 0 is odd or f 0 is even and that P 1 is odd and P 2 is even (with respect to y) and also that γ 1 = 0. To fix ideas, we deal with the case f 0 even, the other one being completely similar. Then, we observe that ϕ ⊥ 1 is odd with respect to y and ϕ ⊥ 2 is even. In the recursion above, we can assume that, for 0 ≤ k ≤ j, ϕ ⊥ k is even/odd if k is even/odd (with respect to y) and that the already known f k are even/odd if k is even/odd. In addition, we assume that γ k = 0 if k is odd and k ≤ j. Using this recursion assumption, we get that, if j is even/odd, then R j (and thus g j ) is odd/even. If j is even, we get g j , f 0 = 0, thus γ j+1 = 0. Then the Fredholm condition (see (4.20)) implies that f j-1 is odd. Coming back to (4. [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF], we check that the other terms in the right hand side are odd with respect to y. We have that ϕ ⊥ j+1 is orthogonal to v ζ 0 , thus we deduce by uniqueness that ϕ ⊥ j+1 is odd. If j is odd, then γ j+1 does not need to be zero and f j-1 is even. We deduce in the same way that ϕ ⊥ j+1 is even. Thus the assertion is proved by recursion.

This analysis provides a quasimode for L h (for all n and J):

u J n (h) = J j=0 ϕ j h j/2 ,
which satisfies

L h - J j=0 γ 2j,n h j u J n (h) L 2 (Ω) ≤ C n,J h J+1 u J n (h) L 2 (Ω) , (4.21)
where we use the notation γ 2j,n for γ 2j to emphasize the dependence on n. Using the spectral theorem, we immediately deduce that:

Proposition 4.
3 For all N ≥ 1 and J ≥ 1, there exist C N,J > 0 and h 0 > 0 such that for all 1 ≤ n ≤ N and 0 < h < h 0 :

dist sp dis (L h ), J j=0 γ 2j,n h j ≤ C N,J h J+1 .
Remark 4.4 In particular, we observe that, for 1 ≤ n ≤ N and h ∈ (0, h 0 ):

0 ≤ s n (h) ≤ hλ n, harm + C N,1 h 2 ≤ C N h. (4.22)

Lower bound

To get a suitable lower bound of s n (h), we will use the so-called Born-Oppenheimer approximation L h,BO with

L h,BO := hD 2 y + W h (y), with W h (y) = µ(ζ 0 + yh 1/2 ) -Θ 0 ≥ 0.
Thus, we have

∀v ∈ D N (L h ), L h v, v ≥ L h,BO v, v . (4.23)

Localization estimates of Agmon type

Let us take N 0 such that 1 ≤ N 0 ≤ N . We are going to prove some localization of the eigenfunctions of L h associated with (s n (h)) 1≤n≤N 0 . For all 1 ≤ n ≤ N 0 , we will consider a normalized eigenfunction u n (h) associated with s n (h) so that the distinct u n (h) are orthogonal. It is convenient to introduce the sum of the first eigenspaces of L h :

E N 0 (h) = span(u 1 (h), . . . , u N 0 (h)). (4.24)
Combining Proposition 2.6 and the scaling (3.3), we have the following localization with respect to the normal variable z: Proposition 4.5 There exist C > 0, γ > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ) and v ∈ E N 0 (h):

Ω e 2γz |v| 2 dydz ≤ C v 2 L 2 (Ω) . (4.25)
Now we improve Theorem 1.3 by proving an optimal localization with respect to y when h goes to 0: Proposition 4.6 There exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ) and v ∈ E N 0 (h):

Ω e 2|y| |v| 2 dydz ≤ C v 2 L 2 (Ω) . (4.26)
Proof: For v = u n (h), we can write:

q h (e Φ v) - Ω |D z Φ| 2 + h|D y Φ| 2 + s n (h) |e Φ v| 2 dydz = 0, (4.27)
where q h is the quadratic form associated to (4.1). Let us choose Φ(y) = |y|. With (4.23), we deduce:

Ω (W h -h -s n (h))|e Φ v| 2 dydz ≤ 0 (4.28)
Let us consider the positivity of:

W h -h -s n (h)
in a region of the type |y| ≥ C 0 with C 0 > 0.

(i) Using the non-degeneracy of the minimum, we know that it exists ε 0 such that:

W h (y) ≥ µ (ζ 0 ) 4 |y| 2 h, for |y| ≤ ε 0 h -1/2 . (4.29) (ii) With η 0 := min{µ(ζ 0 ± ε 0 )} -Θ 0 , we have W h (y) ≥ η 0 > 0 for |y| ≥ ε 0 h -1/2 .
We deduce from (i) and (ii) that

W h (y) ≥ min η 0 , µ (ζ 0 ) 4 C 2 0 h if |y| ≥ C 0 .
Therefore, using Remark 4.4 and choosing C 0 large enough, and h small enough we get the existence of c > 0 such that, for |y| ≥ C 0 :

W h (y) -h -s n (h) ≥ ch. (4.30)
Combining this with (4.28), we obtain:

ch |y|≥C 0 |e Φ v| 2 dydz ≤ |y|≤C 0 |W h -h -s n (h)| |e Φ v| 2 dydz.
Then we take advantage of Remark 4.4 and bound W h (y) for |y| ≤ C 0 by C 0 h for a suitable constant C 0 to deduce finally

ch |y|≥C 0 |e Φ v| 2 dydz ≤ Ch v 2 L 2 (Ω) .
Thus we have proved (4.26

) for v = u n (h), 1 ≤ n ≤ N 0 .
Using the orthogonality of the eigenvectors u n (h), we obtain (4.26) for v ∈ E N 0 (h).

Combining Propositions 4.5 and 4.6, we get the following two corollaries:

Corollary 4.7 There exist C > 0, δ > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ) and v ∈ E N 0 (h):

Ω e δ(|y|+z) |v| 2 dydz ≤ C v 2 L 2 (Ω) ,
and in particular:

Ω (1 + z|y| 3 + y 6 )|v| 2 dydz ≤ C v 2 L 2 (Ω) . (4.31)
Corollary 4.8 For all ε 0 > 0, there exist h 0 > 0, γ > 0 and C > 0 such that, for all h ∈ (0, h 0 ) and v ∈ E N 0 (h):

|y|≥ε 0 h -1/2 (1 + |y| 4 )|v| 2 dydz ≤ Ce -γh -1/2 v 2 L 2 (Ω) .

Approximation of eigenvectors

We will consider the projection

Π 0 : L 2 (Ω) → L 2 (R) ⊗ span{v ζ 0 } defined by w -→ w, v ζ 0 z v ζ 0 .
The aim of the following proposition is to approximate u n (h) by a tensor product: Proposition 4.9 There exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ):

Q u n (h) -Π 0 u n (h) ≤ Ch 1/2 u n (h) 2 L 2 (Ω) , Q D y u n (h) -Π 0 D y u n (h) ≤ Ch 1/4 u n (h) 2 L 2 (Ω) , Q yu n (h) -Π 0 yu n (h) ≤ Ch 1/2 u n (h) 2 L 2 (Ω) ,
where Q is the quadratic form of

Id L 2 (R) ⊗ (H ζ 0 -Θ 0 ).
Proof: 1) We first notice that, since Θ 0 is the first eigenvalue of H ζ 0 , the quadratic form Q is non-negative:

Q(w) ≥ 0 for all w ∈ L 2 (R) ⊗ B 1 (R + ).
2) We also notice that, since v ζ 0 generates the kernel of H ζ 0 -Θ 0 , there holds

Q(w) = Q(w -Π 0 w), ∀w ∈ L 2 (R) ⊗ B 1 (R + ).
Hence we only have to bound Q u n (h) , Q D y u n (h) , and Q yu n (h) .

3) Using the equation satisfied by u n (h):

L h u n (h) = s n (h)u n (h), (4.32)
and taking the scalar product with u n (h), we find the identity

h D y u n (h) 2 L 2 (Ω) + Q(u n (h)) -2h 1/2 (z -ζ 0 )yu n (h), u n (h) + h yu n (h) 2 L 2 (Ω) = s n (h) u n (h) 2 L 2 (Ω) .
With (4.22) and (4.31), we deduce

Q(u n (h)) ≤ Ch 1/2 u n (h) 2 L 2 (Ω) .
4) Considering again the scalar product of identity (4.32) with u n (h), we observe that, using (4.22) and (4.23):

D y u n (h) 2 L 2 (Ω) ≤ C u n (h) 2 L 2 (Ω) . (4.33)
Moreover, calculating the derivative with respect to y of (4.32), we obtain:

L h D y u n (h) + 2ih 1/2 (z -ζ 0 -yh 1/2 )u n (h) = s n (h)D y u n (h). (4.34)
Taking the scalar product with D y u n (h), we get:

D 2 y u n (h) 2 L 2 (Ω) ≤ Ch -1/2 u n (h) 2 L 2 (Ω) , (4.35)
where we have used (4.31) to control the commutator term.

Considering once more the scalar product of (4.34) with D y u n (h), we infer:

h D 2 y u n (h) 2 L 2 (Ω) + Q(D y u n (h)) + 2ih 1/2 (z -ζ 0 -yh 1/2 )u n (h), D y u n (h) -2h 1/2 (z -ζ 0 )yD y u n (h), D y u n (h) + h yD y u n (h) 2 L 2 (Ω) = s n (h) D y u n (h) 2 L 2 (Ω) .
With the help of (4.31), (4.33), (4.35) and integration by parts, we find

Q(D y u n (h)) ≤ Ch 1/4 u n (h) 2 L 2 (Ω) .
5) Similarly, we multiply (4.32) by y and find the identity

L h (yu n (h)) + 2h∂ y u n (h) = s n (h)yu n (h), (4.36)
from which we deduce

h D y (yu n (h)) 2 L 2 (Ω) + Q(yu n (h)) + 2h ∂ y u n (h), yu n (h) -2h 1/2 (z -ζ 0 )y 2 u n (h), yu n (h) + h y 2 u n (h) 2 L 2 (Ω) = s n (h) yu n (h) 2 L 2 (Ω) .
We obtain finally Q(yu n (h)) ≤ Ch 1/2 u n (h) 2 L 2 (Ω) , which concludes the proof. Corollary 4.10 There exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ) and v ∈

E N 0 (h): v -Π 0 v H 1 (Ω) + yv -Π 0 yv L 2 (Ω) ≤ Ch 1/8 v L 2 (Ω) . (4.37) Proof: Let us assume that w ∈ L 2 (R) ⊗ B 1 (R + ) is such that: w, v ζ 0 L 2 (R +,z ) = 0. Then, we get: Q(w) ≥ (µ 2 (ζ 0 ) -Θ 0 ) w 2 L 2 (Ω)
, where µ 2 (ζ 0 ) denotes the second eigenvalue of H ζ 0 . Therefore the left-hand side of (4.37) is bounded by (µ 2 (ζ 0 ) -Θ 0 ) -1 Q(w) + Q(D y w) + Q(yw) . Then, the conclusion follows from Proposition 4.9.

Corollary 4.11 There exists h 0 > 0 such that for h ∈ (0, h 0 ), the projection Π 0 is an isomorphism from E N 0 (h) onto its range.

Conclusion

For all v ∈ E N 0 (h), we have

(hD 2 y + µ(ζ 0 + yh 1/2 ) -Θ 0 )v, v ≤ s N 0 (h) v 2 L 2 (Ω) .
We recall (1.3) and we have, with Corollary 4.8:

|y|≥ε 0 h -1/2 µ(yh 1/2 + ζ 0 ) -Θ 0 -h µ (ζ 0 ) 2 y 2 v 2 dydz = O(h ∞ ) v 2 L 2 (Ω) .
A Taylor approximation (using that µ is smooth) gives:

|y|≤ε 0 h -1/2 µ(yh 1/2 + ζ 0 ) -Θ 0 -h µ (ζ 0 ) 2 y 2 v 2 dydz ≤ C(ε 0 ) y 3 h 3/2 v 2 L 2 (Ω) ≤ C(ε 0 )h 3 v 2 L 2 (Ω) ,
the last inequality coming from (4.31). We get:

h D 2 y + µ (ζ 0 ) 2 y 2 v, v -Ch 3/2 v 2 L 2 (Ω) ≤ s N 0 (h) v 2 L 2 (Ω) .
Applying Corollary 4.10, we obtain for all v ∈ E N 0 (h):

h D 2 y + µ (ζ 0 ) 2 y 2 Π 0 v, Π 0 v -Ch 9/8 Π 0 v 2 L 2 (Ω) ≤ s N 0 (h) Π 0 v 2 L 2 (Ω) .
With Corollary 4.11 and the min-max principle, we infer that:

h(2N 0 -1) µ (ζ 0 ) 2 -Ch 9/8 ≤ s N 0 (h)
and thus k(n) = n in Proposition 4.3. This ends the proof of Theorem 4.1.

Eigenvector asymptotics

From Theorem 4.1, we obtain that the gap between the eigenvalues is of order h. Thus, combining the construction of Section 4.1 with the spectral theorem (see [24, Lemmas 12-13, Section 9]), we can deduce approximation results for the eigenfunctions u n (h) of L h . Let us denote by ũ n (h) and ũ n (h) the n-th quasimode with one and two terms, respectively, constructed in Section 4.1: We recall

ũ n (h)(y, z) = f n (y) v ζ 0 (z), ũ n (h)(y, z) = f n (y) v ζ 0 (z) + h 1/2 yf n (y) ∂ ζ v ζ ζ=ζ 0 (z). (4.38)
Here f n is the n-th eigenfunction of the harmonic oscillator H harm cf. (4.16). It is straightforward that we can take

f n (y) = ψ n-1 µ (ζ 0 ) 2 1/4 y , n ≥ 1, (4.39)
with ψ m the Hermite function of rank m. Then there exists an eigenmode u n (h) such that

u n (h) -ũ n (h) L 2 (Ω) ≤ C h 1/2 u n (h) L 2 (Ω) , and
u n (h) -ũ n (h) L 2 (Ω) ≤ C h u n (h) L 2 (Ω) . ( 4
u n,θ -ũ n,θ L 2 (Ω) ≤ C θ 1/2 u n,θ L 2 (Ω) and u n,θ -ũ n,θ L 2 (Ω) ≤ C θ u n,θ L 2 (Ω) . ( 4

.42)

Remark 4.12 For the same reason (cf. [START_REF] Vi Šik | Regular degeneration and boundary layer for linear differential equations with small parameter[END_REF]), we get an approximation at any order in power of θ 1/2 of u n,θ when θ goes to 0 (and which is given by the asymptotic series defining the quasimode).

Finite element computation of eigenpairs

In this section, we show computations of the eigenvalues σ n (θ) which display numerically the asymptotics given by Theorem 1.7. We also illustrate the results given by Theorems 1.3, 1.4 on isotropic and anisotropic decay of the eigenvectors of L θ . The simulations have been realized with the Finite Element Library M ÉLINA, see [START_REF] Mélina | bibliothèque de calculs éléments finis[END_REF].

Eigenvalues

We illustrate here the behavior of the eigenvalues of L θ . The operator L θ is defined on the infinite domain Furthermore, by similar arguments, we obtain the monotonicity of σ n (θ; a, b, c) according to each variable a, b or c. The method consists in computing for several sets of values of (a, b, c) with several combinations of rectangular finite elements of different degrees until convergence is found.

Ω = R × R + . We
Figure 1 gives an approximation of the first 4 eigenvalues of L θ below 1. For this, the final choice of (a, b, c) is (100, 100, 100) with rectangular elements of degree Q 10 and 15 elements in each direction. By looking at Figure 1, we can conjecture that the derivative of σ 1 (θ) tends to zero when θ goes to π 2 . This is true indeed, as we prove in the following proposition:

Proposition 5.1 For all θ ∈ 0, π 2 , we have:

σ 1 (θ) cos θ -σ 1 (θ) sin θ > 0.
Moreover, we have:

lim θ→ π 2 θ< π 2 σ 1 (θ) = 0.
Proof: For γ ≥ 0, we introduce the operator:

L(θ, γ) = D 2 s + D 2 t + (t(cos θ + γ) -s sin θ) 2
and we denote by σ 1 (θ, γ) the bottom of its spectrum. Let ρ > 0 and α ∈ (0, π 2 ) satisfy cos θ + γ = ρ cos α and sin θ = ρ sin α.

We perform the rescaling t = ρ -1/2 t, s = ρ -1/2 ŝ and obtain that L(θ, γ) is unitarily equivalent to: ρ(D 2 ŝ + D 2 t + ( t cos α -ŝ sin α) 2 ) = ρL α . In particular, we observe that σ 1 (θ, γ) = ρσ 1 (α) is a simple eigenvalue: there holds σ 1 (θ, γ) = (cos θ + γ) 2 + sin 2 θ σ 1 arctan sin θ cos θ + γ .

(5.1)

Performing the rescaling t = (cos θ + γ)t, we get the operator L(θ, γ) which is unitarily equivalent to L(θ, γ) :

L(θ, γ) = D 2 s + (cos θ + γ) 2 D 2 t + ( t -s sin θ) 2 .
We observe that the domain of L(θ, γ) does not depend on γ ≥ 0. Denoting by ũθ,γ the L 2 -normalized and positive eigenfunction of L(θ, γ) associated with σ 1 (θ, γ), we write:

L(θ, γ)ũ θ,γ = σ 1 (θ, γ)ũ θ,γ .
Taking the derivative with respect to γ, multiplying by ũθ,γ and integrating, we get the Feynman-Hellman formula:

∂ γ σ 1 (θ, γ) = 2(cos θ + γ) Ω |D t ũθ,γ | 2 dsdt ≥ 0.
We deduce that, if ∂ γ σ 1 (θ, γ) = 0, then D t ũθ,γ = 0 and ũθ,γ only depends on s, which is a contradiction with ũθ,γ ∈ L 2 (Ω). Consequently, we have ∂ γ σ 1 (θ, γ) > 0 for any γ ≥ 0. An easy computation using formula (5.1) provides:

∂ γ σ 1 (θ, 0) = σ 1 (θ) cos θ -σ 1 (θ) sin θ.
As recalled in Subsection 1.1, the function σ 1 is analytic and increasing. Thus we deduce:

∀θ ∈ 0, π 2 , 0 ≤ σ 1 (θ) < cos θ sin θ σ 1 (θ).
We get:

0 ≤ lim inf θ→ π 2 θ< π 2 σ 1 (θ) ≤ lim sup θ→ π 2 θ< π 2 σ 1 (θ) ≤ 0,
which ends the proof. Figure 2 gives an approximation of all eigenvalues of L θ for small θ. For this, the final choice of (a, b, c) is (100, 100, 50) with rectangular elements of degree Q 10 and 20 elements in each direction. The figure corroborates the densification of the spectrum in [Θ 0 , 1] described in Section 3.

We now illustrate formula (1.8). According to this formula, we have the convergence:

σ n (θ) -Θ 0 a 1 θ → 2n -1 as θ → 0, with a 1 = µ (ζ 0 ) 2 , (5.2)
for all n ≥ 1. Using numerical computations for H ζ , we find good approximations of Θ 0 and a 1 , cf. Tables 1 We can see that the ratio ρ n,1 (θ) converges to 2n -1, corroborating formula (5.2). We observe that the ratio ρ n,2 (θ) converges to a numerical limit ȃn,2 as θ → 0 for any n = 1, . . . , 7, see Table 1. We can still determine numerically the next term of the expansion. On Figure 5

Computations displayed in

n

Eigenvectors

In this section, we highlight the isotropic exponential decay of the eigenvectors of L θ and the anisotropic one given respectively in Theorems 1.3 and 1.4. Figure 6 illustrates the anisotropic decay for θ close to π/2 : we compute the first eigenpair of L θ (5, 15, 75) on [-5, 15] × [0, 75] for θ = ϑπ/2, ϑ = 0.9, 0.85, 0.8, 0.7 with unit square elements of degree Q 2 . The first eigenvector is localized along the line V θ = 0 and we see the exponential decay far away from this line. When θ is close to π/2, the eigenvector spreads along the line V θ = 0 and the exponential decay far away from the origin is not predominant.

σ1 (θ)

1.0001656284 0.99987798948 0.99910390126 0.99445407220 When θ = ϑπ/2 with ϑ ∈ {0.1, . . . , 0.6}, we observe equivalently the decay far away from the line V θ = 0 and the origin (see (1.4) and (1.5)). Figure 7 gives an approximation of the first eigenvector and eigenvalue of L θ on the computational domain [-15, 25] × [0, 15] with unit square elements of degree Q 6 . We observe also that the first eigenvector spreads less and less along the line V θ = 0 when θ is decreasing and the decay becomes essentially radial.

R

  ψ n (y)ψ m (y) dy = R -ψ n (y)ψ m (y) dy = R (2n + 1 -y 2 )ψ n (y)ψ m (y) dy = -R y 2 ψ n (y)ψ m (y) dy.

Proposition 3 . 3

 33 Let ζ > 0 and n be an integer such that µ(ζ) cos θ + (2n + 1) sin θ < 1.

  bound this infinite domain by a large box R a,b,c := (-a, b)×(0, c) to perform numerical approximations. We compute the eigenvalues denoted by σ n (θ; a, b, c) of the operator L θ (a, b, c) = -∆ + V θ on R a,b,c with Neumann condition on t = 0 and Dirichlet conditions on the artificial boundary {s = -a} ∪ {s = b} ∪ {t = c}. Using the inclusion of the form domain of L θ (a, b, c) in that of L θ , we prove σ n (θ) ≤ σ n (θ; a, b, c).

Figure 1 :

 1 Figure 1: σ n (θ; 100, 100, 100) for n = 1, . . . , 4 (ordinates) versus ϑ = 2θ/π (abscissa). Sampling: ϑ = k/100, 1 ≤ k ≤ 99.

Figure 2 :

 2 Figure 2: σ n (θ; 100, 100, 50) for n = 1, . . . , 17 (ordinates) versus ϑ = 2θ/π (abscissa). Sampling: ϑ = k/200, 2 ≤ k ≤ 20.

  -2 in[START_REF] Ël | Harmonic oscillators with Neumann condition on the halfline[END_REF]: Θ0 = 0.590106125 and ȃ1 0.7651881.Let us denote by σn (θ) the n-th computed eigenvalue of L θ , with a convenient choice of the computational domain (-a, b) × (0, c). On Figure3we represent the functions log 10(2θ/π) -→ ρ n,1 (θ) := σn (θ) -Θ0 ȃ1 θ (5.3)

Figures 4 and 5

 5 allow to evaluate the next terms of the asymptotic expansion for σ n (θ). Indeed on Figure4we represent the functions log 10(2θ/π) -→ ρ n,2 (θ) := Θ0 + ȃ1 (2n -1)θ -σn (θ)

Figure 3 :

 3 Figure 3: Convergence of ρ n,1 (θ) to 2n -1 as θ → 0, n = 1, . . . , 7 (bottom to top).

Figure 5

 5 Figure 5 agrees with the first terms of the asymptotics proved in Theorem 1.7 :σ n (θ) = Θ 0 + a 1 (2n -1)θ -a n,2 θ 2 -a n,3 θ 3 + o(θ 3 ).

Figure 4 :

 4 Figure 4: ρ n,2 (θ) (ordinates) versus log 10(2θ/π) (abscissa). n = 1, . . . , 7 (bottom to top).

Figure 5 :

 5 Figure 5: ρ n,3 (θ) (ordinates) versus log 10(2θ/π) (abscissa). n = 1, . . . , 7 (bottom to top).

Figure 6 :

 6 Figure 6: First eigenmode of L θ for θ = ϑπ/2 with ϑ = 0.9, 0.85, 0.8 and 0.7.

Figure 8 :

 8 Figure 8: First 8 computed eigenvectors and eigenvalues of L θ for θ = 0.0125π.

Figure 9 :

 9 Figure 9: Quasimodes ũ n,θ and quasieigenvalues Θ0 + ȃ1 (2n -1)θ for θ = 0.0125π.

Figure 10 :

 10 Figure 10: Quasimodes ũ n,θ and quasieigenvalues Θ0 + ȃ1 (2n -1)θ for θ = 0.0125π.

  = tan θ and (s, t) given by the change of variables (3.3), we obtain quasimodes for L θ which satisfy, for suitable eigenvectors u n,θ of L θ

		.40)
	Setting
	ũ n,θ (s, t) = ũ n (h)(y, z) and ũ n,θ (s, t) = ũ n (h)(y, z),	(4.41)
	with h

Table 1 :

 1 Numerical limits ȃn,2 .

	1	2	3	4	5	6	7
	ȃn,2 0.32616 1.1577 2.8206 5.3148 8.6402 12.797 17.784

To compute these quasimodes, we approximate the one-dimensional eigenvector v ζ0 with a finite difference method for the operator

D 2 t + (t -ζ 0 ) 2 on [0,[START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF] with Dirichlet condition on t = 10, and ζ 0 = 0.76818365314 according to computations of[START_REF] Ël | Harmonic oscillators with Neumann condition on the halfline[END_REF].


Christophe Krzeminski

Xiaohui Tang

Nicolas Reckinger

Vincent Bayot

Emmanuel Dubois

Process Optimization and Downscaling of a Single Electron Single Dot Memory

Keywords: Process modeling, flash memories, non volatile memories, quantum dot, single-electron device, device scaling, scaling limits, silicon on insulator technology (SOI)

This paper presents the process optimization of a single electron nano-flash electron memory. Self aligned single dot memory structures have been fabricated using a wet anisotropic oxidation of a silicon nano-wires. One of the main issue was to clarify the process conditions for the dot formation. Based on the process modeling, the influence of various parameters (oxidation temperature, nano-wire shape) have been investigated. The necessity of a sharp compromise between these different parameters to ensure the presence of the memory dot has been established. In order to propose an aggressive memory cell, the downscaling of the device has been carefully studied. Scaling rules show that the size of the original device could be reduced by a factor two. This point has been previously confirmed by the realization of single electron memory devices.

I. INTRODUCTION

Conventional flash memory devices are believed to be hardly scalable since the design space vanishes below the 40 nm technology node [START_REF]The International Technology Roadmap for Semiconductors[END_REF]- [START_REF] Lu | Future challenges of flash memory technologies[END_REF]. Since a few years, most of the area reduction has been achieved by the downscaling of isolation and interconnection regions [START_REF] Blauwe | Nanocrystal nonvolatile memory devices[END_REF], [START_REF] Fazio | Future directions of non-volative memory technologies[END_REF]. Alternative and innovative storage structures for nonvolatile memories are strongly needed as the number of stored electrons is declining to dangerous small numbers. Many emerging research devices are under investigation using a large diversity of materials and approaches: phase memory changes [START_REF] Lankhorst | Low-cost and nanoscale non-volatile memory concept for future silicon chips[END_REF], polymer memory [START_REF] Ouyang | Programmable polymer thin film and non-volatile memory device[END_REF], nanocrystal memory [START_REF] Tiwari | A silicon nanocrystals based memory[END_REF]. Among all, Phase Change nanocrystal Memory (PCM) is now acknowledged to be the leading candidate [START_REF] Hudgens | The future of phase change semiconductor memory devices[END_REF]. However silicon nanocrystal remains an interesting approach due to a close compatibility with some standard industrial process and for 10 nm devices [START_REF] Lammers | Flash vendors facing scaling challenges[END_REF]. In this paper, the fabrication of single electron nanoflash memories with silicon nanocrystal is under investigation. Numerous studies have been published on multinanocrystal memories generated by low-pressure chemical vapor deposition [START_REF] Baron | Growth of si nano-crystals on alumina and integration in memory devices[END_REF], [START_REF] Lombardo | Silicon nanocrystal memories[END_REF] by implantation and annealing step [START_REF] Bonafos | Si nanocrystals by ultra-low-energy ion beam-synthesis for nonvolatile memory applications[END_REF] or by aerosol deposition techniques [START_REF] Blauwe | A novel, aerosol-nanocrystal floating-gate device for non volatile memory applications[END_REF]. One strong issue faced by the multi-dots approach is the impact of the dot distribution fluctuations on the electrical properties [START_REF] Pernolia | Modeling of the programming window distribution in multinanocrystals memories[END_REF]. A relatively different approach is followed here since the idea is to improve and investigate the limits of a single dot memory.

The fabrication of this alternative memory is based on an original combination of classical microelectronics process DRAFT steps. The main idea is to separate a silicon dot from the channel using anisotropic oxidation. It has already been shown that pattern dependent oxidation (PADOX) could be used to fabricate small silicon single-electron transistors (SETs) [START_REF] Ono | Fabrication of single electron transistors and circuits using sois[END_REF]. Self-limited oxidation effects and strain have been carefully engineered to generate silicon nanostructures embedded in silicon dioxide using pattern conversion [START_REF] Uematsu | Two-dimensional simulation of pattern-dependent oxidation of silicon nanostructures on silicon-on-insulator substrates[END_REF]. A different approach has been undertaken in the present work. The anisotropic character is generated using the strong dependence of the oxidation kinetics on arsenic doping profile [START_REF] Ho | Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels[END_REF], [START_REF]Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels ii[END_REF]. One issue is to improve the understanding of the conditions for the dot formation.

Based on that background, the standard process is first summarized in section II. Process simulation and models limitations are discussed in section III. Section IV reports on the conditions for the dot formation and on the influence of the oxidation temperature and of the nanowire width. Device downscaling is explored in the last section.

II. DEVICE ARCHITECTURE AND PROCESSING

This section briefly recalls the main process steps since a detailed overview can be found in reference [START_REF] Tang | Self-aligned silicon-on-insulator nano flash memory device[END_REF]. The critical step corresponds to the implantation through a thin SiO 2 film of a high arsenic dose (1×10 15 cm 2 ). The main objective is to create a sharply localized and a very doped area in the silicon active layer (200 nm thick) of a SOI wafer. A nitride layer is deposited on the substrate. E-beam lithography and reactive ion etching are used to pattern the nanowires. Fig. 1 presents the layout of the device. The central part of the device consists of a 130 nm by 130 nm square. Two 100 nm wide constrictions are created in order to realize the connection between the source and drain regions and the central part. The different dimensions are defined so that a dot can be created in the central part of the device while being separated from the source/drain regions. Finally, a wet oxidation step is performed in order to create the dot. Fig. 2 shows a SEM photography of the central part where a single dot memory is obtained. A nano-floating gate is clearly observed on top of the channel.

III. PROCESS SIMULATIONS MODELS

Based on process simulation, the aim of this study is to clarify the conditions that lead to dot memory existence.

Two steps have been identified to play a key role: the wet oxidation step and the lateral overetching effect during patterning. The first one is critical for the generation and the separation of the dot/channel structure. The second one helps indirectly to achieve the desired pattern conversion. The main objective of this section is to discuss the limitations of process simulation with these two process steps.

A. Modeling of the oxidation step

To date, the seminal model of Deal and Grove remains the most popular approach used in process simulators to model the oxide growth [START_REF] Deal | General relationship for the thermal oxidation of silicon[END_REF]. The kinetic of oxide growth can be simply described by the following equation :

X = A 2 • 1 + t A 2 /4B -1 . (1)
The oxide thickness X is described by a linear-parabolic relationship as a function of the oxidation duration t.

The term B A characterizes the growth rate regime:

DRAFT X ∼ B A t for t ≪ A 2

4B

(2)

whereas B gives the parabolic rate constant that governs the diffusion limited regime in the limit of very long oxidation time :

X ∼ √ Bt for t ≫ A 2 4B . (3)
The diffusivity of the oxidant species and the reaction rate k Si can directly be deduced from the linear parabolic constants:

         D = B • N 1 2C * k Si = B A • N 1 C * (4)
where C * is the oxidant solubility in the oxide and N 1 is the number of oxidant molecules incorporated into a unit volume of the oxide.

The well-known breakdown of the Deal and Grove model [START_REF] Massoud | Thermal oxidation in dry oxygen growth-rate enhancement in the thin regime[END_REF]- [START_REF] Krzeminski | Silicon dry oxidation kinetics at low temperature in the nanometric range: Modeling and experiment[END_REF] in the ultra-thin regime is not a significant issue here since only very long wet oxidation are considered. However, three points have to be carefully addressed : the mesh refinement, the influence and the diffusion of arsenic on the oxidation reaction rate and the global influence of strain on oxidation.

1) Grid refinement: One critical issue is the mesh refinement since the dot radius is less than 20 nm. The refinement must be as dense as possible in the region of high doping and more relaxed in other parts of the system (nitride mask, buried oxide). The second aspect is that the grid refinement must be preserved during the simulation of the oxidation process. One solution to preserve the quality of the initial grid during oxidation is to perform simulation using very small time-steps (less than 1 ms) and regenerate and adapt the grid frequently.

2) Influence of arsenic: The influence of arsenic on the reaction rate is described by the theory of Ho and Plummer [START_REF] Ho | Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels[END_REF], [START_REF]Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels ii[END_REF]. The physical effect behind the oxidation rate enhancement is that the silicon Fermi level is shifted by doping. The change of the Fermi level enhances the concentration of vacancy that naturally provides more reaction sites for the oxidation mechanism. The linear oxidation rate in presence of dopants can be expressed by:

B A doped = B A 0 • 1 + γ • (C v -1) (5)
where

B A 0
is the intrinsic linear reaction rate for wet oxidation which depends on many other parameters (temperature, pressure), γ is an empirical temperature dependent parameter and C v is the normalized total vacancy concentration. Since dopants are assumed to influence only the oxidation rate, it should be noticed that the anisotropy will be maximum in the reaction limited regime (Eq. 2) and negligible in the diffusion limited regime (Eq. 3). Fig.

DRAFT [START_REF] Lu | Future challenges of flash memory technologies[END_REF] shows the influence of the linear oxidation rate as a function of the arsenic concentration predicted by the Ho and Plummer model [START_REF] Ho | Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels[END_REF], [START_REF]Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels ii[END_REF]. The concentration should exceed 3×10 19 at/cm 3 in order to observe an increase.

Another effect related to arsenic is the thermal diffusion during the oxidation process. A reduction of the initial arsenic gradient probably takes place during the oxidation process. In order to describe the diffusion of arsenic during this step, a standard five stream model has been used [START_REF] Dunham | A quantitative model for the coupled diffusion of phosphorus and point defects in silicon[END_REF]. The main issue is to have a correct description of the amplitude and the shape of the arsenic peak. A correct description of the arsenic tail and TED (Transient Enhanced Diffusion) effects is not crucial here.

3) Influence of strain: In our specific case, the presence of arsenic has a major impact on the reaction rate.

On the other hand, strain are expected to reduce the oxidation rate. The presence of a nitride mask on top of the nanowire and the concave shape of the nanowire generates strain during the oxidation process. In most of process simulators, stress can be included in the basic Deal and Grove model by considering stress dependent parameters.

As proposed by Kao et al. [START_REF] Kao | Two-dimensional thermal oxidation of silicon: Modeling stress effects in wet oxides[END_REF], [START_REF]Two-dimensional thermal oxidation of silicon: Experiments[END_REF] and Sutardja and Oldham [START_REF] Sutardja | Modeling of stress effects in silicon oxidation[END_REF], the hydrostatic pressure P is often assumed to influence the oxidation diffusivity by:

         D σ 0 = D 0 • exp - P V d k B T for P > 0 D σ 0 = D 0 for P ≤ 0 (6)
On the other hand, a compressive normal stress component at the Si/SiO 2 interface reduces the oxidation rate:

     B A σ = B A • exp - σ nn V k k B T f or σ nn < 0 B A σ = B A for σ nn ≥ 0 (7)
where k B is the Boltzmann constant and T is the oxidation temperature in Kelvin, P is the hydrostatic pressure:

P = - 1 2 (σ xx + σ yy ) (8)
and σ nn is the stress normal to the Si/SiO 2 interface, V d and V k are the corresponding activation volumes. A viscoelastic approach has been used in order to take into account strain relaxation in the oxide [START_REF] Senez | Analysis and application of a viscoelastic model for silicon oxidation[END_REF] 4) Model limitations: As previously described, dopant diffusion and oxidation are strongly related in our specific case. The use of a simulator where all these different models are implemented is required. Moreover, severe constraints are imposed on the grid refinement. Both DIOS [START_REF] Dios | SYNOPSYS TCAD Manuals[END_REF] and TSUPREM4 [START_REF]TSUPREM-4[END_REF] process simulators have been used. Relatively close results have been obtained as far as the kinetics of the dot formation is investigated.

Thank to the DIOS features advanced mesh refinement it has been possible also to investigate the oxidation of the dot.

In order to access the evolution of the various parameters of the memory cell, a monitoring tool has been developed [START_REF] Krzeminski | Optimisation and simulation of a alternative nano-flash memory: the sasem device[END_REF]. The main limitation in our study is the description of the mechanical strain field in process simulators. Fig. 4 illustrates this point where the strain field in the oxide layer is reported for the simulation of the standard process described in the previous section. Two contours (± 700 Mpa) of the hydrostatic pressure are reported based on DRAFT TSUPREM4 simulation. Very large compressive stress levels are observed in the vicinity of the dot whereas the nitride mask is subject to a high tensile stress. As far as we approach the outside silicon dioxide interface, stress decreases as the relaxation is much more important. Such a large level of stress at the nanometer scale indicates that we are too close to the limits for a finite elements simulations to expect a correct description of the mechanical stress around the dot. The use of coupled atomistic/continuous models could be a solution [START_REF] Curtin | Atomistic/continuum coupling in computational materials science[END_REF]. The second limitation is the use of a viscoelastic approach to describe the oxide deformation [START_REF] Senez | Analysis and application of a viscoelastic model for silicon oxidation[END_REF]. It has already been shown by Rafferty et al. [START_REF] Rafferty | Plastic analysis of cylinder oxidation[END_REF] that linear viscous approach strongly overestimates the stress generated during cylinder oxidation at low oxidation temperature and large deformation 1 . In principle, an elastoplastic model would be necessary to properly estimate the stress field surrounding the silicon dot for low oxidation temperatures [START_REF] Rafferty | Plastic flow during thermal oxidation of silicon[END_REF]. The third limitation is the modeling of the pile-up of arsenic at the Si/SiO 2 interface [START_REF] Steen | Characterization of the pile-up of as at the sio2/si interface[END_REF]. Despite these fundamental modeling restrictions, it will be shown that interesting trends can be extracted based on the relatively good description of experimental oxidation kinetics in the presence of arsenic by the coupling of the Ho and Plummer model [START_REF] Ho | Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels[END_REF], [START_REF]Si sio 2 interface oxidation kinetics: A physical model for the influence of high substrate doping levels ii[END_REF] and the Deal and Grove models [START_REF] Deal | General relationship for the thermal oxidation of silicon[END_REF].

B. Modeling of the overetching step

Due to the strong concentration of dopants, it has been observed experimentally that the top of the mesa structure is overetched during its creation. This overetching effect has been observed and explained by Winters et al. [START_REF] Winters | Influence of doping on the etching of si(111)[END_REF].

The concentration of negative etching ions is increased due to the electrostatic interaction between fluorine ions from the reactive plasma at the surface and the dopant charges in the silicon. The increase of the etching rate in the high doping region leads to a trapezoidal shape for the mesa-structure with a neck at the top concentration. This effect has a major impact experimentally on the final device structure and facilitates the creation of the dot. The SEM image (Fig. 5) clearly shows the presence of overetching in the presence of strong arsenic doping. This effect must be discussed and incorporated in our two dimensional simulations. An empirical approach has been adopted in order to match as finely as possible the experimental shape. Fig. 6 shows the shape of the mesa-structure matching the experimental configuration of Fig 5.b). The neck at the top is approximated by a simple trapezoidal shape. Three etching steps are used in the process simulator to generate the full structure. The nitride mask etching is simulated by a perfect anisotropic etching. Next 90 nm of the silicon layer are etched with an angle of 104 • with respect to the horizontal direction to simulate the overetching and the presence of the neck. Finally, a classical anisotropic etch with an angle of 60 • (from the horizontal direction) is performed to match the shape of the mesa-structure at the bottom. All of this process leads to an accurate shape of the mesa-structure matching the experimental one. 1 A two dimensional implementation of an elastoplatic approach is very complex and subject to strong numerical issues. A rough estimation can be obtained by the application of the Rafferty model [START_REF] Rafferty | Plastic analysis of cylinder oxidation[END_REF] to our system. Considering a silicon cylinder of 20 nm and a oxide cylinder of 50 nm gives a radial compressive strain component of 1.88 Gpa and an hydrostatic pressure of 454 Mpa at the Si/SiO 2 interface between the two cylinders DRAFT

IV. CONDITIONS OF THE DOT FORMATION

Experimentally, the creation of a silicon dot has always been observed for 80 minutes of wet oxidation at 800 • C.

Process variations show that the oxidation duration should be defined very finely in order to completely liberate the dot to avoid a complete consumption by the oxidation mechanism. Based on process simulation, the objective is therefore to investigate the best process conditions for dot formation. Two parameters have been identified for their influence on the dot formation: the oxidation temperature and the nanowire geometry.

A. Influence of the oxidation temperature 1) Lifetime of the silicon dot: The most important factor for the reliability of our device is the time window during which the dot is formed but not yet consumed. This point is governed by the difference between the oxidation time to create the dot and that to consume it. Fig. 7 reports the evolution of these parameters as a function of the oxidation temperature for various thermal budgets. First of all, an hyperbolic decrease (∝ t -1 ) is observed for both the creation and consumption time as a function of oxidation temperature. For example, the creation time is 160 min. at 750 • C and reduces to 24 min. at 900 • C. This result tells us that the oxidation temperature must be higher than 750 • C to keep a reasonable processing time. On the other hand, it has been observed that the difference between the creation time and the consumption time vanishes rapidly with increasing temperature. At The oxidation temperature has also clear impact on the size of the dot. The evolution of the dot shape as a function of the oxidation temperature has been simulated. Fig. 8 reports both the channel and dot cross sections. It can be observed that the size of the dot is strongly reduced by increasing the temperature. Fig. 8 shows that the dot maximal half width is about 25 nm at 800 • C and reduces to less than 5 nm for 850 • C. For a higher temperature than 900 • C, no dot is created.

Obviously, the oxidation temperature has a clear impact on the process reliability. It directly influences the oxidation parameters such as the oxidation rate or the strain relaxation mechanism. However, the main striking effect is the reduction of the arsenic gradient responsible for the dot creation, which in turn affects the dot formation. The evolution of the arsenic profile upon annealing in an inert ambience is reported. in Fig. 9 for fixed temperatures and durations corresponding to the onset of the dot creation, as obtained from Fig. 7. For temperature below 850 • C, the arsenic profile remains essentially unaffected by thermal diffusion. Dopant diffusion becomes significant at 900 • C with a strong broadening of the arsenic profile 23 . It is clear that the broadening and the flattening of the arsenic 2 A relevant comparison can be performed between our simulation and the experiments on arsenic TED by Solmi et al. [START_REF] Solmi | Transient enhanced diffusion of arsenic in silicon[END_REF]. It has been observed that effectively, the maximum does not diffuse at 750 • C and 800 • C. When the temperature reaches 900 • C, a clear reduction of the maximum level of doping is observed for a 30 min. annealing.

3 It can also be noticed that the Arsenic maximum (2 × 10 20 at/cm 3 ) is just below the concentration where the diffusivity dramatically increases with doping concentration [START_REF] Larsen | Heavily doping effects in the diffusion of group iv and v impurities in silicon[END_REF].

DRAFT profile reduce the anisotropic character of the oxidation step and explains the difficulty to observe the dot creation at temperatures above 850 • C.

B. Influence of the nanowire width 1) Critical linewidth (experiment) :

The main objective is to estimate the influence of the nanowire initial width (at the top of the mesa-structure) on the dot existence. All the other process parameters related to the implantation or the oxidation step are those reported in section II. In the present experimental study, the top linewidth of the mesa-structure defined by the lithography step varies from 200 nm down to 50 nm. Fig. 10 presents some cross sections after the oxidation step for various linewidths (80, 120 and 200 nm). Precise data about the dot size could not be estimated from SEM analysis but the nanofloating gate diameter is about 20 nm with a significant error bar [START_REF] Tang | Fabrication and room temperature single charging behavior of self-aligned single dot memory devices[END_REF]. The critical experimental linewidth that ensures the dot existence for this set of experiments is observed for 120 nm ± 10 nm. The dot is not separated from the channel for a linewidth larger than 130 nm which is therefore favorable to generate the source/drain regions. On the other hand, for a narrower linewidth than 100 nm, no dot is observed from SEM cross section analysis and the configuration seems suitable for the constriction regions. However, it must be kept in mind that the experimental technique used to observe the dot could weaken the structure since some of the SiO 2 cover is removed and that a very small dot could be present but not observed.

2) Critical linewidth (modeling aspects):

A theoretical study has been undertaken in order to quantify the influence of this parameter. The top linewidth of the mesa-structure used in simulation varies from 200 nm down to 50 nm. The overetching effect is taken into account as described in section III.B. to the constriction, central part and source/drain regions respectively. The direct comparison between the simulated configuration (Fig. 11) and their experimental counterpart (Fig. 10) is of particular interest. It can be noticed that there is a global agreement. Moreover, the triangular shape of the channel as well as the dot geometry (shape, position) is well described. The dot-channel distance is estimated above 45 nm and can be compare to the experimental value of ∼ 40nm in Fig. 11. The main discrepancy is observed for the bottom channel since clearly less oxidation is predicted 4 . Fig. 12 presents also a nice results on the dot creation line as simulation predicts almost exactly the experimental linewidth where a dot is first observed. From the simulation point of view, the dot is just separated from the channel for a 125 nm nanowire linewidth. On the other hand, the theoretical critical range for the dot existence is clearly much larger 40 nm compared to the experimental one. Two possible explanations can be proposed about this difference. The first one is related to the limits of the experimental technique used to observe the single dot as discussed just before-mentioned. The other one is related to the models limitations discussed in section III.A.

Stress levels with decreasing silicon dot size are probably overestimated which could lead to the underestimation of wet oxidation kinetics and a larger critical range for the dot existence. On the other hand with respect to the 4 Since the evolution of all the other geometrical parameters are reasonably described, such a large difference could be reasonably explained reasonably by the presence experimentally of a thinner silicon after etching compare to the configuration of Fig. 5. This layer plays a major role as it tends to delay the oxidation of the bottom part of the channel DRAFT experimental limits, simulation confirms that the silicon dot is completely consumed for a 80 nm wide nanowire configuration and that self-limited oxidation effects [START_REF] Coffin | Oxidation of si nanocrystals fabricated by ultralow energy ion implantation in thin sio 2 layers[END_REF] are not sufficient to stop the reaction in our present case.

3) Simulation of the dot characteristics: Process simulation can help to have an estimate of the variation of the dot (shape, size, position)and the parameter range that allow for dot formation. The dot generated by our process has a triangular shape with a dot height often more important than the dot width. In order to simplify the discussion, the use of a mean radius which matches as closely as possible the dot geometry is proposed 5 . The variation of the dot properties as a function of the linewidth are reported Fig. 12. The dot radius r dot decreases linearly with the initial linewidth. It can be described by the following relationship with a small set of parameters:

r dot = 0.33(d -d max ) + r max (nm) ( 9
)
where d is the linewidth, d max is the critical line for the dot creation (125 nm) and r max is the associated maximum radius (20 nm).

Another interesting parameter is the variation of the dot position with the nanowire initial width. In general, the dot center is located 30 nm below the top silicon surface close to the midpoint between the top surface and the arsenic profile maximum. It can be observed that the thickness of the oxide separates the dot from the channel increases as the initial linewidth decreases (Fig. 12). As clearly shown in Fig. 14, the dot position approaches 15 nm for a linewidth around 90 nm. This trend can be simply described by the following linear empirical law as a function of the difference between the linewidth and d max = 125 nm :

r center = 0.484(d -d max ) (nm) (10)
This effect can be explained by the fact that, for thinner structures, the oxidation is more important in the region of strong arsenic gradient since there is a smaller amount of silicon to oxidize for the same thermal budget.

Finally, the evolution of the tunnel oxide thickness t ox (the dot-channel distance) is also reported. The tunnel oxide increases almost linearly with downscaling as the thermal budget is kept constant and not adjusted for the critical linewidth study.

t ox = 26.59 + 2.616(d -d max ) (nm) (11)

C. Conclusion

The existence of two critical linewidth (125 nm and 80 nm) for the dot existence has been clearly established by the process simulation in agreement with the experiments. The different structures are correctly described by process simulation despite the limitations in section III. The difference (15 %) observed on the critical line for the dot consumption could be explained by the limits of the experimental technique and also by the limitations related to strain effects at the nanometer scale, two dimensional arsenic dopant diffusion and segregation. Process simulation also provides interesting trends for dot geometry characteristics. The mean radius of the nanofloating gate scales down almost linearly with the linewidth. On the other hand, the dot-channel distance and the dot center position increases due to a strong oxidation rate at the bottom of the nanofloating gate. These variations have been quantified and empirical variation laws extracted.

V. DOWNSCALING

One challenge faced by alternative nanometric devices is their scalability. In addition, aggressive size reduction would open the perspective to operate close to single electron mode of operation. The main objective of this section is to evaluate the possibility of downscaling this alternative device using the same simulation strategy that was above detailed.

A. Scaling issues

Downscaling of the standard device by a factor 2 is our target. The first issue is to adapt our process for the implementation on a thinner silicon film of 100 nm. The film thickness reduction mainly impacts the implantation conditions. The objective is to create a sharp arsenic gradient in a more localized area. First simulations show that the arsenic peak should not be too close to the silicon surface in order to create the dot since oxidation takes also place under the nitride mask. Therefore, the arsenic maximum has to be located into the top third of the silicon film as in the standard configuration. A reduction of the implantation energy around 60 keV is therefore recommended.

The scaling of the dose must be investigated but other implantation conditions (tilt angle) could remain unchanged.

The second issue is that in order to be able to form the constrictions, the region where the channel will be formed should be significantly thicker that where the dot will.

B. Linewidth variation

In the downscaling study, the initial configuration shown in Fig. 13 has been used. Arsenic implantation has been performed with the parameters chosen in the previous section (Energy 60 keV, Arsenic dose 1e15 at/cm 2 ). A nitride film has been deposited on top of the oxide layer. No overetching is considered since the main objective is to identify the trends. Moreover it will complexity the discussion of the results. A standard sidewall has been chosen to define the shape at the bottom of the mesa-structure.

An extensive set of simulations has been performed as a function of the initial linewidth. A low temperature wet oxidation performed at (800 • C), the optimal temperature. The final resulting configurations (Fig. 15) are reported, DRAFT close to the dot creation. First, large configurations (80 nm to 60 nm) have been simulated. From the simulation standpoint, it clearly appears that downscaling is feasible. The final structure is well organized since the dot is in the top half of the film and the constriction region in the bottom half of the device. Thinner structures have also been investigated (from a 50 nm configuration down to 20 nm). Very surprisingly, the structure dot-channel has been observed to be scalable down to 20 nm with a dot around 5 nm as shown in Fig. 16.

C. Evolution of dimensional properties of the dot

Top Fig. 16 summarizes the evolution of two parameters (the dot mean radius and the thermal budget of the wet oxidation step)as a function of the nanowire linewidth. For a large configuration of 80 nm, the mean radius approaches the 10 nm target. As expected, the objective to reduce the size of the dot by a factor two has been achieved. The dot radius reduces for narrower configurations down to 5 nm for a 20 nm configuration. This figure emphasizes the interest of this approach to generate a large spectrum of dot using single parameter variation. The bottom graph in Fig. 16 shows the evolution of the oxidation time at the onset of the dot creation. The various final structures presented just before obtained by setting the oxidation duration close to the creation time for each linewidth. As shown in Fig. 16, the thermal budget downscales almost linearly with decreasing linewidth. For a large configuration, the oxidation time is about 80 min. and is less than 20 min. for the narrowest device. The impact of dose reduction by a factor 2 has also been evaluated. The linewidth range for the dot existence is clearly much sharper indicating that the doping gradient should not be further reduced.

D. Conclusion

Based on the downscaling study, the fabrication of a single electron memory cell has been achieved with a reduction by a factor 2 with respect to the original design [START_REF] Tang | Fabrication and room temperature single charging behavior of self-aligned single dot memory devices[END_REF]being kept constant. The silicon film thickness has been reduced to 100 nm and the implantation energy reduced to 55 keV (close to the best theoretical value of 60 keV). The formed dot memory has been observed through the gate and has a diameter of about 5-10 nm. The downscale device proposed by simulation is the 60 nm configuration of Fig. 15 with a dot size of 10 nm, a gate oxide of about 20 nm and a tunnel oxide of 15 nm. A preliminary device simulation study has shown that the device can operated for relatively thick oxides (15 nm for the gate oxide and 35 nm the for tunnel oxide) [START_REF] Tang | Two-dimensional self-consistent simulation of a triangular p-channel soi nano-flash memory device[END_REF]. As already reported in a previous paper, single hole effects have been observed by several electrical characterizations at room temperature in the implementation of the downscale device [START_REF] Tang | Fabrication and room temperature single charging behavior of self-aligned single dot memory devices[END_REF]. As the process space allows some process modifications, more work should be done in order to optimise the device space for this device.

VI. CONCLUSION

Important conclusions can be drawn on the two major topics. First, the concept of single silicon dot generated by oxidation is of interest. It has been shown that a large range of single dots can be generated from 20 nm nanometer down to 5 nm. This method is very complementary to other techniques for which smaller dot are often obtained.

Single dots generated with this technique hold the noteworthy property of tight dot size control. The proposed DRAFT flash memory design and the corresponding process open new perspectives for the integration of this concept in self-assembled nanowires to generate mass flash memory or the design of complex device with several single dots in the central part to fabricate single electron pumps [START_REF] Zorin | Coulomb blockade and cotunneling in single electron circuits with on-chip resistors: Towars the implementation of the r pump[END_REF].

Secondly, on the process modeling point of view, this study is one example of the interest of process modeling to speed-up the development of nanodevices. The impact of various parameters and configurations has been For a linewidth larger than 130 nm, the dot is not liberated from the channel. A dot is observed for 120 nm ± 10 nm. at/cm 2 ) and with a lower dose (5× 10 14 at/cm 2 ) is reported. It can be observed that the thermal budget downscales linearly with decreasing linewidth and that reducing the dose by a factor two is not suitable because the range for the dot existence is clearly sharper [60-40] nm only.

DRAFT

  900 • C, the dot is created and consumed at the same time, therefore the recommended wet oxidation temperature range is [775 • C-850 • C] in order to ensure the dot existence. The best compromise is around 800 • C which ensures the presence of the dot in the central region and allows the full consumption of silicon in the constriction regions.

Fig. 11 presents

 11 Fig. 11 presents the simulation of the oxidation step for various linewidths (80, 120 and 200 nm) corresponding

Fig. 1 .

 1 Fig.1. Layout of the nanodevice. In order to generate the structure made of a single silicon dot in a channel separated by oxidation from the source/drain contact, different regions have to defined by a lithography step. The central part of the nanodevice where a single memory silicon dot separated from the channel is implemented is defined by a square area of 130 nm 2 . This central part is connected to thin constriction regions (with less volume to oxidize) promoting the isolation of the dot from the source/drain regions. Finally, larger source/drain contact regions are fabricated in order to control the transport in the channel as a classical FET.

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .

 2345 Fig. 2. SEM photography of the central part of the nanoflash memory.

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 67891011 Fig. 6. The simulated counterpart for a 130 nm configuration. The arsenic gradient in the silicon layer is shown.

Fig. 12 .Fig. 15 .Fig. 16 .

 121516 Fig. 12. Evolution of the main parameters (dot size, dot position, tunnel oxide thickness) between the two critical linewidths.

The mean radius is extracted from the calculation of the dot surface defined by the integration of the various finite elements. DRAFT