Frédéric Protin

LS condition for filled Julia sets in C

Keywords: LS condition, Green function, pluricomplex Green function, complex dynamics, filled Julia set, potential theory Mathematics Subject Classification (2000) 37F50, 37F10, 31C99, 32U35

In this article we derive an inequality of Lojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by dist the euclidian distance in C, we show that the Green function G K of the filled Julia set K of a polynomial such that K = ∅ satisfies the so-called LS condition

Relatively few examples of compact sets satisfying the LS condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. For instance, this is the case for the filled Julia sets of quadratic polynomials of the form z → z 2 + a, provided that the parameter a is hyperbolic, Siegel or parabolic. In particular, this latter case provides an example of a non semi-algebraic compact set in C which has a cusp and satisfies however the LS condition. In order to prove our main result, we define and study the set of obstruction points to the LS condition. We also prove, in dimension n ≥ 1, that for a polynomially convex and L-regular compact set of non empty interior, these obstruction points are rare, in a sense which will be specified.

Introduction

We call pluricomplex Green function G A of a compact set A ⊂ C n , n ≥ 1, the plurisubharmonic function defined as

G A := sup * v ∈ P SH(C n ) : v| A ≤ 0, v(z) ≤ 1 2 log(1 + z 2 ) + O(1) ,
INSA de Toulouse, Institut Mathématique de Toulouse 135 avenue de Rangueil, 31400 Toulouse, France Tel.: +33-684729787 E-mail: fredprotin@yahoo.fr where sup * denotes the upper semi-continuous regularization of the upper envelope, and P SH(C n ) denotes the set of plurisubharmonic functions in C n . The set A is called L-regular if G A is continuous. In this case, the set {G A = 0} is the polynomially convex envelope  of A. We also consider, for an open bounded set U ⊂ C n , the Green function of A ⊂ U relative to U defined by

G A,U := sup * v ∈ P SH(U ) : v ≤ 1, v| A ≤ 0 .
The reader should pay attention to the fact that in [START_REF] Klimek | Pluripotential theory[END_REF] the Green function of A relative to U is defined as G A,U -1.

Let U a := {G A < a} for a ∈ R + \ {0}. If A is not pluripolar and  ⊂ U a , then a relation between G A and G A,U a holding in U a is given by Proposition 5.3.3 in [START_REF] Klimek | Pluripotential theory[END_REF]:

G A = G A,U a . (1)
A compact A ⊂ C n is said to satisfy the LS condition if there exists an open set U containing it and two constants c, c > 0 such that its pluricomplex Green function G A verifies the following regularity condition :

∀z ∈ U, G A (z) ≥ c • dist(z, A) c ,
where dist denotes the euclidean distance (see for instance [START_REF] Gendre | Inégalités de Markov singulières et approximation des fonctions holomorphes de la classe M[END_REF] or [START_REF] Belghiti | Éléments pour une théorie constructive des fonctions lisses[END_REF]).

On a compact set A ⊂ C n verifying the LS condition, as well as the HCP condition (i.e. the Hölderian continuity of G A , for example a semi-algebraic compact set), we have the rapid approximation property of continuous functions by polynomials. Relatively few examples of compacts satisfying the LS condition are known. Some examples are given in [START_REF] Pierzcha La | An estimate for the Siciak extremal function : subanalytic geometry approach[END_REF]. Let us also note that Pierzcha la showed in [START_REF] Pierzcha La | Markov's inequality in the o-minimal structure of convergent power series[END_REF] that a compact verifying the LS condition is polynomially convex. Bia las and Kosek [START_REF] Bia Las-Cież | Iterated function systems and Lojasiewicz-Siciak condition of Green's function[END_REF] construct such sets using holomorphic dynamics.

Along the same vein, we show that the so-called filled Julia sets in C satisfy the LS condition. More precisely, our main goal is to show the following result concerning the filled Julia set of a polynomial f : C → C, i.e. the set of points z ∈ C whose orbit (f n (z)) n is bounded :

Theorem. The filled Julia set of a polynomial f : C → C of degree ≥ 2, if its interior is non empty, satisfies the LS condition.

Recall that a compact set in C is polynomially convex if and only if its complement is connected, so the filled Julia set of a polynomial is polynomially convex. The differentials operators operators ∂ and ∂ will be understood in the sense of currents. Recall that a continuous function u from an open set of C n into R is pluriharmonic (harmonic if n = 1) if and only if ∂∂u = 0 (see for example Theorem 2.28 in [START_REF] Lelong | Entire functions of several complex variables[END_REF]).

In Section 2, we recall some definitions and elementary facts about holomorphic dynamics in one dimension, and we give a useful lemma concerning the regularity of filled Julia sets. More precisely, this lemma shows that the filled Julia set K of a polynomial of degree d ≥ 2 with non-empty interior satisfies K = K. In Section 3, we define in C n , n ≥ 1, the set of obstruction points to the LS condition, and we prove that the complement of this set is big, in a sense which will be specified. We also study explicitly the LS condition on several examples of compact sets in C. Section 4 is devoted to the proof of the main theorem previously stated.

Dynamics in C

We start by recalling some definitions related to one-dimensional holomorphic dynamics. Let us consider a polynomial f :

C → C of degree d ≥ 2.
We call Fatou set of f , denoted F, the largest open subset in which the family of iterations f n is equicontinuous.

The Julia set of f , denoted J, is the complement of F in C. Let us note for what follows that J is not a polar set.

We call filled Julia set of f the set K of points z ∈ C whose orbit (f n (z)) n is bounded. Note that K is compact, as ∞ is a superattractive fixed point of f , hence belonging to F. The complement of K is the basin of attraction of infinity. We have ∂K = J and G K = G J .

There are many situations where the set K is of non-empty interior. Consider, for instance, the case where f (z) = z 2 + a with a ∈ C. By Sullivan's classification theorem (see e.g. Theorem 2.1 in [START_REF] Carleson | Complex dynamics[END_REF] or Theorem 3.2 of [START_REF] Mcmullen | Complex dynamics and renormalization[END_REF]), we can distinguish three cases where K = ∅. The first case is when a is chosen in the interior of the Mandelbrot set in such a way that f is hyperbolic in the sense of [START_REF] Carleson | Complex dynamics[END_REF] p. 89. By Theorem 4.7 in [START_REF] Mcmullen | Complex dynamics and renormalization[END_REF], f is hyperbolic if and only if some iterate

f k of f has a fixed point z 0 ∈ C for which |(f k ) (z 0 )| < 1.
The second case is when a is chosen on the boundary of the Mandelbrot set such that some iterate f k of f has a fixed point z 0 ∈ C for which (f k ) (z 0 ) is a root of the unity. By Theorem 6.5.10 of [START_REF] Beardon | Iteration of rational functions[END_REF] and Theorem 4.8 of [START_REF] Mcmullen | Complex dynamics and renormalization[END_REF], this corresponds to the parabolic case in the Sullivan's classification. By Theorem 4.8 of [START_REF] Mcmullen | Complex dynamics and renormalization[END_REF], the last case is when a is chosen on the boundary of the Mandelbrot set such that K contains a Siegel disk and all its preimages in the sense of Definition 7.1.1 of [START_REF] Beardon | Iteration of rational functions[END_REF].

We construct the subharmonic function G : C → R + , limit in L 1 loc of the sequence (log(1 + |f n |)/d n ) n (see [START_REF] Guedj | Dynamics of polynomial automorphisms of C[END_REF] for a general construction). It is known that G is continuous (and even Hölderian [START_REF] Kosek | Hölder continuity property of fillin-in Julia sets in C n[END_REF], see also Theorem 3.2 of [START_REF] Carleson | Complex dynamics[END_REF]), harmonic in F, that it verifies G(z) = 0 if and only if z ∈ K, and also that G(z)-log |z| = O(1) at infinity. By uniqueness, G is therefore the pluricomplex Green function of K (and of J). It satisfies by construction the invariance property

G • f = d • G. (2)
The measure i π ∂∂G is a probability measure of support exactly J (see e.g. [START_REF] Guedj | Propriétés ergodiques des applications rationnelles[END_REF]). We will use the following preliminary lemma about filled Julia sets.

Lemma 1 The filled Julia set K of a polynomial of degree d ≥ 2 with non-empty interior satisfies K = K.

Proof Suppose, by contradiction, that there exists x ∈ ∂K having a neighborhood U which does not intersect K. Then there exists n 0 ∈ N such that K ⊂ f n 0 (U ) (see for example Theorem 4.2.5. of [START_REF] Beardon | Iteration of rational functions[END_REF]). But this contradicts the fact that f n 0 (U ∩K) ⊂ ∂K. Thus every open subset of C intersecting J = ∂K also intersects K. In other words, K = K.

Study of the obstruction to the LS condition

For n ≥ 1, let

O c := {z ∈ C n : dist(z, A) < 1, G A (z) < c • dist(z, A) 1/c }. (3)
Note that the sequence of open sets O c is increasing with c for c < 1. The LS condition is satisfied by a compact non-pluripolar set A ⊂ C n , L-regular and polynomially convex, if and only if the set

I := c>0 O c ⊂ ∂A (4)
is empty. We call I the set of obstruction points to the LS condition.

Example 1 ([3])

If A is the union of two disks of radius 1, tangent to each other at the origin, then it does not satisfy the LS condition; the set of obstruction points to the LS condition is

I = {0} = ∅.

Example 2

The previous set A is mapped by the function g : z → z 2 onto a filled cardioid C, and we have g -1 (C) = A. We deduce from Theorem 5.3.1 of [START_REF] Klimek | Pluripotential theory[END_REF] that the set of obstruction points to the LS condition for C is

I = {0} = ∅.
Example 3 (see also [START_REF] Gendre | Inégalités de Markov singulières et approximation des fonctions holomorphes de la classe M[END_REF]). For ε ∈ ]0, 1[ fixed, consider the sets

L ε := {(1+i)t, t ∈ [-ε, ε]}, L ε := {(1 -i)t, t ∈ [-ε, ε]}, and X ε := L ε ∪ L ε ⊂ B(0, 2ε).
We show that X ε satisfies the LS condition, i.e. I = ∅. Indeed, the function g

: C → C defined by g(z) = i 2 z 2 maps X ε onto [-ε 2 , ε 2 ]. On the other hand, g -1 ([-ε 2 , ε 2 ]) = X ε . Theorem 5.3.1 of [9] implies G X ε = G [-ε 2 ,ε 2 ] • g. Since the segment [-ε 2 , ε 2 ]
is convex, it satisfies the LS condition (see [START_REF] Gendre | Inégalités de Markov singulières et approximation des fonctions holomorphes de la classe M[END_REF]). Moreover, it follows from Theorem 1 in [START_REF] Dubinin | On the finite-increment theorem for complex polynomials[END_REF]

that ∀z ∈ C, dist g(z), [-ε 2 , ε 2 ] ≥ 1 4 |z|dist (z, X ε ) ≥ 1 4 dist (z, X ε ) 2 .
We deduce that X ε also satisfies the LS condition.

The following result provides more insight into the structure of the complement of O c . We prove it for any n ≥ 1. Recall that, given an open set U ⊂ C n , a set E ⊂ U is called pluripolar if for each a ∈ E there exist a neigborhood V ⊂ U of a and a plurisubharmonic function v : V → R∪{-∞} such that E ∩V ⊂ {v = -∞}. Proposition 1 Let A ⊂ C n , n ≥ 1, be a non-pluripolar, L-regular and polynomially convex compact set. Suppose that the pluricomplex Green function G A is pluriharmonic outside of A (harmonic if n = 1). Then, there exists

c 0 > 0 such that ∀c ∈ ]0, c 0 ], ∂A is included in the boundary of the open set {z ∈ C n : G A (z) > c • dist(z, A) 1/c }. Proof Let µ denote the positive measure i π ∂∂G A ∧ ω n-1 on C n , where ω := i 2π ∂∂ log(1 + z 2 )
is the Fubini-Study form. Note that the support of the measure µ is exactly ∂A. Indeed, supp(µ) ⊂ ∂A since i π ∂∂G A = 0 in C n \ ∂A by hypothesis. On the other 4 Proof of the main theorem

We will need the following result of Poletsky (Corollary p. 170 in [START_REF] Poletsky | Plurisubharmonic functions as solutions of variational problems[END_REF], see also [START_REF] Poletsky | Holomorphic currents[END_REF], or Theorem 2.2.10 and Corollary 2.2.13 in [START_REF] Stout | Polynomial convexity[END_REF]), generalized by Rosay ([17]). Let U be a connected complex manifold of dimension n ≥ 1. We denote by H z,U the set of holomorphic functions h :

V h → U from a neighbourhood V h of ∆ = {|z| ≤ 1} ⊂ C
(possibly depending on h) into U such that h(0) = z. We also denote by P SH(U ) the set of plurisubharmonic functions defined on U .

û ≤ v ≤ u.
Since û is maximal among the subharmonic functions which are ≤ u in B and equal to u on ∂B, we conclude that û = v, and hence ũ = û. Thanks to Theorem 3.1.4 in [START_REF] Klimek | Pluripotential theory[END_REF], the conclusion is the same if B is a ball in C n , when substituting the expression "harmonic function" by "maximal plurisubharmonic function", and the expression "antisubharmonic function" by "antiplurisubharmonic function".

Let U ⊂ C n , n ≥ 1, be a bounded open set. Denote by λ the normalized Lebesgue measure on the unit circle ∂U ⊂ C. Denote also by Λ z,U the set of measures of the form h * λ(•) := λ(h -1 (•)), where h : V h → U is an holomorphic function defined in a neigborhood V h (possibly depending on h) of the closed unit disk U, such that h(0) = z. Note that the Dirac measure δ z belongs to Λ z,U (this corresponds to the case where the function h is constant, equal to z). An immediate consequence of Proposition 2 is the following corollary, where 1 G denotes the characteristic function of G ⊂ C n :

Corollary 1 Let U ⊂ C n be a bounded open set, and A ⊂ U a L-regular non- pluripolar compact set satisfying Å = A. Then 1 2π inf f ∈H z,U 2π 0 -1 Å • f (e iθ )dθ = -sup µ z ∈Λ z,U µ z ( Å) = G A,U (z) -1.
Recall that we denote by K the filled Julia set of a polynomial application f : C → C of degree ≥ 2, and dist(•, •) the euclidean distance on C n . Let us prove the main result stated in the introduction : Theorem 1 Let K ⊂ C be the filled Julia set of a polynomial f : C → C of degree d ≥ 2, of non-empty interior. Then K satisfies the LS condition. [START_REF] Belghiti | Éléments pour une théorie constructive des fonctions lisses[END_REF]. Denote by C a the annulus U a \ f -1 (U a ). There exists δ ∈]0, 1[ such that

Proof For b ∈ R + \ {0}, denote U b := {G K < b} ⊂ C. For l ∈ R + \ {0}, denote also K l := {z ∈ C | dist(z, K) ≤ l}. Then choose a > 0 such that K 2 ⊂ f -1 (U a ). Note that f -1 (U a ) = U a d ⊂⊂ U a by
G K 2,Ua ≥ δG K,U a on C a . (6)
There exists also e ∈]0, 1[ such that e • dist(•, K) < 1 on U a . We change slightly (3) and define O c by

O c := {z ∈ C n : dist(z, K) < 1, G A (z) < c • e 1 c • dist(z, K) 1/c },
that do not change the set I defined in [START_REF] Gendre | Inégalités de Markov singulières et approximation des fonctions holomorphes de la classe M[END_REF]. Take c ∈ 0, δ 2a sufficiently small to have O c ⊂ f -1 (U a ) and 1 c 2 c < 2, as well as

G K ≥ c • e 1 c • dist(•, K) 1 c on C a . (7)
We have ∀ε ∈]0, 2], ∀y ∈ U a , c • dist(y, K)

1 c ≥ inf µ y ∈Λ y,Ua U a c • dist(•, K) 1 c dµ y ≥ inf µ y ∈Λ y,Ua U a \ Kε c • dist(•, K) 1 c dµ y ≥ min U a \K ε c • dist(•, K) 1 c inf µ y ∈Λ y,Ua U a \ Kε dµ y = cε 1 c G K ε ,U a (y).
The first inequality comes from the fact that the Dirac measure δ y belongs to Λ y,U a . The last inequality comes from Corollary 1, whose application is allowed by Lemma 1, and from Corollary 4.5.9 in [START_REF] Klimek | Pluripotential theory[END_REF]. Then taking ε = 1

c 2 c < 2, we obtain in U a : c • dist(•, K) 1 c ≥ 1 c G K ε ,U a . (8)
Now suppose, by contradiction, that O c = ∅ (see Equation (3) for definition). Recall that c < δ 2a . We can then choose

x ∈ O c \ {G K < 2ac 2 δ dist(•, K) 1 c }.
Let us control the growth of the iterates of f . Note that z ∈ O c implies a "slow growth" of (f n (z)) n , in the sense that ∀n ≥ 1 such that

f n (z) ∈ {dist(•, K) < 1} \ O c , we have 1 d n c • e 1 c • dist(f n (z), K) 1 c ≤ G K (z) < c • e 1 c • dist(z, K) 1 c , and hence dist(f n (z), K) < d nc dist(z, K). (9)
Moreover, by a similar reasoning, Equation [START_REF] Guedj | Propriétés ergodiques des applications rationnelles[END_REF] implies that for all z ∈ O c and n ≥ 1 such that f n (z) ∈ U a \ {dist(•, K) < 1}, we have

dist(f n (z), K) < d nc dist(z, K). (10)
Finally, by ( 11) and ( 10), for all z ∈ O c and n ≥ 1 such that

f n (z) ∈ U a \ O c , we have dist(f n (z), K) < (d n ) c dist(z, K). (11)
Conclusion. Recall that we have choosed 2), there exists N > 0 such that f N (x) ∈ C a . Equations ( 11), ( 8), ( 6), (1), then (2), give c • dist(x, K)

x ∈ O c \ {G K < 2ac 2 δ dist(•, K) 1 c }. Since U a \ K = i≥0 f -i (C a ) by (
1 c ≥ c d N dist f N (x), K 1 c ≥ 1 cd N G K ε ,U a • f N (x) ≥ δ cd N G K,U a • f N (x) = δ ca G K (x).
But this contradicts our assumption x / ∈ {G K < 2ac 2 δ dist(x, K) 1 c }. We conclude that O c = ∅. In other words, K satisfies the LS condition.

Remark 2 We note that if f is assumed to be hyperbolic, that is to say if f do not have critical points in J, there exist a constant b > 0 and a neighborhood of K in which dist (f (•), K) ≥ b • dist(•, K).

Indeed, it is sufficient to establish this inequality outside K. Let then V be a neigborhood of K in which |f | ≥ a for some a > 0, let z ∈ V \ K, and z 0 ∈ J such that f (z 0 ) ∈ J achieves the distance dist(f (z), J). Then Theorem 1 of [START_REF] Dubinin | On the finite-increment theorem for complex polynomials[END_REF] shows the existence of a constant k > 0 (depending only on the degree of f ) and of a point z 1 ∈ J = ∂K, such that

dist(f (z), K) = dist (f (z), f (z 0 )) ≥ a • k • dist(z, z 1 ) ≥ a • k • dist(z, K).
In the particular case where b ≥ 1 in (12), we obtain a simpler proof of Theorem 1, and a more quantitative estimation for c in Equation

Proposition 2 Remark 1

 21 Let u : U → R be an upper semi-continuous function. With the previous notations, the function defined by (e iθ ))dθ, if it is not everywhere equal to -∞, belongs to P SH(U ) and verifies ũ ≤ u. Moreover, this function ũ is maximal among all the functions in P SH(U ) verifying this inequality. We deduce from Proposition 2 the following property of antisubharmonic functions, i.e. functions with subharmonic opposite. Let B := B(a, r) ⊂ C be an open ball, u : B → R a continuous function, antisubharmonic in B. Then û : B → R is an harmonic function, with the same boundary values as u, in the sense that limz→z 0 û = u(z 0 ) for z 0 ∈ ∂B.Indeed, given a continuous function g : B → R, denote by g : B → R the solution of the Dirichlet problem in B with boundary condition g | ∂B , that is to say, the unique continuous function defined on B which is harmonic in B and equal to g on ∂B. Then v := max(ũ, û) is a subharmonic function with the same values as u on ∂B. Since u is antisubharmonic, we have ũ ≤ u. Thus

( 3 )

 3 . Indeed, suppose O c = ∅ with O c ⊂⊂ V . We can choose x ∈ O c such that f (x) / ∈ O c . Then, (11) together with (12) give c > log b log d .

Acknowledgements We thank Laurent Gendre and Taïb Belghiti for their reading and suggestions, as well as Marta Kosek and the whole organization committee of the conference 'On Constructive Theory of Functions" for their invitation to expose a preliminary version of this article in Poland. We also thank the anonymous referee for constructive suggestions.

hand, if there existed x ∈ ∂A \ supp(µ), then G A would be (pluri)harmonic in a neighborhood of x, hence null in this neighborhood, which can not happen because A is polynomially convex.

Let us suppose by contradiction that

Thus we can take c ∈ 0, 1 4n , x ∈ ∂A, and r > 0, such that

Denote r 0 := r 2 . Let us establish the following Chern-Levine-Nirenberg-type inequality :

for some constant k > 0 independent of r, r 0 , x and c . Let indeed ξ : C n → R + be a positive test function ≡ 1 in B(0, 1) and having its support in B(0, 2). There exists a decreasing sequence (G n ) n of C ∞ plurisubharmonic functions converging towards G A (Theorem 2.9.2 in [START_REF] Klimek | Pluripotential theory[END_REF]). Theorem 3.4.3 in [START_REF] Klimek | Pluripotential theory[END_REF] and Stokes' theorem imply that ∀r < r 0 , ∀x ∈ B(x , r 0 ) ∩ ∂A,

Then the monotone convergence theorem implies that ∀r < r 0 , ∀x ∈ B(x , r 0 )∩∂A,

where k depends only on the sum of the supremum norms of the coefficients of the differential form ∂∂ξ. Therefore, (5) holds.

With the notation ν := µ µ B(x , r 0 ) 1 B(x ,r 0 ) , where 1 B(x ,r 0 ) is the characteristic function of B(x , r 0 ), the measure ν is a probability measure, and we can rewrite [START_REF] Carleson | Complex dynamics[END_REF] :

Then, by Frostman Lemma (see for example Lemma 10.2.1 in [START_REF] Beardon | Iteration of rational functions[END_REF]), the Hausdorff dimension of ∂A ∩ B(x 0 , r 0 ) is strictly greater than 2n for our choice c < 1 4n , which gives a contradiction. (Recall that Frostman Lemma ensures that, if m is a probability measure on a metric space E verifying m B(x, r) < q • r α for all x ∈ E, r > 0, with fixed q > 0, α > 0, then the Hausdorff dimension of E is greater than α).

We thus conclude that ∃c 0 > 0, ∀c ∈]0, c 0 ], ∀x ∈ ∂A, ∀r > 0:

which proves the statement.


Borinic acid Catalysed Peptide Synthesis

Tharwat Mohy El Dine, Jacques Rouden, and Jérôme Blanchet*

The catalytic synthesis of peptides is a major challenge of modern organic chemistry hindered by the well established use of stoichiometric coupling reagents. Herein, we describe the first borinic acid able to catalyse this reaction under mild conditions with an improved activity compared to our recently developed thiophene-based boronic acid. This catalyst is particularly efficient for peptide bond synthesis affording dipeptides in good yields without detectable racemization.

The synthesis of peptides via the assembly of chiral N-protected αamino acids and α-aminoesters is a well-known methodology. [START_REF] Kent | For recent reviews about peptide synthesis[END_REF] This essential transformation could be easily realized with high level of predictability owing to the progress made with the use of stoichiometric peptide coupling reagents (Figure 1). Despite those great accomplishments, a catalyst that would be able to achieve similar efficiency is still to be discovered. Albeit the catalytic synthesis of amides from simple carboxylic acids and amines has been reviewed several times 3 , almost no information concerning the assembly of two α-amino acids derivatives is available. Compared to simple carboxylic acids and amines, protected α-amino acids are challenging substrates, due to their weaker nucleophilic character of the amine function and increased steric hindrance caused by the side chain. Inspired by Yamamoto's pioneering work 4 , Hall 5 (3a-b, Figure 2), Tam 6 (4, Figure 2) and our laboratory (5, Figure 2) 7 have recently reported boronic acids able to promote room temperature coupling of non-activated carboxylic acids and amines. Regarding peptide synthesis, 25 mol% of boronic acid 5 was found to promote the formation of Boc-Phe-Val-OMe in 50% yield with no detectable racemization, thus achieving the first synthesis of a dipeptide using a simple chemical catalyst.

Previously, the coupling of protected α-amino acids has been explored by Whiting 8 using stoichiometric quantities of a combination of aryl boronic acids 1-2 (Figure 2). However, low to moderate yields of the corresponding dipeptides were obtained. Interestingly, a cooperative effect of combined catalyst systems was suggested to be beneficial when 50 mol% of 1a and 50 mol% of 1b were used together for the coupling of hindered valine derivatives. Although peptides are of great value to pharmaceutical industry, their catalysed synthesis is still unknown. Therefore, the next challenge lies in identifying an efficient catalyst that would directly couple N-protected and C-protected α-amino acids to provide the desired dipeptides in reasonable yields without racemization. Since boronic acid 5 has shown moderate activity in peptide coupling, we decided to shift to a potentially more reactive boron derivative. We hypothesized that the use of borinic acids, which possess an amplified Lewis acidity at the boron centre 10 , can provide an improved catalytic system. Additionally, they offer an improved structural flexibility for further tuning of the electronic and steric properties with two aromatic moieties being attached to the same boron atom. In this context, we report the use of borinic acids for catalysed peptide synthesis. We started our investigations by synthesizing a library of biaryl borinic acids according to known procedures (See Supporting Information). The reactivity of these borinic acids was compared to a set of commercially available boronic acids in the model reaction carried out between phenylacetic acid 6 and benzylamine 7 (Table 1). Interestingly, higher yields were systematically obtained with the borinic acids. With electron donating group substituted borinic acids 9c (4-t-Bu), 9d (4-MeO) and 9e (3-MeO) 39-59% isolated yields were obtained when the corresponding boronic acids were found to be essentially unreactive (Table 1, Entries 3 and4). a Reaction conditions: Phenylacetic acid 6 (0.55 mmol), benzylamine 7 (0.50 mmol) and molecular sieves (1g of 5 Å activated powder) were stirred in dry CH2Cl2 (7 mL) with borinic acids 9a-e or boronic acid (10 mol%) under inert atmosphere for 48h at room temperature.

Encouraged by these results, the screening of various aryl borinic acids was undertaken. Based on Hall's previous work, 5 we focused our attention on halogen substituted borinic acids.

The presence of a fluorine atom at the ortho position in 9f decreased the yield to 63% compared to the 81% yield obtained with the fluorine at the para position 9g (Table 2, Entries 1 and 2). A different picture was observed with the introduction of a chlorine atom. Indeed, the 2-chloro substituted 9h was found to be very effective delivering a quantitative yield of the amide 8 (Table 2, Entry 3) and suggesting that increased electronegativity at the ortho position in 9f lowered the reactivity of the catalyst. The importance of the ortho-chloro substituent was shown by the lower yields (76% and 46% respectively) obtained with the meta 9i and para 9j substituents (Table 2, Entries 4 and5). 2-Bromo substituted 9k led to a similar reactivity as 9h (Table 2,Entry 6), however, with a much less pronounced availability. [START_REF]The synthesis of 9k is remarkably low yielding[END_REF] These results suggest the need for a specific electronic requirement in the ortho position for an optimal catalytic performance.

Hall and co-workers have reported a dramatic improvement of the performance of boronic acid 3a (Figure 2) when an electron donating group was introduced para to the halogen atom. In their study, the 5-methoxy substituted 3b was identified to provide faster reactions thus supporting the role of the ortho iodo substituent as a hydrogen-bond acceptor in the proposed transition state. 5a Based on these results, we prepared the 5-methoxy substituted borinic acid 9l. However, in our hands, catalyst 9l was less effective affording amide 8 with 80% yield (Table 2, Entry 7). Substituting the methoxy group with a chlorine or fluorine atom significantly decreased the catalyst performance (9m and 9n, 37% and 18% yields respectively, Table 2, Entries 8 and 9). Interestingly, switching the fluorine position para to the boron atom in 9o enhanced the catalytic activity with 97% conversion being attained within 4 hours (99% yield, Table 2, Entry 10) compared to 99% conversion attained within 6 h with catalyst 9h (99% yield, Table 2,Entry 3). These results illustrate the importance of fine-tuning the electronic properties of the catalyst in terms of electron-density of the orthohalogen and boron Lewis acid property.

Having identified 9o as a useful catalyst, we monitored the reaction using 19 F-NMR to collect some insights regarding the mechanism of the reaction. Indeed, it was important to rule out any boronic acid catalysis consecutive to a side protodeboronation reaction facilitated by the increased Lewis acidity of borinic acids. During the whole course of the reaction, a single signal at -114.5 ppm in 19 F NMR (Figure 3) was observed, with no intermediates being detected. This signal matched the isolated 9o, while the corresponding boronic acid was measured at -106.7 ppm, suggesting the stability of the catalyst 9o during the reaction. While 9o was found to be slightly more reactive than 9h, the combined stability and higher availability of 9h led us to select it as the optimal catalyst for the further investigations. Before exploring dipeptide synthesis, a control reaction was carried out between phenyl acetic acid 6 and phenylalanine methyl ester 10d to ascertain the absence of racemization (Scheme 1). The use of less reactive α-amino ester 10d required the use of 25 mol% of catalyst 9h and a switch to a different low polarity solvent and a higher temperature (fluorobenzene, 65 °C).

Scheme 1. Coupling between phenyl acetic acid 6 and phenylalanine methyl ester 10.

Amide 11 was previously reported by Whiting for its propensity to provoke racemization, providing an enantiomeric excess of 68% under similar reaction conditions with catalysts 1a and 1b. 8 This result was previously improved using our catalyst 5 (ee: 94%).

7

Interestingly, 9h afforded the desired amide 11 in 96% yield with the complete retention of enantiopurity (ee > 99%, Scheme 1). First attempts of coupling two protected α-amino acids at room temperature resulted in the total absence of conversion. However using previously optimized conditions (Scheme 1), Boc-protected phenylalanine (Boc-Phe) was efficiently coupled with various methyl α-amino esters derived from hindered Valine, Alanine, Glycine and phenylalanine in average 47-61% yields (Table 3, Entries 1-4).

Similarly, Z-proline (Z-Pro) was coupled with various α-amino esters (Table 3, Entries 5-11) in 47-73% yields. While the modification of the nature of the ester group led to marginal change when valine αamino ester was used (Table 3, Entries 7-8), a higher yield was obtained when an ethyl glycine ester (H-Gly-OEt) was used compared to benzyl or methyl counterparts (Table 3, Entries 9-11). Highest 80-72% yields were also obtained with the less hindered Boc-glycine (Boc-Gly-H) as the nucleophilic component of the reaction (Table 3, Entries 12-13) while Z-methionine (Z-Met) led to a non-optimized 40% yield (Table 3,Entry 14). Careful examination of the 1 H NMR data of all the synthesized dipeptides incorporating two chiral centres showed that a single diastereomer was obtained. Based on Hall's mechanism with boronic acids, we rationalized that the N-protected α-amino acid is activated in the form of the mixed boron anhydride A (Figure 4). This step is greatly facilitated by the increased Lewis acidity of the borinic acid 9h. The next step involves the additions of the α-amino ester to generate a tetravalent intermediate B. The collapse of B is then assisted by water and the Lewis basicity of ortho chlorine substituents to yield eventually the desired dipeptide. To clarify if both chlorine atoms are involved, the unsymmetrical borinic acid 9p was prepared and tested (Scheme 2). With this catalyst bearing a single chlorine atom, the yield was reduced from 99% with 9h (Table 2) to 27%.

Scheme 2. Reactivity of borinic acid 9p in the model reaction between phenylacetic acid 6 and benzylamine 7.

Accordingly, the specific reactivity of 9h and 9k tends to indicate that both halogen atoms are probably involved in the ratedetermining step and that the intermediate B is formed rather than B' (Figure 4). Indeed, mild hydrogen-halogen bondings might stabilize the transition state and favour the collapsing of B. Additionally we suggest that the formation of anhydride A is pre-organized by similar hydrogen-halogen bonding.

To summarize, we have uncovered the unique reactivity of borinic acids for amide and peptide syntheses. Fourteen dipeptides were prepared in synthetically useful yields without detectable racemization. Importantly, no stoichiometric peptide coupling reagents were used and thus minimal waste was generated with water being the sole side product of the reaction. However, there is still room for improvements; progress is necessary to match the requirements of solid phase peptide synthesis such as very high yields and short reaction times. We believe that those results will shed light on new strategies aiming at developing the catalysed peptide synthesis, which exhibits an utmost importance for the pharmaceutical industry. We gratefully acknowledge ANR "NeoACAT" (JC 2012 program) for a fellowship to TM and région Basse-Normandie for financial supports.

Notes and references

2Figure 1 .

 1 Figure 1. Representative coupling reagents in amide and peptide synthesis.

9Figure 2 .

 2 Figure 2. Aryl boronic acids investigated by Whiting, Hall, Tam and Blanchet.

12 Figure 3 .

 123 Figure 3. Monitoring the catalyst 9o behaviour using 19 F probe (375 MHz; CDCl3; 25 ᵒC). A: isolated 2-Cl-4-F-C6H4-B(OH)2. B: isolated 9o. C: NMR monitoring of the reaction corresponding to entry 10, table 2, after 6h (97% conversion).

Figure 4 .

 4 Figure 4. Proposed mechanism for the amide synthesis catalysed by the borinic acid 9h.

Table 1 .

 1 Borinic acids vs boronic acids in the coupling of acid 6 and amine 7. a

	Entry	Ar	Ar2B(OH)	Yield with ArB(OH)2 (%)	Yield with Ar2B(OH) (%)
	1		9a	13	19
	2		9b	5	15
	3		9c	0	59
	4		9d	4	45
	5		9e	5	39

Table 2 .

 2 Aryl borinic acid screening. a

	Entry	Ar2B(OH)	Ar	Isolated Yields (%)
	1	9f		63
	2	9g		81
	3	9h		99 (99 b ) d
	4	9i		76
	5	9j		46
	6	9k		98 (99 b ) d
	7	9l		80
	8	9m		37
	9	9n		18
	10	9o		99 (97 c ) d
	a Reaction conditions: Phenyl acetic acid 6 (0.55 mmol), benzylamine 7 (0.50
	mmol) and molecular sieves (1 g of 5 Å activated powder) were stirred in dry
	CH2Cl2 (7 mL) with borinic acids 9f-o (10 mol%) under inert atmosphere for 48h at
	room temperature. b 1 H NMR conversion after 6 hours. c 1 H NMR conversion after
	4 hours.

d

Conversions calculated using 1,3,5-trimethoxybenzene (10 mol%) as an internal standard.

Table 3 .

 3 Scope of the reaction: peptide synthesis. a

	Entry	α-Amino-acid 12		α-Amino-ester 10		Dipeptide 13		Yield (%)
	1	Boc-Phe-H	12a	H-Val-OMe	10a	Boc-Phe-Val-OMe	13a	51
	2	Boc-Phe-H	12a	H-Ala-OMe	10b	Boc-Phe-Ala-OMe	13b	47
	3	Boc-Phe-H	12a	H-Gly-OMe	10c	Boc-Phe-Gly-OMe	13c	55
	4	Boc-Phe-H	12a	H-Phe-OMe	10d	Boc-Phe-Phe-OMe	13d	61
	5	Z-Pro-H	12b	H-Phe-OMe	10d	Z-Pro-Phe-OMe	13e	60
	6	Z-Pro-H	12b	H-Leu-OMe	10e	Z-Pro-Leu-OMe	13f	60
	7	Z-Pro-H	12b	H-Val-OMe	10a	Z-Pro-Val-OMe	13g	58
	8	Z-Pro-H	12b	H-Val-OBn	10f	Z-Pro-Val-OBn	13h	59
	9	Z-Pro-H	12b	H-Gly-OEt	10g	Z-Pro-Gly-OEt	13i	73
	10	Z-Pro-H	12b	H-Gly-OBn	10h	Z-Pro-Gly-OBn	13j	47
	11	Z-Pro-H	12b	H-Gly-OMe	10c	Z-Pro-Gly-OMe	13k	51
	12	Boc-Gly-H	12c	H-Gly-OBn	10h	Boc-Gly-Gly-OBn	13l	80
	13	Boc-Gly-H	12c	H-Phe-OMe	10d	Boc-Gly-Phe-OMe	13m	72
	14	Z-Met-H	12d	H-Phe-OMe	10d	Z-Met-Phe-OMe	13n	40
	a Reaction conditions: N-protected α-amino acid 12 (0.46 mmol), α-aminoester 10 (0.46 mmol) and molecular sieves (1 g of 5 Å activated powder) were stirred in dry
	PhF (6.8 mL)with the borinic acid catalyst 9h (25 mol%) under inert atmosphere for 32-48h at 65 ᵒC.


Alex Masolin
email: alex.masolin@imec.be

Pierre-Olivier Bouchard

Roberto Martini

Marc Bernacki

Thermo-mechanical and fracture properties in single crystal silicon

Keywords: Single Crystal Silicon, Thermo-mechanical properties, Fracture properties, Anisotropic fracture, Brittle-Ductile transition

Single crystal silicon is extensively used in the semiconductor industry. Even though most of the steps during processing involve somehow thermo-mechanical treatment of silicon, we will focus on two main domains where these properties play a major role: cleaving techniques used to obtain a thin silicon layer for photovoltaic applications and MEMS. The evolution and validation of these new processes often rely on numerical simulations. The accuracy of these simulations, however, requires accurate input data for a wide temperature range. Numerous studies have been performed, and most of the needed parameters are generally available in the literature but unfortunately, some discrepancies are observed in terms of measured data regarding fracture mechanics parameters. The aim of this paper is to gather all these data and discuss the validity of these properties between room temperature and 1273 K. Particular attention is given to silicon fracture properties depending on crystallographic orientations, and to the brittle-ductile temperature transition which can strongly affect the quality of silicon layers.

Introduction

Nowadays silicon is the most employed material in semiconductor industry. Integrated circuits, solar cells and Micro-ElectroMechanical Systems (MEMS) industries extensively use this material both as single crystal silicon (also called monocrystalline silicon), which consists of silicon where the crystal lattice of the entire solid is continuous, with no misorientation, and polycrystalline, which consists of a collection of grains of single-crystal silicon separated by grains boundaries. Because of its wide use, silicon properties have been thoroughly investigated in the past from an electrical and mechanical point of view.

In the last decades, thermo-mechanical properties of single crystal silicon have gained more and more interest due to its use in solar cell and MEMS industries. Common processes in these industries involve very high temperatures and an assessment of both stresses induced in silicon during these processes and the residual stresses after the processes is paramount to analyze the feasibility of these processes without breaking the sample. MEMS are sensors and actuators where sensing or actuating parts consist of micrometers-scaled structures, e.g. cantilevers, bridges and plates, usually made of silicon. The mechanical properties of these microstructures have to be tailored and the residual stresses after the fabrication have to be assessed to design MEMS with certain properties. A considerable number of papers have been published on the design of MEMS which cover a wide range of MEMS, such as microphones, accelerometers, pressure sensors, switches and micro-grippers. In the solar cell industry, mechanical properties of silicon are important to estimate the final bowing of very thin wafers after the contact formation. Further interest in mechanical properties of silicon and, more precisely, in its post-elastic behavior at very different temperature is due to the cleaving technology to manufacture thin silicon foils. Various new experimental techniques have been proposed to produce such thin silicon wafers without kerf loss [START_REF] Gordon | Three novel ways of making thin-film crystalline-silicon layers on glass for solar cell applications[END_REF].

Applications exist where the thermo-mechanical and fracture properties of silicon are changed in order to obtain a weak layer, such as [START_REF] Henley | Kerf-free wafering: Technology overview and challenges for thin PV manufacturing[END_REF][START_REF] Masolin | Method for fabricating thin photovoltaic cells[END_REF]. Since the presence of such weak layers inherently changes the thermo-mechanical properties of bulk silicon they will not be reviewed in this paper.

The first work reported in literature about cleaving silicon wafer by using pure thermo-mechanical properties dates from 1975 [START_REF] Hillberry | Method for fracturing crystalline materials[END_REF]. This patent describes an idea on how to control the propagation of a crack in crystalline materials to produce thin wafers. The first step is to introduce a preselected stress concentration into the crystal, e.g. by means of a notch or a scribe line. Subsequently an internal tensile stress, acting in normal direction, may be accomplished by tensile, compressive, shear forces or by a bending or torsional moment. Finally, the fracture can be achieved e.g. by a wedge, expanding material in the notch, a stress wave, and impact load.

Later, at the beginning of the eighties, Wilkes [START_REF] Wilkes | Process for cleaving crystalline materials[END_REF] proposed a process for cleaving boules of single crystal material by creating an inward-directed radial stress concentration completely around a boule which intersects its crystallographic plane of minimum bond strength. Then, triggering the cleavage via a shock wave applied.

In 1986, Tanielian et al. [START_REF] Tanielian | Method of making thin free standing single crystal films[END_REF] proposed a method to produce foils by sputtering deposition of a layer of metal onto a single crystal substrate. Then, the assembly is treated to stress the metal layer which then can peel off with a part of the single crystal substrate still attached. Free standing foils thus produced have typical thicknesses in the order of tens of micrometers.

A few years later, Owens [START_REF] Owens | Method and apparatus for cleaving brittle materials[END_REF][START_REF] Owens | Method for cleaving brittle materials[END_REF] and Takeguchi [START_REF] Takeguchi | Method and apparatus for cutting polycrystalline silicon rods[END_REF] invented a tool to cleave brittle materials into thin sections using the same principle of the aforementioned Hillberry [START_REF] Hillberry | Method for fracturing crystalline materials[END_REF], namely the use of a wedge to induce a pure opening mode into the crystal.

Almost two decades later, Yamaguchi [START_REF] Yamaguchi | Cutting method and apparatus for ingot, wafer, and manufacturing method of solar cell[END_REF] re-proposed to cleave a wafer from an ingot in a two step approach: generation of a line defect on the surface by means of ion beam along a direction defined by crystal axes and then cleaving the ingot applying a shock in the same point by means of a knife-edge. A few years later, Baer [START_REF] Baer | Cleaving wafers from silicon crystals[END_REF] chose a two-step process where the first is the creation of a notch at a given depth. The crack is propagated then by applying light at a wavelength absorbed at the same given depth of a notch. The heat generated by absorption of such light, which is scanned along the desired direction, is claimed to be sufficient to propagate the crack.

In 2007, Dross et al. [START_REF] Dross | Stress-induced large-area lift-off of crystalline si films[END_REF][START_REF] Dross | Method for the production of thin substrates[END_REF][START_REF] Dross | Method for the production of thin substrates[END_REF] presented the SLIM-Cut process, which consists in inducing a tensile stress in the silicon substrate in order to initiate [START_REF] Qian | Crack initiation for Kerf-Loss-Free wafering[END_REF] and to propagate a crack at a given depth . In order to generate such a tensile stress field, a metallic stressinducing layer is deposited and the system brought at high temperature. During the cooling stage, the mismatch between the Coefficient of Thermal Expansion (CTE) of the metal and the silicon induces a tensile stress field that can be high enough to initiate and propagate a crack all along the silicon substrate [START_REF] Vaes | Slim-cut thin silicon wafering with enhanced crack and stress control[END_REF]. The temperature range in which lift-off occurs in the SLIM-Cut process, may include the silicon brittle-ductile transition temperature: specific attention to this brittle-ductile temperature transition must be paid if one wants to obtain sound defect-free silicon layer after fracture [START_REF] Masolin | Thermal curing of crystallographic defects on a SLIM-Cut silicon foil[END_REF][START_REF] Masolin | Evidence and characterization of crystallographic defect and material quality after slimcut process[END_REF]. Alternatively, the stress-inducing layer can also be a polymer-based material, where the process involves a much lower thermal budget and peak temperatures, assuring brittle crack propagation [START_REF] Martini | Polymer-induced spalling of silicon[END_REF].

A company [START_REF] Mathew | Method of forming an electronic device using a separationenhancing species[END_REF][START_REF] Rao | A novel low cost 25 um thin exfoliated monocrystalline si solar cell technology[END_REF] is marketing solar cells using the same principle meanwhile also IBM [START_REF] Bedell | Spalling for a semiconductor substrate[END_REF][START_REF] Bedell | Spalling for a semiconductor substrate[END_REF][START_REF] Bedell | Kerf-Less removal of si, ge, and III-V layers by controlled spalling to enable Low-Cost PV technologies[END_REF] claims being able to produce multiple high quality thin silicon layers from a single substrate.

The set up and optimization of the aforementioned manufacturing processes imply the use of numerical modeling, which in turns requires accurate input data in terms of thermo-mechanical behavior of silicon. A considerable number of papers have been published about thermo-mechanical properties and fracture properties of silicon, but they are spread all over the literature and they sometimes contradict each other. In this paper, the mechanical properties of single crystal silicon between 293 K and 1273 K will be firstly presented and discussed, a second section will focus on its thermal properties in the same temperature range, while a third section will discuss about the fracture properties of single crystal silicon.

Mechanical properties of single crystal silicon

Silicon, like carbon and germanium, crystallizes at common pressures in a diamond cubic crystal structure with a density of 2.329 g•cm -3 at 298 K. Therefore, silicon is an anisotropic material whose properties depend on its relative orientation to the crystal lattice as well as an orthotropic material, i. e. a crystal with at least two orthogonal planes of symmetry. Silicon is a brittle material at room temperature, which means that its behavior is purely elastic until failure.

Elastic constants

In an anisotropic material, Hooke's law involves a fourth rank tensor (either the stiffness C or the compliance S) to describe the elastic relationship between the second rank stress σ and strain ǫ tensors:

σ ij = C ijkl ε ij and ε ij = S ijkl σ kl (1)
In silicon, the combination of cubic symmetry and the equivalence of the shear conditions enable specifying the fourth rank tensor with only three independent elastic constants. These tensors are given with respect to a specific basis, which in the case of the cubic structure of silicon is commonly given for the <100> directions.

σ ii = C 11 ε ii + C 12 (ε jj + ε kk ) (2)
σ ij = C 44 ε ij
The tensor can then be easily rotated in the orientation of interest. Up to now, the best measurement of the elastic constant is achieved using acoustic waves propagation in the solid. Even if the values from Mason [START_REF] Mason | Physical acoustics and the properties of solids[END_REF] are often cited in the literature, the measurement performed a decade later by Hall [START_REF] Hall | Electronic effects in the elastic constants of n-type silicon[END_REF] In the context of orthotropic materials, as for silicon, it is possible to give, for the axes of interest, the elastic properties in terms of orthotropic material constants involving the Young's modulus E, the Poisson's ratio n and the shear modulus G.

The Young's modulus is a parameter to characterize the stiffness of an elastic material. It can be measured from the slope of the linear portion of the stress-strain curve recorded from an experiment where the specimen undergoes to a uniaxial load.

Otherwise the Young's modulus E can be calculated from the general formulae for cubic crystal [START_REF] Nye | Physical properties of crystals: their representation by tensors and matrices[END_REF] :

1 E hkl = S 11 -2 S 11 -S 12 - 1 2 S 44 m 2 n 2 + p 2 n 2 + m 2 p 2 (3)
where m, n, p are the "direction cosines" i. e. the cosine of the angle between the [hkl ] direction and the 3 basis axes (the <100> directions). For a better understanding, three different cases, at room temperature, are illustrated in the Table 2 where a classical [110] direction is assumed for the primary flat. The first column corresponds to the case where the used basis is equal to the <100> directions and so, for each subscript x, y or z, two of the "direction cosines" are null. Therefore,

Ex = Ey = Ez = 1 /S11 ≈ 130GPa (4)
For the second column, the subscript x (resp. y) corresponds to the [110] (resp.[ 110]) directions. Therefore,

Ex = Ey = S 11 -2 S 11 -S 12 - 1 2 S 44 1 √ 2 2 1 √ 2 2 -1 (5) = 2 S 11 + S 12 + 1 2 S 44 -1 ≈ 169GPa Ez = 1 /S11 ≈ 130GPa (6)
Same methodology can be used for the third column in the context of a (111)-wafer. Brantley et al. [START_REF] Brantley | Calculated elastic constants for stress problems associated with semiconductor devices[END_REF] reports maximum and minimum values of Young's modulus for other directions lying in important crystal planes. Concerning Poisson's ratio and shear modulus, the cubic crystal configuration allows also using the following formulae:

υ αβ = S 12 + S 11 -S 12 -1 2 S 44 m 2 α m 2 β + n 2 α n 2 β + p 2 α p 2 β S 11 -2 S 11 -S 12 -1 2 S 44 (m 2 α n 2 α + p 2 α n 2 α + m 2 α p 2 α ) (7)
G ij = 1 /Sij (8)
with and two orthogonal directions and m , n , p , the "direction cosines" of the angle between the direction and the basis axes.

Finally, the bulk modulus B can be also obtained from:

B = C 11 + 2C 12 3 (9)
At room temperature (298 K), these formula leads to a bulk modulus B of 0.9781•10 11 Pa (therefore a compressibility K of 1.0221•10 -11 Pa -1 ), which is in very good agreement with B ∼0.995 ± 0.005•10 11 Pa measured in real experiments [START_REF] Balamane | Comparative study of silicon empirical interatomic potentials[END_REF][START_REF] Haynes | Linear-scaling methods in ab initio quantum-mechanical calculations[END_REF]. The values of the Poisson's ratio and shear modulus for the different configurations considered are summarized in Table 2 For simplified analyses, or analytic expressions, a single isotropic elasticity value may be used. To insure accuracy, the choice of this value must depend on the orientation and loading of the structure. Since the crystal structure of silicon has a cubic symmetry, computations in configurations presenting orthogonal shapes and loads will be reasonably accurate, as long as the appropriate elasticity value for the direction family is used. But for more complex cases with off-axis orientations or non-rectilinear structures, the use of the full orthotropic description will have significant benefits for the accuracy of the results.

Hardness

Hardness may be defined as the resistance of a material to permanent penetration by another material. The most important and comprehensive work about nanoidentation experiments to measure silicon hardness was performed by Bhushan et al. [START_REF] Bhushan | Nanoindentation hardness measurements using atomic force microscopy[END_REF][START_REF] Bhushan | Micromechanical and tribological characterization of doped singlecrystal silicon and polysilicon films for microelectromechanical systems[END_REF]. Even though hardness values are dependent on the normal load, the indentation depth and crystallographic orientation, a value of 12±1 GPa could be considered as average for all the cases. The only exception is a p-type silicon, boron-doped: it is shown that the doping using thermal diffusion with boron ions softened the silicon surface down to ∼7 GPa [START_REF] Bhushan | Micromechanical and tribological characterization of doped singlecrystal silicon and polysilicon films for microelectromechanical systems[END_REF] .

Temperature effects on elastic constants

The silicon Young's modulus evolves with temperature. This thermal dependency is traditionally described for each elastic constant C 11 , C 12 , S 11 , . . . with the Thermal Coefficient of Elasticity (TCE) of the considered elastic constant. More precisely, for each of these constants C, its thermal variation between T 0 and T can be described via a power series of coefficients TCE(C) k :

C (T ) = C (T 0 )   1 + k≥1 TCE (C) k (T -T 0 ) k   (10)
Several different measurements of TCE are reported in the literature [START_REF] Mcskimin | Measurement of the elastic constants of silicon single crystals and their thermal coefficients[END_REF][START_REF] Burenkov | Temperature dependence of the elastic constants of silicon[END_REF][START_REF] Bourgeois | Design of resonators for the determination of the temperature coefficients of elastic constants of monocrystalline silicon[END_REF] for the first order temperature coefficients, and so, their values cannot be given definitively. However, as recently reported by Hopcroft et al. [START_REF] Hopcroft | What is the young's modulus of silicon?[END_REF], the results given by Bourgeois et al. [START_REF] Bourgeois | Design of resonators for the determination of the temperature coefficients of elastic constants of monocrystalline silicon[END_REF] seem to come from the most carefully performed experiments as the values proposed include the second order temperature coefficients (see Table 3).

Macroscopic mechanical behavior at high temperature

As stated before, beyond the elastic regime, silicon is a brittle material for low temperatures, but exhibits viscoplastic behavior before ductile failure above the brittleductile temperature T BD . This viscoplastic behavior strongly depends on strain rate and temperature. At high temperature, the stress-strain curve of silicon shows two yield regions. Indeed after elastic domain, and between the upper yield stress and the lower yield stress, silicon exhibits a transient softening effect due to a drastic increase of dislocation density. Then, after the lower yield point, small and then strong workhardening are observed (stage I and II). This behavior is classically modeled in the crystal plasticity framework, accounting for the discrete nature of plastic slip in crystal as in [START_REF] Cacho | A constitutive single crystal model for the silicon mechanical behavior: Applications to the stress induced by silicided lines and STI in MOS technologies[END_REF][START_REF] Cochard | Constitutive modeling of intrinsic silicon monocrystals in easy glide[END_REF], or in a more standard isotropic formulation of plastic flow as in [START_REF] Dillon | Dislocation dynamics during the growth of silicon ribbon[END_REF][START_REF] Moon | A constitutive model for the mechanical behavior of single crystal silicon at elevated temperature[END_REF].

In the latest model, the plastic strain rate produced by a crystal ( γP ) is, in general, determined by the Orowan equation [START_REF] Orowan | Problems of plastic gliding[END_REF]:

γP = ρmbν ( 11
)
where Öm is the mobile dislocations density which corresponds to a part of the Sta- tistically Stored Dislocations (SSDs) density [START_REF] Nye | Physical properties of crystals: their representation by tensors and matrices[END_REF][START_REF] Arsenlis | Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density[END_REF], b is the Burgers vector magnitude, and ν is the average velocity of these dislocations. In accordance with experimental measurement, the dislocation velocity ν is a function of the temperature T (Arrhenius factor) and of the effective shear stress Ø eff (power law)

ν = ν 0 τ ef f τ 0 1 /m exp -U dis kT sign (τ ) (12)
where Ò 0 and Ø 0 are reference values for the dislocation velocity and stress, m is the stress exponent, k is the Botzmann's constant and U dis is the activation energy for dislocation velocity. The effective shear stress is the difference between the internal stress Ø i and the applied stress Ø:

τ ef f = |τ | -τ i , with x = x + |x| 2 (13)
Then, the different constitutive models available in the literature are traditionally derived from the work of Alexander and Haasen [START_REF] Alexander | Dislocations and plastic flow in the diamond structure[END_REF], which is quite successful for initial stages of deformation up to the lower yield point. After this point, more appropriate models, like the ones proposed by Delaire [START_REF] Delaire | Plastic heterogeneities of a copper multicrystal deformed in uniaxial tension: experimental study and finite element simulations[END_REF] or Moon et al. [START_REF] Moon | A constitutive model for the mechanical behavior of single crystal silicon at elevated temperature[END_REF] are needed. In the Alexander and Haasen's model, the mobile dislocations density Öm is supposed to be equal to the total dislocation density Ö, and the internal stress Ø i is given by the following relation, where is a constant and Ñ is the shear modulus:

τ i = τ 0 + αµb √ ρ (14)
The evolution equation for the dislocation density is finally described by the following equation, in which K is a material constant taken into account the creation and annihilation of the SSDs:

ρ = K b τ ef f γP (15)
Note that in the case of the Alexander-Haasen's model, the temperature and strain rate dependence of the upper τuyp and lower τ lyp yield points are straightly described by:

τuyp /lyp = Cuyp /lyp γ1 /(2+m) exp U dis (2 + m) kT (16)
where C uyp/lyp are constants respectively associated to the upper and lower yield points. The following values for the different models constants, in case of undoped silicon, can be found in the literature: If thermal problems have to be faced, thermal conductivity (κ), diffusivity (D), specific heat (Cp) and emissivity (e) of silicon have to be known for different temperatures.

b ν 0 τ 0 U dis m α K 3,
If stresses induced by the change of temperature, i.e. thermal stresses, are of interest, also the coefficient of thermal expansion ( T ) must be evaluated.

Thermal conductivity, diffusivity and specific heat

Above 200 K, the thermal conductivity is largely independent from the particular sample specification and the various reported data obtained through different methods show a rather good agreement [START_REF] Abeles | Thermal conductivity of Ge-Si alloys at high temperatures[END_REF][START_REF] Glassbrenner | Thermal conductivity of silicon and germanium from 3K to the melting point[END_REF][START_REF] Fulkerson | Thermal conductivity, electrical resistivity, and seebeck coefficient of silicon from 100 to 1300K[END_REF][START_REF] Yamamoto | Thermal diffusivity of crystalline and liquid silicon and an anomaly at melting[END_REF][START_REF] Yamasue | Thermal conductivities of silicon and germanium in solid and liquid states measured by non-stationary hot wire method with silica coated probe[END_REF]. The following values are representative measurements from Glassbrenner and Slack [START_REF] Glassbrenner | Thermal conductivity of silicon and germanium from 3K to the melting point[END_REF]. Concerning the thermal diffusivity D of single crystal silicon, the values in Table 5 were measured from room temperature up to 1400 K by Abeles et al. [START_REF] Abeles | Thermal conductivity of Ge-Si alloys at high temperatures[END_REF]. These measurements were found in between those of Glassbrenner and Slack [START_REF] Glassbrenner | Thermal conductivity of silicon and germanium from 3K to the melting point[END_REF] and Shanks et al. [START_REF] Shanks | Thermal conductivity of silicon from 300 to 1400 k[END_REF] which are within the order of 5% accurate below 1000 K, but less accurate above. Specific heat recommended data reported by Hull [START_REF] Hull | Properties of crystalline silicon[END_REF] are also given in Table 5. From these previous experimental data, the conductivity can also be given by the following [START_REF] Prakash | Thermal conductivity variation of silicon with temperature[END_REF]:

T (K) κ (W•cm -1 •K -1 ) D (cm 2 •s -1 ) Cp (J•g -1 •K -1 )
κ = κ 0 1 -B T -T 0 T A , with B =
where κ 0 is the thermal conductivity of silicon (W•cm -1 •K -1 ) at room temperature T 0 .

Emissivity

There have been several studies of the thermal radiation emitted by silicon at elevated temperatures. But since silicon emissivity strongly depends on many factors, such as the sample thickness, doping, surface conditions... experiments must be interpreted with care before applying one of the various measured silicon emissivity values to any kind of model. As first approximation, an emissivity value acceptable regarding the different experimental results is null at room temperature, which then rises smoothly with temperature increase to a maximum value of 0.7 at 1220 K.

Thermal expansion

The single-crystal silicon thermal expansion coefficient T has been measured in the range 120 K to 1500 K by several means such as an interferometric dilatometer [START_REF] Okaji | Absolute thermal expansion measurements of single-crystal silicon in the range 300-1300K with an interferometric dilatometer[END_REF][START_REF] Watanabe | Linear thermal expansion coefficient of silicon from 293 to 1000K[END_REF], X-ray diffractometry [START_REF] Okada | Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500K[END_REF] and other techniques [START_REF] Swenson | Recommended values for the thermal expansivity of silicon from 0 to 1000 K[END_REF]. The reported measurements were found in good agreement between different methods (Table 6). From these measurements, a fitting expression of the linear thermal expansion coefficient T (10 -6 •K -1 ) is given by [START_REF] Okada | Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500K[END_REF]:

α(T ) = 3.725 1 -e -5.88•10 -3(T -124) + 5.548•10 -4 T (18)
with T between 120 K and 1500 K. At room temperature the recommended value is (298.2) = 2.59 ± 0.05 • 10 -6 K -1 . Note that the effect of temperature on silicon density can also be evaluated from these values. [START_REF] George | Dislocation loops at crack tips: nucleation and growth-an experimental study in silicon[END_REF], Hsia and Argon [START_REF] Hsia | Experimental study of the mechanisms of brittle-to-ductile transition of cleavage fracture in Si single crystals[END_REF]. They pointed out that silicon presents a particularly sharp brittle-ductile transition. This transition is indeed associated with a sudden increase in stress to fracture, in order to intercept the yield stress curve. This transition occurs over a very narrow temperature range, typically less than 10 K (Figure 1). The microscopic studies of the fractured samples have shown that there is hardly any dislocation activity at the crack tip below the brittle-ductile transition temperature, few hundred dislocations can be seen from the crack, moving into the bulk, along well-defined crystallographic directions approaching T BD and a huge amount of dislocations nucleate above this critical temperature T BD . It is also important to note that if silicon is pre-deformed to introduce dislocations and dislocations sources, it exhibits a softer transition [START_REF] Warren | The brittle-ductile transition in silicon: the influence of pre-existing dislocation arrangements[END_REF]. illustrates the Arrhenius plot of the most current data on the brittleductile transition temperature. It shows that, although all the lines for intrinsic silicon have the same slope, the intercepts vary widely from one result to another, showing the dependence on the testing methods (especially levels of crack tip perfection). It also points out that p-type dopants do not affect the brittle-ductile transition temperature while n-type dopants decrease it. Moreover, T BD increases with a higher rate of stress intensity. These experiments have determined T BD as a function of, using the activation energy for the brittle-ductile transition U BD (this activation energy was found to be nearly equal to U dis in Eq. 12): where A is a model constant and k is the Boltzmann constant. U BD was measured to be 2.1 eV for intrinsic, and 1.6 eV for n-type silicon by Samuels and Roberts [START_REF] Samuels | The brittle-ductile transition in silicon. I. experiments[END_REF]. The most quantitative model proposed for the brittle-ductile transition in silicon is the one proposed by Hirsch and Roberts [START_REF] Hirsch | The brittle-ductile transition in silicon[END_REF][START_REF] Hirsch | Modelling plastic zones and the brittle-ductile transition[END_REF]. In this model, the shielding of the crack front by dislocations emitted from there with the rise of the stress intensity factor K I to the critical value K Ic . The main feature of the model is that the material becomes ductile only when the emitted dislocations shield every point of the crack front. In this sense, the mobility of the dislocations plays the major role in this model.

T (K) (W•cm -1 •K -1 ) 300
K = A exp -U BD kT BD (19)

Fracture toughness

As for the mechanical properties, the single crystal silicon fracture toughness K Ic depends on the crystallographic orientation. Vickers micro-hardness indentation associated, or not, with four point bending, and double cantilever beam are the most commonly used methods to evaluate this toughness anisotropy. Since the different reported silicon fracture studies have emphasized the fracture anisotropy on the low index planes, we will focus here on the fracture toughness of these planes, i.e. {100}, {110} and {111} planes, although many higher order index planes surface energy values sit between the ones of the low index planes [START_REF] Hesketh | Surface free energy model of silicon anisotropic etching[END_REF]. These results suggest that silicon may also cleave on crystallographic planes other than the low index ones [START_REF] Ebrahimi | Crack path in single crystals[END_REF][START_REF] Ebrahimi | Fracture anisotropy in silicon single crystal[END_REF]. Table 7 summarizes the fractures toughness and fracture energy values at room temperature reported in the literature. As seen in the previous section, these fracture toughness values are valid for temperatures below the brittle-ductile transition one. In case of simulated values, the method is written in parentheses: Molecular Dynamics (MD) or Density Functional Theory (DFT).

The referenced articles might report values either in terms of fracture surface energy γ (hkl) or fracture toughness K Ic(hkl) . The following approximate equation, in which E [hkl] is the Young's modulus in the perpendicular direction to the crack surface, was used to convert them when necessary:

K Ic(hkl) = 2γ (hkl) E [hkl] 1 /2 (20) Fracture toughness K Ic (MPa•m 1/2 ) Fracture energy s (J•m -2 ) Fracture planes {111} {110} {100} {111} {110} {100}
[69] 0.62 [START_REF] Jaccodine | Surface energy of germanium and silicon[END_REF] 0.62 0.71 0.75 [START_REF] St | The brittle-to-ductile transition in pre-cleaved silicon single crystals[END_REF] 0.93 [START_REF] Chen | Fracture toughness of silicon[END_REF] 0.82 0.90 0.95 [START_REF] Messmer | The surface energy of Si, GaAs, and GaP[END_REF] 0.65 0.80 [START_REF] Chen | Crack Growth in Single-Crystal Silicon[END_REF] 1.00 [START_REF] Bhaduri | Fracture surface energy determination in 110 planes in silicon by the double torsion method[END_REF] 0.78 [START_REF] Tsai | Fractal fracture of single crystal silicon[END_REF] 1.05-1.19 [START_REF] Hayashi | Fracture toughness of single crystal silicon[END_REF] 1.14-1.19 1.07-1.18 0.82-1.15 [START_REF] Xin | A technique to generate straight through thickness surface cracks and its application to studying dislocation nucleation in si[END_REF] 0.95 [START_REF] Ebrahimi | Fracture anisotropy in silicon single crystal[END_REF] 1.22 0.81-1.01 0.86-1.25 [START_REF] Hauch | Dynamic fracture in single crystal silicon[END_REF] 0.66 [START_REF] Swadener | Effect of dopants on the fracture toughness of silicon[END_REF] 0.68-0.73 [START_REF] Fitzgerald | Subcritical crack growth in singlecrystal silicon using micromachined specimens[END_REF] 1.15 [START_REF] Tan | Crystallographic cracking behavior in silicon single crystal wafer[END_REF] 1.12 1.29 [START_REF] Cook | Strength and sharp contact fracture of silicon[END_REF] 0.63-0.74 1.11 [START_REF] Pérez | An ab initio study of the cleavage anisotropy in silicon[END_REF] As first remark, the countervailing maxima and minima in the modulus and fracture resistance variations lead to a very small variation in toughness with fracture planes. Silicon is reported to have two principal cleavage planes: {111} planes, usually the easiest cleavage plane and {110} planes. In other words, the cleavage energy of {111} is lower than {100} one and thus, crack will unlikely propagate on the {100} plane.

Different crack propagation directions have been studied for both fracture planes. The <110> propagation directions were seen to be the preferred propagation directions on both cleavage planes. Nevertheless, on the {111} fracture surface, the anisotropy with respect to propagation direction manifests itself only in faint markings along <110> directions. Complementary, cleavage fracture on the {110} plane is very anisotropic. Propagation along the <110> directions is easy and results in nearly perfectly flat fracture surfaces, while along the <100> directions, perpendicular to the preferred direction, the crack deflects onto {111} planes inclined by 35.26°with respect to the original fracture plane [START_REF] Tsai | Fractal fracture of single crystal silicon[END_REF][START_REF] Ebrahimi | Fracture anisotropy in silicon single crystal[END_REF][START_REF] Perez | Directional anisotropy in the cleavage fracture of silicon[END_REF]. In contrast to the results of the {110} fracture planes, the cracks introduced along the {100} planes were observed to deviate from these planes. These results can be understood by the fact that the fracture toughness of the {100} planes is almost the same as those of the higher order planes near {100}. Cracks following the {100} planes even deflect onto {110} planes, inclined by 45°with respect to the {100} planes, since these second planes exhibit the minimum fracture toughness value among the possible deflecting planes [START_REF] Tan | Crystallographic cracking behavior in silicon single crystal wafer[END_REF][START_REF] Tanaka | Orientation dependence of fracture toughness measured by indentation methods and its relation to surface energy in single crystal silicon[END_REF].

Aside, there is an experiment of Deegan et al. [START_REF] Deegan | Wavy and rough cracks in silicon[END_REF] who observed that cracks which deviate from the <110> plane can travel in arbitrary directions, moreover these directions can fluctuate wildly creating a fractal fracture surface. This influence of the crack propagation direction in a given fracture plane, and the fact that cracks often deflect from the original fracture plane, are therefore responsible for the large scatter in the measured toughness value for each fracture plane, as clearly pointed out in Table 7. Many other parameters contribute to this scatter, including the testing method, specimen surface preparation, and the crack length measurement in case of indentation fracture method.

Even though some ambiguities exist in the literature regarding the exact value of the fracture toughness of single crystal silicon, it appears that the earliest measurements of silicon fracture toughness [START_REF] Gilman | Direct measurements of the surface energies of crystals[END_REF][START_REF] Jaccodine | Surface energy of germanium and silicon[END_REF], using the well-defined double-cantilever beam geometry, are the least ambiguous from a testing geometry perspective, and in best agreement [START_REF] Clarke | The Mechanical Properties of Semiconductors[END_REF] with both molecular dynamics calculations, based on known bond-rupture energies, and experimental scaling of fracture resistance with band-gap in elemental and compound semiconductors. Over the past 40 years, subsequent measurements using smaller cracks from indentation fractographic methods seem to always overestimate fracture toughness while providing critical information on the orientation dependence of fracture toughness. In Table 8 is summarized the range of values reported for fracture toughness and fracture energy and some recommended values based on aforementioned considerations. Both values for fracture toughness and fracture energy are reported for reader's convenience, using equation [START_REF] Mathew | Method of forming an electronic device using a separationenhancing species[END_REF]

Crack speed

The development of high-speed data acquisition has extended studies to dynamic crack propagation at crack driving forces greater than the equilibrium fracture resistance.

Different experiments [START_REF] Hauch | Dynamic fracture in single crystal silicon[END_REF][START_REF] Cramer | Crack velocities during dynamic fracture of glass and single crystalline silicon[END_REF][START_REF] Cramer | Energy dissipation and path instabilities in dynamic fracture of silicon single crystals[END_REF][START_REF] Sherman | From crack deflection to lattice vibrations-macro to atomistic examination of dynamic cleavage fracture[END_REF][START_REF] Sherman | Macroscopic and microscopic examination of the relationship between crack velocity and path and rayleigh surface wave speed in single crystal silicon[END_REF] show that cracks propagate with velocities of about 1 to 3.5 km•s -1 . There is therefore an apparent speed gap between 0 and ∼1-2 km•s -1 for crack driving forces just exceeding the fracture resistance [94,[START_REF] Holland | Cracks and atoms[END_REF]. A possible explanation of this phenomenon is described by many scholars [START_REF] Thomson | Lattice trapping of fracture cracks[END_REF][START_REF] Lawn | An atomistic model of kinetic crack growth in brittle solids[END_REF][START_REF] Marder | Molecular dynamics of cracks[END_REF] and recently by Bernstein and Hess [START_REF] Bernstein | Lattice trapping barriers to brittle fracture[END_REF] where they indicate the presence of lattice trapping barriers as major player for the propagation of a brittle fracture, i.e. the fracture crack might lead to a configuration where the stress could be below or above the Griffith stress but the crack is stable [START_REF] Thomson | Lattice trapping of fracture cracks[END_REF].

Deegan [START_REF] Deegan | Wavy and rough cracks in silicon[END_REF] reports that, depending on the speed of the crack propagation, transitions from straight to wavy to multiply branched cracks are possible and could be discontinuous, bistable, and hysteretic. At large crack driving forces, the velocities approach an apparent upper limit approximately equal to 75% of the Rayleigh wave speed (c R ∼4.6 km•s -1 ) depending on the direction of crack propagation [100, 101].

Conclusion

Single crystal silicon has been extensively used in the electronic industry, and therefore numerous studies have also been performed and most of the needed parameters for the computation are available in the literature. These data have been gathered and compared here for a large temperature range.

Due to its crystalline structure, silicon is a strongly anisotropic material whose properties depend on orientation relative to the crystal lattice, especially regarding its fracture behavior. Several toughness values have been found in the literature. However the variation of fracture toughness between each orientation planes remains small.

More importantly, silicon is a brittle material at room temperature, which means that its behavior is purely elastic until it fails. But it also exhibits a sharp brittle-ductile transition at a precise temperature.

  1.093 and A = 0.7805 T < 1000K B = 0.93795 and A = 0.42 T > 1000K

Fig. 1

 1 Fig. 1 Sharp brittle-ductile transition in silicon. The stress intensity at fracture rises abruptly at T BD

Fig. 2

 2 Fig.2Brittle-ductile transition temperature in silicon. Data from: ABC[START_REF] Brede | The brittle-to-ductile transition in doped silicon as a model substance[END_REF], D[START_REF] St | The brittle-to-ductile transition in pre-cleaved silicon single crystals[END_REF], EF[START_REF] George | Dislocation loops at crack tips: nucleation and growth-an experimental study in silicon[END_REF], GH[START_REF] Samuels | The brittle-ductile transition in silicon. I. experiments[END_REF]. Doping levels: ABCH n-type, and DEFGH intrinsic

Table 1

 1 Elastic constants of silicon at 298K (C: 10 9 Pa, S: 10 -12 Pa)

	reports slightly better accuracy

Table 2

 2 .

	wafer	(100)		(111)
	in GPa
	Ex	130	169	174
	Ey	130	169	174
	Ez	130	130	188
	Òyz	0.278	0.362	0.166
	Òzx	0.278	0.362	0.166
	Òxy	0.278	0.064	0.241
	Gyz	79.6	79.6	60.5
	Gzx	79.6	79.6	60.5
	Gxy	79.6	50.9	70.0

Approximate values of elasticity in the reference frame of standard silicon wafers

Table 3

 3 Temperature coefficients of the elastic constants

	TCE	p-type (4 Ï cm, B)	n-type (0.05 Ï cm, P)	p-type (4 Ï cm, B)	n-type (0.05 Ï cm, P)
		First-order (x 10 -6 /K)	Second-order (x 10 -6 /K 2 )
	TCE S11	64.73 ± 0.29	63.60 ± 0.60	61.19 ± 1.1	60.51 ± 0.35
	TCE S12	51.48 ± 1.5	45.79 ± 2.8	72.26 ± 5.1	75.70 ± 6.1
	TCE S44	60.14 ± 0.20	57.96 ± 0.17	54.90 ± 1.7	57.31 ± 1.4
	TCE C11	-73.25 ± 0.49	-74.87 ± 0.99	-49.26 ± 4.8	-45.14 ± 1.4
	TCE C12	-91.59 ± 1.5	-99.46 ± 3.5	-32.70 ± 10.1	-20.59 ± 11.0
	TCE C44	-60.14 ± 0.20	-57.96 ± 0.17	-51.28 ± 1.9	-53.95 ± 1.8

Table 4

 4 Models constant values3 Thermal properties of single crystal silicon

	83e -10 m	4.3e -4 m/s	5.5 MPa	2.35 eV	0.91	0.3-2.0	2e -4 m/N

Table 5

 5

	200	2.66		0.557
	300	1.56	0.86	0.713
	400	1.05	0.52	0.785
	500	0.80	0.37	0.832
	600	0.64	0.29	0.849
	700	0.52	0.24	0.866
	800	0.43	0.19	0.883
	900	0.356	0.16	0.899
	1000	0.31	0.14	0.916
	1100	0.28	0.13	0.933
	1200	0.261	0.12	0.950
	1300	0.248	0.12	0.967
	1400	0.237	0.12	0.983
	1500	0.227		1.000
	1600	0.219		1.017

Single crystal silicon thermal conductivity (κ), diffusivity (D) and specific heat (Cp)

Table 6

 6 Single crystal silicon thermal expansion coefficient values 4 Fracture properties of single crystal silicon 4.1 Brittle-ductile transition Single crystal silicon is a brittle material at room temperature, in which cracks propagate without any appreciable plastic deformation. Nevertheless, it exhibits a ductile behavior above a certain temperature T BD , for a given loading rate (or increase rate of stress intensity ) and doping level. Single crystal silicon brittle-ductile transition experiments were carried out especially by St. John [57], Brede and Haasen [58], Hirsch et al. [59, 60, 61], George and Michot

		2.616
	400	3.253
	500	3.614
	600	3.842
	700	4.016
	800	4.151
	900	4.185
	1000	4.258
	1100	4.323
	1200	4.384
	1300	4.442
	1400	4.500
	1500	4.556

Table 7

 7 Reported silicon fracture toughness and fracture energy values. MD = Molecular Dynamics. DFT = Density Functional Theory.

	(DFT)

Table 8

 8 to convert them. Summary of reported and recommended silicon fracture toughness and fracture energy values

	Fracture planes	{111}	{110}	{100}
	Reported experimental range
	Fracture toughness	0.62-1.22	0.68-1.19	0.75-1.29
	K Ic (MPa•m 1/2 )
	Reported simulated range
	Fracture energy	1.19-1.45	1.50-1.73	1.56-2.26
	s (J•m -2 )
	Recommended value
	Fracture toughness	0.62	0.71	0.75
	K Ic (MPa•m 1/2 )
	Recommended value
	Fracture energy	1.022	1.483	2.163
	s (J•m -2 )

Acknowledgements

The authors wish to thank EC for the financial support for this research (SUGAR project FP7 nº 256752). A special thank to Guillaume Lebret.