Dominique Azé
email: dominique.aze@math.univ-toulouse.fr

Jean-Noël Corvellec
email: corvellec@univ-perp.fr

Nonlinear Error Bounds via a Change of Function

Keywords: Nonlinear error bounds, Asymptotically well-behaved functions, Descent methods in optimization Mathematics Subject Classification 49J53, 49J52, 90C30

This work can be seen as a sequel to our previous paper, which dealt with local nonlinear error bounds for lower semicontinuous functions on complete metric spaces, based on estimates of the strong slope of the function through the distance to a sublevel set. Here, we consider estimates of the strong slope through the values of the function, and we provide characterizations of nonlinear local and global error bounds that are again obtained by a reduction to the linear case. Our main tool is a simple chain rule for the strong slope. We provide several examples showing that recent results in the literature can be recaptured from our general framework.

Introduction

In our previous paper [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF], we developed a general approach to nonlinear local error bounds, where the basic assumptions are estimates of the strong slope (of a lower semicontinuous function defined on a complete metric space) through the distance to a sublevel set. The main tool was a so-called change-of-metric principle that allows reducing the nonlinear case to the linear one (systematically studied in [START_REF] Azé | Characterizations of error bounds for lower semicontinuous functions on metric spaces[END_REF]), and the results were refinements of those of [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF], where the local case was not addressed in a satisfactory manner.

In this paper, we develop the theory of nonlinear error bounds based on assumptions that are estimates of the strong slope through the values of the function outside a sublevel set. This was also treated in [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF] (and related therein to the notion of asymptotically well-behaved function) in the global case, through an ad hoc argument that is again unsatisfactory, in particular since it does not extend to the local case. Here, we use a simple calculus lemma, a chain rule for the strong slope (Lemma 4.1), that reduces the nonlinear problem to a linear one, both in the local and global case (and which accounts for the title of the paper). We thus start with a presentation of the characterization of error bounds in the linear case, which was already (partly) addressed more than ten years ago in our papers [START_REF] Azé | Characterizations of error bounds for lower semicontinuous functions on metric spaces[END_REF][START_REF] Azé | Variational pairs and applications to stability in nonsmooth analysis[END_REF]. However, not only was the local case not treated in a systematic way in these papers, but we shall here do things in a way that fits the mentioned reduction procedure, and that takes into account more recent developments (both ours and in the literature).

Technically, the results in this paper are obtained in a simpler way than those of [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF][START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF], but it could be said that they do not serve the same purposes. Roughly speaking, the latter are related to the notion of metric (sub) regularity and to applications to sensitivity analysis in optimization, while the former fit problems of convergence in descent methods. As a matter of fact, an important motivation for our present study comes from the papers [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF][START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF][START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF], which basically deal with that kind of problem, and we conclude this work with an example of convergence of a descent method in optimization, following [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF][START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF].

The paper is organized as follows. In Sect. 2, we recall the notion of strong slope and its relationship with subdifferential notions, while specifying our main notations. In Sect. 3, we establish the characterization of local and global linear error bounds through the variational principle. In Sect. 4, we deduce the nonlinear results through a chain rule for the strong slope. In Sect. 5, we focus on so-called Hölderian error, which has been mainly considered in the literature, some examples of which we describe. In Sect. 6, we make a few remarks in the specific case of convex functions (on Banach spaces). In Sect. 7, we compare our main assumption with that of [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF], we give a new result under the former assumption, and we describe an application to the convergence of a descent method in optimization, as mentioned above.

Basic Notations and Notions

Throughout the paper, we let X denote a metric space endowed with the metric d, and we let f : X → R ∪{+∞}be an extended-valued function that will generally be required to be lower semicontinuous. For x ∈ X and ρ>0, we denote by B ρ (x) (resp. B ρ [x]), the open (resp., closed) ball of radius ρ>0 centered at x.ForY ⊂ X ,welet

d(x, Y ) := inf{d(x, y) : y ∈ Y } , with the usual convention d(x, ∅) =+∞.For-∞ < a < b ≤+∞,welet: [ f ≤ a]:={x ∈ X : f (x) ≤ a} , [ f < b]:={x ∈ X : f (x)<b}
denote the sublevel sets of f , and we let

[a< f < b]:=[f <b]\[ f ≤ a]
denote the slice between a and b. When b =+∞, we write

dom f := [ f <+∞] , [ f > a]:=[a< f < +∞].
As usual, for a ∈ R,weleta + := max{a, 0}.

Given x ∈ X with a := f ( x) ∈ R, we say that the function f has a nonlinear local error bound at x if there exist γ :]0, +∞[→]0, +∞[ continuous and ρ>0 such that

f (x) ≥ a + γ(d(x, [ f ≤ a])) for every x ∈ B ρ ( x)\[ f ≤ a] .
We say that f has a nonlinear global error bound with respect to

[ f ≤ a] if the inequality holds for all x ∈[f >a] (that is, B ρ ( x) is replaced by X ).
Recall from [START_REF] De Giorgi | Problemi di evoluzione in spazi metrici e curve di massima pendenza[END_REF] that for x ∈ dom f ,thestrong slope of f at x is defined by

|∇ f |(x) := ⎧ ⎨ ⎩ 0 , if x is a local minimum point of f lim sup y→x f (x) -f (y) d(x, y) , otherwise .
(For x / ∈ dom f ,weset|∇ f |(x) := +∞). The strong slope allows developing general results in variational analysis, while it easily compares with (sub)derivative notions in the case when (X, • ) is a (real) normed vector space-so that these results are both enlightening from the point of view of the abstract theory, and useful from the point of view of applications.

Let us recall a few such comparisons (some of which will be referred to later). Denoting by (X * , • * ) the topological dual of (X, • ),forx ∈ dom f and u ∈ X \{0}, set:

f ′ (x; u) := lim inf t→0 + ,v→u f (x + tv) -f (x) t , ∂ F f (x) := x * ∈ X * : lim inf y→x f (y) -f (x) -x * , y -x y -x ≥ 0 ,
which are the lower contingent derivative of f at x in the direction u and the Fréchet subdifferential of f at x, respectively. From the definitions, it is easy to see that

|∇ f |(x) ≥ sup u =1 -f ′ (x; u) (1)
(with equality if X is finite dimensional), and

|∇ f |(x) ≤ d * (0,∂ F f (x)) := inf{ x * : x * ∈ ∂ F f (x)} , (2)
with equality if f is convex, in which case, if x is not a minimum point of f we also have

|∇ f |(x) = sup y =x f (x) -f (y) x -y .
Roughly speaking, estimate (1)isofprimal type (in applications, it is related to results involving tangent cones), while estimate (2)isofdual type (related to results involving normal cones). In general, a major estimate is

|∇ f |(x) ≥ lim inf (y, f (y))→(x, f (x)) d * (0,∂f (y)) , (3)
where ∂ is a subdifferential operator. From known considerations in "classical" nonsmooth analysis, given a specific subdifferential operator ∂,( 3) holds for any lower semicontinuous f whenever (X, • ) belongs to some appropriate class of Banach spaces: See, e.g., [START_REF] Azé | Characterizations of error bounds for lower semicontinuous functions on metric spaces[END_REF]Section 4], and the beginning of Section 6 in [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF] for details and references. For instance, if

∂ = ∂ F , then (3)

Characterization of Linear Error Bounds

In order to obtain sufficient conditions in our error bound results, we use the following straightforward consequence of the variational principle (for which a major reference is [START_REF] Ekeland | On the variational principle[END_REF]). See [1, Proposition 2.1] for a proof, and [1, Remark 2.1] for some comments.

Proposition 3.1 Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, and U be a (nonempty) open subset of X . Then:

f (x) -inf U f ≥ inf U |∇ f | d(x, X \U ) for every x ∈ U . (4)
Theorem 3.1 Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, C ⊂ X, -∞ < a < b ≤+∞, and σ, ρ > 0. Assume that:

|∇ f |(x) ≥ σ for every x ∈ B ρ (C) ∩[a< f < b] . ( 5
)
The following two properties hold:

(a) If [ f ≤ a] =∅, then f (x) -a ≥ σ d(x, [ f ≤ a]) for every x ∈ C ∩[a< f < b] with d(x, [ f ≤ a]) ≤ ρ ; (b) If C ∩[f <a+σρ] =∅, then [ f ≤ a] =∅, and f (x) -a ≥ σ d(x, [ f ≤ a]) for every x ∈ C ∩[a < f < a+σρ] . Proof Let x ∈ C ∩[a< f <b],set X := [ f ≤ f (x)]
,let f denote the restriction of f to X , and set

U := B ρ (C) ∩[ f >a], so that d(x, X \U ) ≥ min{ρ,d(x, [ f ≤ a])} .
On the other hand,

f (z) = f (z) and |∇ f |(z) =|∇f |(z) for every z ∈ X , so that f (x) -a ≥ σ d(x, X \U )
according to [START_REF] Azé | Variational pairs and applications to stability in nonsmooth analysis[END_REF][START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF].

Thus, if d(x, [ f ≤ a]) ≤ ρ, then d(x, X \U ) ≥ d(x, [ f ≤ a]), which yields (a). If f (x) -a <σ ρ , then d(x, X \U )<ρ , which shows that d(x, X \U ) = d(x, [ f ≤ a]) indeed, yielding (b) (in particular, [ f ≤ a] =∅). ⊓ ⊔ Theorem 3.2 (Characterization of linear local error bound) Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, -∞ < a < b ≤+∞, x ∈[f ≤ a],

and σ>0. Consider the following statements:

(a) There exists r > 0 such that

|∇ f |(x) ≥ σ for every x ∈ B r ( x) ∩[a < f < b]; (6) (b) There exists ρ>0 such that f (x) -c ≥ σ d(x, [ f ≤ c]) for every c ∈[a, b[ and x ∈ B ρ [x]∩[c< f <b] . (7
x ∈ B ρ [x]∩ [a< f <b] and σ ∈]0,σ[ .Forn ∈ N such that c n := f (x)-1/n ≥ a,letx n ∈[f ≤ c n ] be such that f (x) -c n ≥σ d(x, x n ).
Then, 0 < d(x, x n ) → 0, so that x is not a local minimum point of f , and As far as we know, the first example of (linear) error bound appeared in Rosenbloom [START_REF] Rosenbloom | Quelques classes de problèmes extrémaux[END_REF]. Anticipating the well-known result by Hofmann [START_REF] Hoffman | On approximate solutions of systems of linear inequalities[END_REF], [START_REF] Rosenbloom | Quelques classes de problèmes extrémaux[END_REF]Lemme 3] provides an error bound for the distance to a convex polyhedral cone, and [START_REF] Rosenbloom | Quelques classes de problèmes extrémaux[END_REF]Lemme 4] provides an error bound for the distance to a convex polyhedron, in a special case. In the general nonconvex case, the first sufficient conditions for local error bounds date back to [START_REF] Ioffe | Regular points of Lipschitz functions[END_REF] (more recently, see also [START_REF] Wu | First-order and second-order conditions for error bounds[END_REF]). The above three theorems gather and synthesize the approach developed in our earlier papers [START_REF] Azé | Characterizations of error bounds for lower semicontinuous functions on metric spaces[END_REF][START_REF] Azé | Variational pairs and applications to stability in nonsmooth analysis[END_REF] where indeed, the local case was not presented as such (the emphasis was put on (local) metric regularity, see the coming remark about this notion). We thus provide here a unified and direct presentation of both cases.

f (x) -f (x n ) d(x, x n ) ≥ f (x) -c n d(x, x n ) ≥σ, so that |∇ f |(x) ≥σ , whence |∇ f |(x) ≥ σ .
Remark 3.2 Given f : X → R ∪{+∞}, consider the multifunction epi f := {(x, c) ∈ X × R : f (x) ≤ c} , that is, epi f (x) =[f (x), +∞[ if x ∈ dom f and epi f (x) =∅otherwise.
Under the assumptions (and using the notations) of Theorem 3.3,[6, Theorem 2] states:

The following properties are equivalent and imply that

[ f ≤ a] =∅: (i) |∇ f |(x) ≥ σ for every x ∈[a < f < b]; (ii) ( f (x) -c) + ≥ σ d(x, [ f ≤ c]) for all (x, c) ∈ U := [a < f < b]× ]a, b[; (iii) epi fi sσ -metrically regular on U .
Property (iii) means that for every ( x, c) ∈ U , there exists ε>0 such that

d(c, epi f (x)) ≥ σ d(x,(epi f ) -1 (c)) for all (x, c) ∈ U ∩ (B ε ( x)×]c -ε, c + ε[ ).
Clearly, for (x, c) ∈ X × R we have

(epi f ) -1 (c) =[f ≤ c] and d(c, epi f (x)) = ( f (x) -c) + ,
so that (iii) reads: For every ( x, c) ∈ U we have

(iii) ′ ( f (x) -c) + ≥ σ d(x, [ f ≤ c]) for all (x, c) ∈ U ∩ (B ε ( x)×]c -ε, c + ε[ ),
which is aprioriweaker than (ii). Moreover, while (i)

⇒ (ii) is (a) ⇒ (b) in Theorem 3.3 (yielding also [ f ≤ a] =∅ ), (iii) ′ ⇒ (i) is similar to (b) ⇒ (a) in Theorem 3.2, that
is, it depends only on the definition of the strong slope: There is no need to use Zorn's lemma as in [START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF]. In the sequel, we shall refer several times to [START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF], which contains many interesting observations on the matter we are dealing with. We already mention that, similarly as what is observed in [6, Remark 3], Theorem 3.3(b) is equivalent to

(iv)σd H ([ f ≤ c 2 ], [ f ≤ c 1 ]) ≤ c 2 -c 1 for all a ≤ c 1 < c 2 < b ,
where d H is the Hausdorff-Pompeiu metric:

d H ([ f ≤ c 2 ], [ f ≤ c 1 ]) := max{e H ([ f ≤ c 2 ], [ f ≤ c 1 ]), e H ([ f ≤ c 1 ], [ f ≤ c 2 ])} = e H ([ f ≤ c 2 ], [ f ≤ c 1 ]):= sup{d(x, [ f ≤ c 1 ]:x ∈[f ≤ c 2 ]}. Remark 3.3 In [15, Definition 2],
Chao and Cheng introduce the notion of subslope which is denoted and defined by

† |∇ f |(x) := sup y∈D(x) f (x) -f (y) + d(x, y) for x ∈ dom f , and † |∇ f |(x) := +∞ for x / ∈ dom f , where D(x) := {y ∈ X : d(y, [ f ≤ 0]) ≤ d(x, [ f ≤ 0])} .
In [START_REF] Chao | Linear and nonlinear error bounds for lower semicontinuous functions[END_REF]Theorem 2], the authors establish the following result (we use our notations):

Let (X, d) be a complete metric space, let f : X → R ∪{+∞}be proper and lower semicontinuous, and let σ>0. The following are equivalent:

(i) † |∇ f |(x) ≥ σ for every x ∈[f >0]. (ii) [ f ≤ 0] =∅, and f (x) ≥ σ d(x, [ f ≤ 0]) for every x ∈[f >0]. Indeed, (i) ⇒ (ii) is contained in [14, Theorem 3] (see also [2, Remark 2.2 (b)]), while (ii)
⇒ (i) readily follows from the definition of the subslope. Since it does not involve all values c ≥ 0, this statement is formally simpler than that of Theorem 3.3. However, since the subslope is a global notion (as is the global slope introduced in [START_REF] Ngai | Error bounds in metric spaces and applications to the perturbation stability of metric regularity[END_REF]), we prefer to stick with results based on the strong slope, sharing Ioffe's opinion at the end of [17, Section 5], that the interest of the sufficient condition in Theorem 3.3 lies in the fact that it "gives a global estimate based on purely infinitesimal information."

We conclude this section with a "bilateral" result, where for a ∈ R,

[ f = a]: = {x ∈ X : f (x) = a}.
Proposition 3.2 Let (X, d) be a complete metric space with connected balls, and f : X → R be continuous with

|∇ f |(x) =|∇(-f )|(x) for every x ∈ X . (a) Let x ∈ X with a := f ( x) ∈ R, and σ, ρ > 0. Assume that |∇ f |(x) ≥ σ for every x ∈ B 2ρ ( x)\[ f = a] . Then, | f (x) -a|≥σ d(x, [ f = a]) for every x ∈ B ρ ( x). (b) Let -∞ < a < b ≤+ ∞with [a < f < b] =∅and σ>0.
The following are equivalent:

(i) |∇ f |(x) ≥ σ for every x ∈[a < f < b]; (ii) [ f = c] =∅for every c ∈[a, b[ , and | f (x) -c|≥σ d(x, [ f = c]) for all (c, x) ∈[a, b[×[a< f <b] .
Proof Since X has connected balls and

f is continuous, for c ∈ R such that [ f ≤ c] = ∅ and for x ∈[f > c] we have d(x, [ f ≤ c]) = d(x, [ f = c]).
Thus, the conclusions are obtained applying Theorems 3.2 and 3.3 to both f andf . ⊓ ⊔ Remark 3.4 Observe that (ii) above is equivalent to

(v)σd H ([ f =c 2 ], [ f =c 1 ]) ≤ c 2 -c 1 for every a ≤ c 1 < c 2 < b
(recall the end of Remark 3.2), which was observed in [6, Proposition 6], from which we borrowed the assumption that X has connected balls.

Characterization of Nonlinear Error Bounds

In this section, we establish a general (though elementary) chain rule for the strong slope that is used to reduce our main results on nonlinear error bounds to the linear ones in the previous section.

Lemma 4.1 Let (X, d) be a metric space, f : X → R ∪{+∞}be lower semicon- tinuous, -∞ < a < b ≤+ ∞ , and ϕ :]a, b[→ R be of class C 1 with ϕ ′ > 0.
Then,

|∇(ϕ • f )|(x) = ϕ ′ ( f (x))|∇ f |(x) for every x ∈[a< f <b] . Proof Of course, ϕ • f :[ a< f <b]→R.L e tx ∈[ a< f <b]. We first show that |∇(ϕ • f )|(x) ≤ ϕ ′ ( f (x))|∇ f |(x).
We may assume that |∇ f |(x)<+∞ and that |∇(ϕ • f )|(x)>0. Thus, there exists a sequence (x n ) converging to x such that

|∇(ϕ • f )|(x) = lim n→∞ ϕ( f (x)) -ϕ( f (x n )) d(x, x n ) > 0 , so that f (x n )< f (x) (because ϕ is increasing) and f (x n ) → f (x) (because |∇ f |(x)<∞). Let z n ∈]f (x n ), f (x)[ be such that ϕ( f (x)) -ϕ( f (x n )) = ϕ ′ (z n )( f (x) -f (x n )) , then |∇(ϕ • f )|(x) = lim n→∞ ϕ ′ (z n ) f (x) -f (x n ) d(x, x n ) = ϕ ′ ( f (x)) lim n→∞ f (x) -f (x n ) d(x, x n ) ≤ ϕ ′ ( f (x))|∇ f |(x).
We then show that

|∇(ϕ • f )|(x) ≥ ϕ ′ ( f (x))|∇ f |(x). We may assume that |∇ f |(x)> 0 and that |∇(ϕ • f )|(x)<+∞.
Thus, there exists a sequence (x n ) converging to x such that

|∇ f |(x) = lim n→∞ f (x) -f (x n ) d(x, x n ) > 0 , so that (for n large enough) 0 < f (x n )< f (x) (because f is lower semicon- tinuous at x). From which ϕ( f (x n )) < ϕ( f (x)) (because ϕ is increasing) and ϕ( f (x n )) → ϕ( f (x)) (because |∇(ϕ • f )|(x)<∞), so that f (x n ) → f (x) (because ϕ is increasing, again). Let z n ∈]f (x n ), f (x)[ be such that ϕ( f (x)) -ϕ( f (x n )) = ϕ ′ (z n )( f (x) -f (x n )) , then |∇(ϕ • f )|(x) ≥ lim n→∞ ϕ ′ (z n ) f (x) -f (x n ) d(x, x n ) = ϕ ′ ( f (x))|∇ f |(x).
ϕ α (t) := t 0 ds α(s) < +∞ . Thus, if α ∈ b , then ϕ α :]0, b[→]0, +∞[ is of class C 1 , is increasing, extends continuously to [0, b[ with ϕ α (0)
= 0, and we may set:

ϕ α (b) := sup 0<t<b ϕ α (t).
Keeping this notation in mind, the following are the two main results of this paper. (a) There exists r > 0 such that

|∇ f |(x) ≥ α( f (x) -a) for every x ∈ B r ( x) ∩[a< f <b] . (8)
|∇(ϕ α • f )|(x) = ϕ ′ α ( f (x) -a)|∇ f |(x) = |∇ f |(x) α( f (x) -a) ≥ 1 for every x ∈ B r ( x) ∩[a< f <b]. Since ϕ α :[0, b -a[→ [0,ϕ α (b -a)
[ is one-to-one increasing, the result readily follows from Theorem 3.2 applied to ϕ α .

⊓ ⊔

In a similar way, we deduce from Theorem 3.3 the following characterization of a nonlinear global error bound.

(c) d H ([ f ≤ c 2 ], [ f ≤ c 1 ]) ≤ ϕ α (c 2 -a) -ϕ α (c 1 -a) for all a ≤ c 1 < c 2 < b .
The main part in the above two theorems is of course the sufficient condition (a) ⇒ (b), with c := a in (b). In the case of Theorem 4.2, it reads (i) ⇒ (ii) with:

(i) |∇ f |(x) ≥ α( f (x) -a) for every x ∈[a< f <b] ; (ii) [ f ≤ a] =∅, and f (x) -a ≥ ϕ -1 α (d(x, [ f ≤ a])) for every x ∈[a< f <b] .
This was already proved in [3, Theorem 6.1], through a different argument, essentially based on property (c) above (an argument previously used in [4, Theorem 4.2] in a particular case). Here, we provide a more satisfactory (truly variational) approach, by reducing the nonlinear case to the linear one through Lemma 4.1, which naturally yields a local result as well (see [3, Remark 6.1 (b)]). We note that in the quoted results of [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF][START_REF] Azé | Variational pairs and applications to stability in nonsmooth analysis[END_REF], the function α is not assumed continuous, but it is assumed nondecreasing, in which case the function ϕ α is concave. A typical such situation is ϕ α (t) := 1 σ t γ with σ>0 and 0 <γ <1 ,

which we shall emphasize in the following section. As pointed out in [3, Section 6], the existence of a (continuous) nondecreasing α :]0, ba[→]0, +∞[ such that (i) above holds is equivalent to the fact that f is so-called asymptotically well behaved in [a< f <b], that is: For every sequence

(x k ) ⊂[a< f <b], |∇ f |(x k ) → 0 ⇒ f (x k ) → a .
See

( f (x) -a)|∇ f |(x) ≥ 1 for every x ∈ B r ( x) ∩[a< f <b] (10)
(with ϕ = ϕ α ). According to the terminology used in [START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF] (also used in [START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF]), inequality (10) could be termed a generalized Kurdyka-Łojasiewicz inequality. Indeed, such inequality was considered by Kurdyka in [START_REF] Kurdyka | On gradients of functions definable in o-minimal structures[END_REF] in the case when f is a C 1 function defined on a Hilbert space X . The original inequality was established by Łojasiewicz in the 1960s, in the case when f is (sub)analytic on R n and ϕ is of the type ( 9), an inequality that was used to establish the convergence of the trajectories of dynamical systems, see, e.g., the short survey [START_REF] Łojasiewicz | On semi-analytic and subanalytic geometry[END_REF].

Error Bounds of Order γ>0 and Examples from the Literature

In this section, we specialize the sufficient parts of Theorems 4.1 and 4.2 to the case when α(s) := τ s θ with τ>0 and θ<1. This is the case that has been essentially considered in the literature, which is natural due to its simplicity and its applicability. We thus give several examples of recent results that can be recovered from our general approach.

Corollary 5.1 Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, -∞ < a < b ≤+∞, and σ, γ > 0.

(a) Let x ∈[f ≤ a], and ρ>0. Assume that

|∇ f |(x) ≥ σ γ ( f (x) -a) 1-γ for every x ∈ B 2ρ ( x) ∩[a < f < b] .
Then,

( f (x) -a) γ ≥ σ d(x, [ f ≤ a]) for every x ∈ B ρ [x]∩[a < f < b] .
(b) Assume that [a< f <b] =∅, and that

|∇ f |(x) ≥ σ γ ( f (x) -a) 1-γ for every x ∈[a < f < b] . Then, [ f ≤ a] =∅and ( f (x) -a) γ ≥ σ d(x, [ f ≤ a]) for every x ∈[a< f <b] .
Example 5.1 Let (X, • , • ) be a real Hilbert space, let A : X → X be a bounded self-adjoint operator, and define f :

X → R ∪{+∞}by f (x) := Ax, x if x =1 , f (x) := +∞ otherwise. It is immediate that for x ∈ dom f : ∂ F f (x) = 2 Ax + Rx
(of course, we identify X with its dual), from which it is readily seen that

d(0,∂ F f (x)) = 2 Ax -Ax, x x =2 Ax -f (x)x .
Let λ 0 = inf X f be the lower bound of the spectrum of A. When λ 0 is an eigenvalue of A,argminf is the intersection of the associated eigenspace X 0 with the unit sphere.

Assume that λ 0 is isolated in the spectrum of A,letX 1 := X ⊥ 0 , and let λ 1 be the lower bound of the spectrum of the restriction of A to X 1 , so that λ 0 <λ 1 . We have (see, e.g., [20, (10.11)]):

Ax -λ 0 x 2 ≥ (λ 1 -λ 0 ) Ax -λ 0 x, x =(λ 1 -λ 0 )( f (x) -λ 0 ) for x ∈ dom f , so that for x ∈[f <λ 1 ] we obtain d(0,∂ F f (x)) 2 = 4 Ax -f (x)x 2 = 4 Ax -λ 0 x -( f (x)x -λ 0 x) 2 = 4( Ax -λ 0 x 2 -( f (x) -λ 0 ) 2 ) ≥ 4(λ 1 -f (x))( f (x) -λ 0 )
(see [START_REF] Lyubich | The general theory of relaxation processes for convex functionals[END_REF]Lemma 10.2]). Thus, given λ ∈]λ 0 ,λ 1 [ ,wehave

d(0,∂ F f (x)) 2 ≥ 4(λ 1 -λ)( f (x) -λ 0 ) for every x ∈[f ≤ λ] ,
or, according to (3):

|∇ f |(x) ≥ 2(λ 1 -λ) 1 2 ( f (x) -λ 0 ) 1 2 > 0 for every x ∈[λ 0 < f ≤ λ] .
In all the results quoted from the literature, in the examples below, we freely use our notations in order to provide a more straightforward comparison with our results. We note that all these results were obtained using the variational principle.

Example 5.2 In [21, Corollary 2 i), ii)],

Ngai and Théra establish the following, which is their main result on nonlinear error bounds:

Let X be an Asplund space, and let f : X → R∪{+∞} be lower semicontinuous.

Then: i) If there are σ, γ > 0 such that γ x * f (x) γ -1 ≥ σ for all x / ∈[f ≤ 0], x * ∈ ∂ F f (x), then d(x, [ f ≤ 0]) ≤ 1 σ f (x) γ + for all x ∈ X .
ii) Let x be in the boundary of

[ f ≤ 0]. Let us suppose that there exist realσ, γ, ρ > 0 such that γ x * f (x) γ -1 ≥σ , for all x ∈ B 2ρ ( x)\[ f ≤ 0], x * ∈ ∂ F f (x), then d(x, [ f ≤ 0]) ≤ 1 σ f (x) γ + for all x ∈ B ρ ( x).
Indeed, as recalled earlier, (3) holds in the above setting, so that for

x / ∈[f ≤ 0] (resp., x ∈ B 2ρ ( x)\[ f ≤ 0])wehave: |∇ f |(x) ≥ lim inf (y, f (y))→(x, f (x)) d * (0,∂ F f (x)) ≥ σ γ f (x) 1-γ ,
yielding the assumptions of Corollary 5.1. Note that the assumption that x belong to the boundary of [ f ≤ 0], though not essential in an abstract result, is the interesting case in practice since it yields that B r ( x) ∩[f >0] =∅for any r > 0.

d(-∇ f (x), N (x; S)) ≥ δ( f (x) -μ) 1-γ for every x ∈[f >a] . ( 11
)
The following statements hold: (a) For every sequence (x k ) ⊂ S,

lim k→∞ d(-∇ f (x k ), N (x k ; S)) = 0 ⇒ lim k→∞ f (x k ) = a . (b) [ f ≤ a] =∅. (c) ( f (x) -a) γ ≥ γ δ d(x, [ f ≤ a]) for every x ∈ S .
Indeed, let g := f + ι S , where ι(x) := 0ifx ∈ S and ι(x) := +∞ if x / ∈ S.From elementary calculus, we have

∂ F g(x) =∇f (x) + N (x; S) for every x ∈ S ,
where N (x; S) := {z ∈ R n : z, yx ≤0 ∀ y ∈ S} is the normal cone to S at x. Thus, d(0,∂ F g(x)) = d(-∇ f (x), N (x; S)), so that |∇g|(x) ≥ δ( f (x)a) 1-γ for every x ∈ S , according to (3) and [START_REF] Rosenbloom | Quelques classes de problèmes extrémaux[END_REF], and (b), (c) follow from Corollary 5.1(b).

Observe that property (a) is immediate from [START_REF] Rosenbloom | Quelques classes de problèmes extrémaux[END_REF]. Facchinei and Pang mention it in order to relate their result with the notion of asymptotical good behavior (of f in [ f >0]) introduced by Auslender and Crouzeix in [START_REF] Auslender | Well behaved asymptotical convex functions[END_REF] for the study of noncoercive convex minimization problems on R n . Such property might be called a concrete asymptotical good behavior property, to be checked in applications (using (sub)differential notions fitting the problem). Note that, due to (3), such concrete property implies the abstract one (using the strong slope) mentioned at the end of Remark 4.1, so that the abstract theory may indeed be applied to the problem under study.

Example 5.4 In [17, Theorem 7], Ioffe establishes the following result:

Let X be a complete metric space, let f : X → R ∪{+∞}be lower semicontinuous, let U be an open subset of X , let β be a 1-Lipschitz function on X which is positive on U , and set

U β := x∈U B β(x) (x).
Assume that for some k ≥ 1 and σ>0:

|∇ (k) f |(x)>σ for every x ∈[f >0]∩U β . Then, [ f ≤ 0] =∅and f (x) 1 k ≥ σ d(x, [ f ≤ 0]) for everyx ∈ U with 0 < f (x)<(σβ(x)) k .
Here,

|∇ (k) f |: =| ∇ [f ] 1 k |, where [ f ] 1 k (x) := | f (x)| 1 k sign( f (x))
. Indeed, in order to obtain that [ f ≤ 0] =∅ ,i tm u s tfi r s tb ea s s u m e dt h a tt h e r eis some x ∈ U with 0 < f (x)<( σβ(x)) k . Then, set ρ := β(x). According to the definition of |∇ (k) f |, we have

|∇[ f ] 1 k |(y) ≥ σ for every y ∈ B ρ (x) ∩[0<[ f ] 1 k <σρ] ,
and the conclusion follows from Theorem 3.1(b), applied to [ f ] 1 k with C := {x}.Note that the inequality need not be strict in the main assumption, while we may consider any constant k > 0, i.e., we need not restrict ourselves to the "concave case" k ≥ 1 (an assumption used in Ioffe's proof for the computation of |∇ (k) f |).

We conclude this section with a nonlinear version of Proposition 3.2(a). Proposition 5.1 Let (X, d) be a complete metric space with connected balls, and f : X → R be continuous with

|∇ f |(x) =|∇(-f )|(x) for every x ∈ X . Let x ∈ X with a := f ( x) ∈ R, and σ, ρ > 0. Assume that |∇ f |(x) ≥ σ γ | f (x) -a| 1-γ for every x ∈ B 2ρ ( x)\[ f = a] .
Then,

| f (x) -a| γ ≥ σ d(x, [ f = a]) for every x ∈ B ρ ( x).
Remark 5.1 Let f be analytic in a neighborhood of 0 ∈ R n , with f (0) = 0. In [START_REF] Łojasiewicz | On semi-analytic and subanalytic geometry[END_REF], Łojasiewicz mentions that the following assertions are equivalent:

(i) There exist θ ∈]0, 1[ and r > 0 such that

∇ f (x) ≥|f (x)| θ for every x ∈ B r (0) ; (ii) There exist σ, γ, ρ > 0 such that | f (x)|≥σ d(x, [ f =0]) γ for every x ∈ B ρ [0]
Of course, implication (ii) ⇒ (i) relies on the fact that f is analytic, and is definitely not true in general, in the continuous case.

Remarks on the Convex Case

In this section, (X, • ) is a Banach space with topological dual (X * , • * ), f : X → R ∪{+∞}is convex and lower semicontinuous, and for x ∈ dom f ,

∂ f (x) := {x * ∈ X * : f (z) -f (x) ≥ x * , z -x }
denotes the Fenchel subdifferential of f at x. Proposition 6.1 Fo r x ∈ dom f , we have

|∇ f |(x) = d * (0,∂f (x)) ,
and if x is not a minimum point of f we have

|∇ f |(x) = sup f (z)< f (x) f (x) -f (z) x -z . Thus, if a ∈ R is such that [ f ≤ a] =∅, we have: |∇ f |(x) ≥ f (x) -a d(x, [ f ≤ a]) for every x ∈[f > a] .
Proof See, e.g., [2, Proposition 3.1] and [3, Proposition 5

.2]. ⊓ ⊔ Let -∞ < a < b ≤+ ∞with [ f ≤ a]
=∅ , and assume that there exists an increasing, continuous

ψ :]0, b -a[→]0, +∞[ such that ψ( f (x) -a) ≥ d(x, [ f ≤ a]) for every x ∈[a < f < b] , so that |∇ f |(x) ≥ f (x) -a ψ( f (x) -a) for every x ∈[a < f < b] ,
according to Proposition 6.1. Thus, if

ϕ(t) := t 0 ψ(s) s ds < +∞ for t ∈]0, b -a[ ,
then f satisfies the inequality:

ϕ ′ ( f (x) -a)|∇ f |(x) ≥ 1 for every x ∈[a< f <b] .
This was observed in [6, Theorem 30], in the case when X is a Hilbert space (the Hilbertian structure being used in the proof), and a := min f . Theorem 6.1 Let X be a Banach space, f : X → R ∪{+∞}be convex and lower semicontinuous, -∞ < a < b ≤+ ∞ , and 0 <γ <1. Consider the following statements:

(a) There exists r > 0 such that

|∇ f |(x) ≥ r ( f (x) -a) 1-γ for every x ∈[a < f < b]; (b) [ f ≤ a] =∅
, and there exists ρ>0 such that Note that in the previous result, and with respect to Corollary 5.1, we restricted γ to be less than 1. The reason is that in the convex case, "the slope is nondecreasing with altitude," so that an inequality of the type

( f (x) -a) γ ≥ ρ d(x, [ f ≤ a]) for every x ∈[a < f < b] .
|∇ f |(x) ≥ α( f (x) -a) for every x ∈[a < f < b]
should involve a nondecreasing function α. Let us be more specific about this. Proposition 6.2 Let X be a Banach space, f : X → R ∪{+∞}be convex and lower semicontinuous, and a ∈ R with

[ f ≤ a] =∅. Then, inf [ f >a] |∇ f |≥ inf [ f = a] |∇ f | . Proof See [2, Proposition 3.2]. (Note that [ f = a] =∅.)
⊓ ⊔ Proposition 6.3 Let X be a Banach space, f : X → R ∪{+∞}be convex and lower semicontinuous, and -∞ < a < b ≤+ ∞ . Assume that there exists a continuous

α :]0, b -a[→]0, +∞[ with t 0 ds α(s) < +∞ for t ∈]0, b -a[ , such that |∇ f |(x) ≥ α( f (x) -a) for every x ∈[a< f <b] .
Then, there exists a nondecreasing, continuous

α :]0, +∞[→]0, +∞[ with t 0 ds α(s) < +∞ for t > 0 , such that |∇ f |(x) ≥α( f (x) -a) for every x ∈[f >a] .
Proof Of course, we assume [a< f <b] =∅.Fors > 0, set

α(s) := inf{|∇ f |(x) : f (x) -a = s} .
Then, α ≡+ ∞(Proposition 3.1 readily implies that dom|∇ f | is dense in dom f ), and α is nondecreasing, according to Proposition 6.2. By definition of α,wealsohave is continuous and increasing.

|∇ f |(x) ≥α( f (x) -a) ≥ α( f (x) -a)>0
Proposition 7.1 Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, and a b, α, ϕ, β be as above.

(a) Assume that for some x ∈[f ≤ a] and ρ>0 we have

|∇ f |(x) ≥ α( f (x) -a) for every x ∈ B 2ρ ( x) ∩[a < f < b] .
Then,

|∇ f |(x) ≥ β(d(x, [ f ≤ a])) for every x ∈ B ρ [x]∩[a < f < b] . ( 12
) (b) Assume that |∇ f |(x) ≥ α( f (x) -a) for every x ∈[a< f <b] . Then, [ f ≤ a] =∅and |∇ f |(x) ≥ β(d(x, [ f ≤ a])) for every x ∈[a < f < b] . ( 13
)
Proof From Theorem 4.1,wehave

f (x) -a ≥ ϕ -1 (d(x, [ f ≤ a])) for every x ∈ B ρ [x]∩[a < f < b] . Thus, since ϕ ′ is nonincreasing, for every x ∈ B ρ [x]∩[a< f <b] we have |∇ f |(x) β(d(x, [ f ≤ a])) = ϕ ′ (ϕ -1 (d(x, [ f ≤ a])))|∇ f |(x) ≥ ϕ ′ ( f (x) -a)|∇ f |(x) = |∇ f |(x) α( f (x) -a) ≥ 1 .
Similarly, (b) follows from Theorem 4.2. ⊓ ⊔ Remark 7.1 Properties ( 12) and ( 13) are the main assumptions in [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF][START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF], where nonlinear error bound results are derived from the linear case through a "change of metric." Though quite natural, that technique is more involved than the one used in this paper, so that the implications above are only fitting. Proposition 7.1(b) was not observed in [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF], nor was it observed there that the global result [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF]Theorem 4.3] indeed holds without assuming that the function β be nondecreasing, a result we now establish, after which we provide some comparison between the two types of results. Proof Assume first that β(0) = 0. We may assume that [a< f <b] =∅ , so that for c ∈]a, b[ sufficiently close to b and for δ>0 small enough, we have

[ f ≤ c]\C δ =∅, where
C δ := {x ∈ X : d(x, [ f ≤ a]) ≤ δ} .
For such δ, define

β δ :[0, ∞[→]0, ∞[ by β δ (s) := β(δ) if s ≤ δ β(s) if s ≥ δ ,
:= [ f ≤ c] \C δ : d(x, [ f ≤ a]) = d(x,[ f ≤ a]) 0 β δ (s) ds ≥ d(x,[ f ≤ a]) δ β(s) ds , and
| ∇ f |(x) =| ∇ f |(x) = |∇ f |(x) β δ (d(x, [ f ≤ a])) = |∇ f |(x) β(d(x, [ f ≤ a])) ≥ 1
f (x) -a ≥ f (x) -inf U f ≥ d(x, X \U ) ≥ d(x, [ f ≤ a]) ≥ d(x,[ f ≤ a]) δ β(s) ds ,
and the conclusion follows by letting δ → 0 first, then c → b.

In the case when β(0)>0, the proof is just simpler, since we may directly use the metric 12) is naturally linked with the notion of metric (sub)regularity, see, e.g., [START_REF] Azé | Nonlinear local error bounds via a change of metric[END_REF], see also [START_REF] Ioffe | Metric regularity. Theory and applications-a survey[END_REF] for a quite recent survey on this notion.

d := d([ f ≤ a],β). ⊓ ⊔ Remark
We conclude the paper by illustrating how the results of Sect. 4 can be applied to the convergence of descent methods. This example is borrowed from the papers [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF][START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF], as we explain in more detail in Remark 7.3 below.

Example 7.1 Let (X, • ) be a Banach space, let U be an open subset of X , and let f : U → R be differentiable, with a uniformly continuous differential Df : U → X * . Set m(s) := sup{ Df(x) -Df(z) * : xz ≤s} , so that m :]0, ∞[→]0, ∞[ is nondecreasing, finite for small s, with m(s) → 0as s → 0. It follows from the mean value inequality that

| f (x) -f (z) -Df(z)(x -z)|≤M( x -z ) whenever [x, z]⊂U , (15)
where M(t) := t 0 m(s) ds ≤ tm(t). Let us consider the following algorithm.

Parameters: τ, θ ∈]0, 1[ .

Step 1. Initialization. Choose x 0 ∈ U and δ>0 such that m(δ) is finite. We assume that

{x ∈ X : d(x, [ f ≤ f (x 0 )]) ≤ δ}⊂U .
Step 2. Stopping test. Given iterate x k ,stopifDf(x k ) = 0. Otherwise, let d k =1 be such that

Df(x k )(d k ) ≤-τ Df(x k ) * . (16)
Step 3.

Linesearch.If f (x k + δd k ) ≤ f (x k ) + θδ Df(x k )(d k ),sett k := δ, x k+1 := x k + t k d k and go to Step 2.
Otherwise, by a classical backtracking argument, find

t k ∈]0,δ/2] such that f (x k + t k d k ) ≤ f (x k ) + θt k Df(x k )(d k ), (17)
f (x k + 2t k d k )> f (x k ) + 2θt k Df(x k )(d k ), (18)
set x k+1 := x k + t k d k and go to Step 2.

From ( 16)to [START_REF] Ioffe | Nonlinear regularity models[END_REF], since t k = x k+1x k we have

f (x k+1 ) ≤ f (x k ) -τθ x k+1 -x k Df(x k ) * , (19)
while from ( 15)to(18), if t k+1 <δwe have 0

< f (x k + 2t k d k ) -f (x k + t k d k ) -θt k Df(x k )(d k ) ≤ M(t k ) + t k (Df(x k + t k d k ) -Df(x k ))(d k ) + (1 -θ)t k Df(x k )(d k ) ≤ 2t k m(t k ) -t k τ(1 -θ) Df(x k ) * , that is, 2m(t k ) ≥ τ(1 -θ) Df(x k ) * . (20)
(a) Assume that the sequence of iterates is infinite [so that the sequence ( f (x k )) is decreasing according to [START_REF] Łojasiewicz | On semi-analytic and subanalytic geometry[END_REF], and has a subsequence converging to some x ∈ [ f < f (x 0 )] (which is the case, e.g., if X = R n and [ f < f (x 0 )] is bounded), and set a := f ( x). Assume further that for some r > 0 and b > a we have: where β := (ϕ -1 α ) ′ . Of course, f (x k ) = a if Df(x k ) = 0. On the other hand, if the sequence of iterates (x k ) is infinite, for any K ∈ N we have (similarly as before):

Df(x) * ≥ α( f (x) -a) for every x ∈ B r ( x) ∩[a< f <b] , (21)
τθ K k=1
x k+1x k ≤ϕ α ( f (x 1 )a)<+∞ due to [START_REF] Łojasiewicz | On semi-analytic and subanalytic geometry[END_REF] and [START_REF] Auslender | Well behaved asymptotical convex functions[END_REF], which shows that (x k ) converges to some x ∈[f <b], and that t k = x k+1x k →0, so that Df( x) = 0 due to [START_REF] Lyubich | The general theory of relaxation processes for convex functionals[END_REF], which shows that f ( x) = a. Remark 7. [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF] The above example synthesizes some results developed in [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]Sects. 3,4.1] and in [START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF]Section. 3] (where X = R n ). In particular, the very nice argument leading to [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems[END_REF] is given in the proof of [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]Theorem 3.2], which applies to Theorem 3.4 therein, which is itself expressed with a condition of the type of [START_REF] Ngai | Error bounds for systems of lower semicontinuous functions in Asplund spaces[END_REF]. Still, in [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]as well as in [START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF], much emphasis is put on the Kurdyka-Łojasiewicz property, which, as already said, is but another formulation of (21) (in the local case), or of ( 23 We note that this condition was used for the study of convergence of descent methods, back in [27, p. 49], with ϕ(t) := σ √ t for some σ>0. We finally observe that in [START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF], it is assumed that the function f is of class C 1,1 . Assuming only that Df is uniformly continuous seems as good and natural. Moreover, with respect to [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF][START_REF] Noll | Convergence of linesearch and trust-region methods using the Kurdyka-Łojasiewicz inequality[END_REF], we added the error bound estimates provided by our abstract results.

Conclusions

With respect to earlier work of ours, we refined the abstract approach to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, in the case when the main assumptions are lower estimates of the strong slope with respect to the values of the function. Such estimates are linked to the notion of asymptotically well-behaved functions, and to the Kurdyka-Łojasiewicz inequality, so that our approach can be used in various settings, as we illustrated through many examples from the literature. We hope this can help develop systematic approaches that may simplify and clarify various theoretical and technical aspects of nonsmooth analysis and optimization.

  ) Then, (a) ⇒ (b) with ρ := r/2, and (b) ⇒ (a) with r := ρ. Proof (a) ⇒ (b): Apply Theorem 3.1(a) with C := B r/2 [x], ρ := r/2, and arbitrary c ∈[a, b[ in place of a. (b) ⇒ (a): This follows from the definition of the strong slope, as showed in [2, Proposition 2.1]. We give the details for the reader's convenience. Let

⊓ ⊔ Theorem 3 . 3 (

 33 Characterization of linear global error bound) Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, -∞ < a < b ≤∞ with [a< f <b] =∅, and σ>0. The following are equivalent:(a) |∇ f |(x) ≥ σ for every x ∈[a < f < b]; (b) [ f ≤ a] =∅,and f (x)c ≥ σ d(x, [ f ≤ c]) for every c ∈[a, b[ and every x ∈[c < f < b] . Proof (a) ⇒ (b): Apply Theorem 3.1(b) with C := X , arbitrary c ∈[ a, b[ in place of a, and arbitrary ρ ∈]0,(bc)/σ [ such that [ f <c+σρ] =∅. (b) ⇒ (a) is obtained like in the previous theorem. ⊓ ⊔ Remark 3.1

⊓ ⊔ Notation 4 . 1

 41 Given b ∈[0, +∞], we denote by b the set of continuous α :]0, b[→ ]0, +∞[ such that for some (hence for all) t ∈]0, b[ we have:

Theorem 4 . 1 (

 41 Characterization of nonlinear local error bound) Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, -∞ < a < b ≤ +∞, x ∈[f ≤ a], and α ∈ b-a . Consider the following statements:

( b )

 b There exists ρ>0 such thatϕ α ( f (x)a) ≥ ϕ α (ca) + d(x, [ f ≤ c])for every c ∈[a, b[ and every x ∈ B ρ [x]∩[c< f <b]. Then, (a) ⇒ (b) with ρ := r/2, and (b) ⇒ (a) with r := ρ. Proof According to Lemma 4.1,(8) reads

Theorem 4 . 2 (

 42 Characterization of nonlinear global error bound) Let (X, d) be a complete metric space, f : X → R∪{+∞} be lower semicontinuous, -∞ < a < b ≤∞ with [a< f <b] =∅, and α ∈ b-a . The following are equivalent: (a) |∇ f |(x) ≥ α( f (x)a) for every x ∈[a< f <b] . (b) [ f ≤ a] =∅, and ϕ α ( f (x)a) ≥ ϕ α (ca) + d(x, [ f ≤ c]) for all c ∈[a, b[ and all x ∈[c< f <b] . Remark 4.1 Recalling the last part of Remark 3.2, note that the assertions of Theorem 4.2 are equivalent to:

Example 5 . 3

 53 In [22, Corollary 6.5.4], Facchinei and Pang establish the following result: Let S ⊂ D ⊂ R n with S closed convex and Dopen, let f : D → R be of class C 1 with a := inf S f ∈ R, and let γ ∈]0, 1] and δ>0. Assume that

  Then, (a) ⇒ (b) with ρ := γ r , and (b) ⇒ (a) with r := ρ. Proof (a) ⇒ (b) is given by Corollary 5.1,(b)⇒ (a) follows from the above considerations with ψ(s) := s γ /ρ.⊓ ⊔

7 A

 7 Further Comparison and an Application to Descent Methods Let -∞ < a < b ≤+∞, and let α ∈ b-a (recall Notation 4.1)benondecreasing, so that ϕ :]0, ba[→]0, +∞[ defined by ϕ(t) := t 0 ds α(s) is of class C 1 , increasing, and concave, so that β :]0,ϕ(ba)[→]0, +∞[ defined by β(s) := (ϕ -1 ) ′ (s)

Theorem 7 . 1 14 )

 7114 Let (X, d) be a complete metric space, f : X → R ∪{+∞}be lower semicontinuous, -∞ < a < b ≤∞ , and β :[ 0, +∞[ → [0, +∞[ be continuous, with β(s)>0 for s > 0, and such that +∞ 0 β(s) ds =+∞. (Assume that [ f ≤ a] =∅, and that |∇ f |(x) ≥ β(d(x, [ f ≤ a])) for every x ∈[a< f <b] . Then, f (x)a ≥ d(x,[ f ≤ a]) 0 β(s) ds for every x ∈[a< f <b] .

  and consider the metric d = d([ f ≤ a],β δ ) given by the so-called change-of-metric principle: See [24, Theorem 4.1], or [25, Theorem 2.2]. Then, d is topologically equivalent to d, (X, d) is complete (due to (14)), and for every x ∈ U

,

  where | ∇•| denotes the strong slope with respect to the metric d, and f denotes the restriction of f to [ f ≤ c]. Thus, applying Proposition 3.1 in ([ f ≤ c], d) to the function f , we obtain that for every x ∈ U :

2 - 1 k=k 1 x

 211 for some nondecreasing α ∈ b-a (see Notation 4.1). If x k ∈ B r ( x) ∩[a< f <b], thenτθ x k+1x k ≤ f (x k )f (x k+1 ) α( f (x k )a) ≤ ϕ α ( f (x k )a)ϕ α ( f (x k+1 )a)due to[START_REF] Łojasiewicz | On semi-analytic and subanalytic geometry[END_REF],[START_REF] Ngai | Error bounds for systems of lower semicontinuous functions in Asplund spaces[END_REF], and the fact that α is nondecreasing. Consequently, given 0 <ρ≤ min{r, b/τ θ },i fk 1 < k 2 are such that ϕ α ( f(x k 1 )a)<τθ ρand x k ∈ B ρ ( x) for k = k 1 ,...,k 2 -1, then τθ k k+1x k ≤ϕ α ( f (x k 1 )a)<τθρ (note that ϕ α ( f (x k )a)<ϕ α ( f (x k 1 )a) for k > k 1 ), which shows that x k 2 ∈ B 2ρ ( x), whence x = lim k→∞ x k .(22)Moreover, it follows from[START_REF] Lyubich | The general theory of relaxation processes for convex functionals[END_REF] thatDf( x) = lim k→∞ Df(x k ) = 0 .Also, Theorem 4.1 and Proposition 7.1 yieldd(x k , [ f = a]) ≤ ϕ α ( f (x k )a), β(d(x k , [ f = a])) ≤ Df(x k ) * ≤ 2m(t k+1 ) τ(1θ)for all k large enough, where β := (ϕ -1 α ) ′ (recall from the proof of Proposition 3.2 that d(x k , [ f ≤ a]) = d(x k , [ f = a])).(b) Assume now that U = X , that f is bounded below, and that, letting a := inf f < b := f (x 0 ), there exists a nondecreasing α ∈ b-a such thatDf(x) * ≥ α( f (x)a) for every x ∈[a < f < b] .(23)Theorem 4.2 and Proposition 7.1 yield that argmin f =[f = a]=[f ≤ a] is nonempty and ϕ α ( f (x)a) ≥ d(x, [ f = a]) for every x ∈[a < f < b] , Df(x) * ≥ β(d(x, [ f = a])) for every x ∈[a< f < b] ,

  ) (in the global case):ϕ ′ ( f (x)a) Df(x) * ≥ 1 for every x ∈[a< f <b] .

  [START_REF] Corvellec | Nonlinear error bounds for lower semicontinuous functions on metric spaces[END_REF] Remark 6.1], and Example 5.3 below, for bibliographical comments on this notion.

	Remark 4.2 A main step in the above proof of Theorem 4.1 is to write assumption (8)
	as
	ϕ ′

  for every x ∈[a< f <b] , This result was obtained in[START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF] Theorem 29] (where it is formulated in terms of the generalized Kurdyka-Łojasiewicz inequality), with X a Hilbert space (and a := inf f ), through more involved arguments. Note that the "standard inf-convolution arguments" mentioned in the above proof can be found, e.g., in[START_REF] Bolte | Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF] Lemma 45].

	Remark 6.1
	whence
	0	t	ds α(s)	≤	0	t	ds α(s)	< +∞ for t ∈]0, b -a[ .
	Standard inf-convolution arguments then produce a continuous, nondecreasing α : ]0, +∞[→]0, +∞[ with α ≤α and t 0 ds α(s) < +∞ for t ∈]0, b -a[ ,
	and the result follows.					⊓ ⊔


Philipp Wagner

Christopher P Ewels

Jean-Joseph Adjizian

Laurence Magaud

Pascal Pochet

Stephan Roche

Alejandro Lopez-Bezanilla

Viktoria V Ivanovskaya

Abu Yaya

Mark Rayson

Patrick Briddon

Bernard Humbert

Band Gap Engineering via Edge-Functionalization of Graphene Nanoribbons

INTRODUCTION

While graphene exhibits great promise for use in electronic devices, [START_REF] Geim | The rise of graphene[END_REF][START_REF] Avouris | Carbon-based electronics[END_REF][START_REF] Novoselov | A roadmap for graphene[END_REF] one of the major limitations of an ideal infinite graphene sheet is its lack of an electronic band gap. [START_REF] Neto | New directions in science and technology: two-dimensional crystals[END_REF] One solution to this problem is to work with thin strips of graphene, so-called graphene nanoribbons (GNRs), [START_REF] Nakada | Edge state in graphene ribbons: Nanometer size effect and edge shape dependence[END_REF] which demonstrate finite band gaps. [START_REF] Son | Energy Gaps in Graphene Nanoribbons[END_REF][START_REF] Barone | Electronic Structure and Stability of Semiconducting Graphene Nanoribbons[END_REF][START_REF] Yang | Quasiparticle Energies and Band Gaps in Graphene Nanoribbons[END_REF] The characteristics of thin, straight GNRs are dominated by the precise nature of their edges. [START_REF] Nakada | Edge state in graphene ribbons: Nanometer size effect and edge shape dependence[END_REF][START_REF] Son | Half-metallic graphene nanoribbons[END_REF][START_REF] Ritter | The influence of edge structure on the electronic properties of graphene quantum dots and nanoribbons[END_REF][START_REF] Terrones | Graphene and graphite nanoribbons: Morphology, properties, synthesis, defects and applications[END_REF][START_REF] Acik | Nature of Graphene Edges: A Review[END_REF][START_REF] Wakabayashi | Nanoscale and edge effect on electronic properties of graphene[END_REF] Edge engineering has been proposed as a route to tailor various properties of graphene [START_REF] Hod | Enhanced Half-Metallicity in Edge-Oxidized Zigzag Graphene Nanoribbons[END_REF][START_REF] Cervantes-Sodi | Edgefunctionalized and substitutionally doped graphene nanoribbons: Electronic and spin properties[END_REF][START_REF] Lee | Electronic structures of zigzag graphene nanoribbons with edge hydrogenation and oxidation[END_REF][START_REF] Lu | Effects of edge passivation by hydrogen on electronic structure of armchair graphene nanoribbon and band gap engineering[END_REF][START_REF] Wagner | Ripple edge engineering of graphene nanoribbons[END_REF] and hexagonal boron nitride nanoribbons. [START_REF] Wu | Chemically decorated boronnitride nanoribbons[END_REF][START_REF] Lopez-Bezanilla | Nitride Nanoribbons Become Metallic[END_REF][START_REF] Lopez-Bezanilla | Structure and Electronic Properties of Edge-Functionalized Armchair Boron Nitride Nanoribbons[END_REF] For (unreconstructed) graphene edges, armchair or zigzag edges are both possible, [START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF][START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Zobelli | A comparative study of density functional and density functional tight binding calculations of defects in graphene[END_REF] giving armchair GNRs (AGNRs) and zigzag GNRs (ZGNRs) with well-defined widths. [START_REF] Nakada | Edge state in graphene ribbons: Nanometer size effect and edge shape dependence[END_REF][START_REF] Son | Energy Gaps in Graphene Nanoribbons[END_REF][START_REF] Wakabayashi | Electronic and magnetic properties of nanographite ribbons[END_REF] A mix of these different edge types results in chiral edges [START_REF] Nakada | Edge state in graphene ribbons: Nanometer size effect and edge shape dependence[END_REF][START_REF] Branicio | Properties on the edge: graphene edge energies, edge stresses, edge warping, and the Wulff shape of graphene flakes[END_REF] and chiral graphene nanoribbons (CGNRs). All of these graphene edge types have been identified in atomic resolution transmission electron microscopy (HRTEM). [START_REF] Liu | Open and Closed Edges of Graphene Layers[END_REF][START_REF] Xie | Graphene Nanoribbons from Unzipped Carbon Nanotubes: Atomic Structures, Raman Spectroscopy, and Electrical Properties[END_REF][START_REF] Suenaga | Atom-by-atom spectroscopy at graphene edge[END_REF] An alternative graphene edge configuration was also proposed and exper-imentally confirmed, the Klein edge. [START_REF] Suenaga | Atom-by-atom spectroscopy at graphene edge[END_REF][START_REF] Klein | Graphitic polymer strips with edge states[END_REF] Further, it has been recently shown theoretically that pairwise reconstruction of Klein edges adds stability, [START_REF] Ivanovskaya | Low-Energy Termination of Graphene Edges via the Formation of Narrow Nanotubes[END_REF] especially when hydrogenated. [START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] Additionally, such graphene edges lead to reconstructed Klein GNRs (RKGNRs). [START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] In 2010, atomically precise AGNRs (with width 7, defined as the number of rows of carbon atoms parallel to the ribbon axis, referred to as 7-AGNR hereafter) were successfully assembled via a bottom-up process by Cai and co-workers. [START_REF] Cai | Atomically precise bottom-up fabrication of graphene nanoribbons[END_REF] The ability to grow a single specific ribbon width is an important prerequisite for designing and hence tuning GNR electronic properties for future carbon-based electronic devices. [START_REF] Lee | Electronic structures of zigzag graphene nanoribbons with edge hydrogenation and oxidation[END_REF][START_REF] Wagner | Ripple edge engineering of graphene nanoribbons[END_REF][START_REF] Dubois | Quantum Transport in Graphene Nanoribbons: Effects of Edge Reconstruction and Chemical Reactivity[END_REF][START_REF] Cocchi | Designing All-Graphene Nanojunctions by Covalent Functionalization[END_REF][START_REF] Selli | Redoxswitchable devices based on functionalized graphene nanoribbons[END_REF] First studies exploring hydroxyl (-OH) functionalized edges (among others) of thin AGNRs have shown the significant influence of chemical addends to structural, electronic, mechanical, and ABSTRACT: Density functional calculations are used to perform a systematic study of the effect of edge-functionaliza-tion on the structure and electronic properties of graphene nanoribbons (GNRs). -H, -F, -Cl, -Br, -S, -SH, and-OH edgefunctionalization of armchair, zigzag, and recon-structed Klein-type GNRs was considered. The most energeti-cally favorable edge structure varies depending on the choice of functional group. It is shown, for the first time, that reconstructed Klein-type GNRs are important stable configurations for several edge-functional groups. Band gaps using three different exchange-correlation functionals are calculated. The band gap for armchair GNRs can be tuned over a range of 1.2 eV by varying the edge-functional groups. In contrast, the band gaps of zigzag and reconstructed Klein edge GNRs are largely insensitive to the choice of edge-functional group, and ribbon width is instead the defining factor. Alternatively, the armchair GNR band gap can be controlled by varying the number of functional groups per opposing edge, altering the GNR "effective" width. Edge-functionalization design is an appropriate mechanism to tune the band gap of armchair GNRs. vibrational properties. [START_REF] Wagner | Ripple edge engineering of graphene nanoribbons[END_REF][START_REF] Rosenkranz | Ab initio calculations of edge-functionalized armchair graphene nanoribbons: Structural, electronic, and vibrational effects[END_REF] More recently, oxygen (-O), nitrogen (-N), fluorine (-F), chlorine (-Cl), and again hydroxyl (-OH)-functionalized AGNRs were studied. [START_REF] Jippo | First-principles study of edge-modified armchair graphene nanoribbons[END_REF][START_REF] Martin-Martinez | Tuning aromaticity patterns and electronic properties of armchair graphene nanoribbons with chemical edge functionalisation[END_REF] Nevertheless, the literature is missing a complete study covering the full range of graphene edge types with different functional addends, at consistent levels of theory. Such a reference, how to tailor ribbon composition toward specific electronic behavior, would serve as an important guide for experimental synthesis groups in the future.

For this reason, in this Article a detailed theoretical investigation of the influence of edge-functionalization is presented, using hydrogen (-H), fluorine (-F), chlorine (-Cl), bromine (-Br), sulfur (-S) atoms, and -SH and -OH groups, on a width 7 armchair GNR, a width 6 zigzag GNR, and for the first time on a width 6 reconstructed Klein GNRn. [START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] An overview is given of the influence of these edge-functional groups attached to these different graphene nanoribbon types, focusing on structural and electronic ribbon properties (especially the band gap). Additionally, the dependence of armchair GNR band gap value on functionalization and theoretical method is explored, to better classify such calculated results. Finally, an alternative route to tune AGNR electronic properties, by introducing an effective reduction of the ribbon width through different hydrogenation of its two opposing edges, is reviewed.

METHOD

Density functional theory (DFT) calculations using LDA-PW92 [START_REF] ;W A N G | A c c u r a t ea n ds i m p l ea n a l y t i c representation of the electron-gas correlation energy[END_REF] and GGA-PBE functionals [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] were performed as implemented in the AIMPRO code. [START_REF] Briddon | LDA Calculations Using a Basis of Gaussian Orbitals[END_REF][START_REF] Rayson | Highly efficient method for Kohn-Sham density functional calculations of 500-10000 atom systems[END_REF][START_REF] Briddon | Accurate Kohn-Sham DFT with the speed of tight binding: Current techniques and future directions in materials modelling[END_REF] The calculations were carried out using supercells, fitting the charge density to plane waves with an energy cutoff of 150 Ha (Ha: Hartree energy). Electronic level occupation was obtained using a Fermi occupation function with kT = 0.04 eV. Relativistic pseudopotentials are generated using the Hartwingster-Goedecker-Hutter scheme. [START_REF] Hartwigsen | Relativistic separable dual-space Gaussian pseudopotentials from H to Rn[END_REF] Periodic boundary conditions have been applied by using supercells. Supercell sizes have been checked and chosen to be sufficiently large (vacuum distance between ribbons larger than 12 Å) to avoid interaction with neighboring ribbons. All calculations were performed using orthorhombic supercells. A fine k-point grid was chosen (armchair GNR, (12/N) × 1 × 1 and zigzag/reconstructed Klein GNR, (18/N) × 1 × 1, where N is the number of fundamental graphene unit cells along the ribbon axis). Energies have converged to better than 10 -7 Ha. Atomic positions and lattice parameters were geometrically optimized until the maximum atomic position change in a given iteration dropped below 10 -5 a 0 (a 0 : Bohr radius).

HSE03 band gap calculations for the width 7 AGNRs have been performed with the code VASP [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF] and the HSE03 functional [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF] with the same supercells and k-point meshes as described before, using the LDA-PW92 geometrically optimized atomic positions.

Width 7 armchair nanoribbons (7-AGNRs) have seven rows of carbon atoms parallel to the ribbon axis. Width 6 zigzag ribbons (6-ZGNRs) have six zigzag rows of carbon atoms (carbon chains) parallel to the ribbon axis. Width 6 reconstructed Klein ribbons (6-RKGNRs) also have six zigzag rows of unfunctionalized carbon atoms parallel to the ribbon axis; that is, sp [START_REF] Novoselov | A roadmap for graphene[END_REF] -C atoms at the edge are not included in the width definition. Ribbon width is discussed further in the relevant sections below.

RESULTS

Armchair Graphene Nanoribbons (AGNRs). While hydrogenated graphene edges are flat, the steric hindrance between larger functional groups -F, -Br, -Cl, -S, -OH, and -SH has been shown to cause armchair edges to form static out-of-plane distorted ripples. [START_REF] Wagner | Ripple edge engineering of graphene nanoribbons[END_REF][START_REF] Rosenkranz | Ab initio calculations of edge-functionalized armchair graphene nanoribbons: Structural, electronic, and vibrational effects[END_REF][START_REF] Jippo | First-principles study of edge-modified armchair graphene nanoribbons[END_REF] More recently, similar edge rippling has been predicted for -OH edge-functionalized BN armchair nanoribbons. [START_REF] Lopez-Bezanilla | Structure and Electronic Properties of Edge-Functionalized Armchair Boron Nitride Nanoribbons[END_REF] In the present study, out-of-plane edge rippling has been taken into account for all types of functional groups attached to AGNRs, shown in Figure 1a. Here, we use free-standing AGNRs with opposing edges terminated identically. Further, for the width 7 AGNR, the influence of the edge-functional groups -H, -2H, -F, -Br, -Cl, -S, -OH, and -SH on ribbon band gap was calculated using different treatments for the exchange-correlation energy (LDA-PW92, GGA-PBE, and HSE03), as shown in Figure 1c. The well-known band gap underestimation in LDA and PBE-GGA calculations is confirmed. [START_REF] Jain | Reliability of Hybrid Functionals in Predicting Band Gaps[END_REF][START_REF] Ferretti | Ab initio complex band structure of conjugated polymers: Effects of hydrid density functional theory and GW schemes[END_REF] While qualitatively all band gaps show similar trends, an average deviation of 20-30% was found using LDA-PW92 and GGA-PBE as compared to calculations using a HSE03 functional, in good agreement with prior literature. [START_REF] Barone | Electronic Structure and Stability of Semiconducting Graphene Nanoribbons[END_REF] Using GW calculations, the gap of a singly hydrogenated width 7 AGNR is found even larger at about ∼3.7 eV. [START_REF] Yang | Quasiparticle Energies and Band Gaps in Graphene Nanoribbons[END_REF][START_REF] Prezzi | Optical properties of graphene nanoribbons: The role of many-body effects[END_REF][START_REF] Ruffieux | Electronic Structure of Atomically Precise Graphene Nanoribbons[END_REF] Experimentally, the band gap of a simple hydrogenated width 7 AGNR on an Au(111) surface was measured to be ∼2.3 eV. [START_REF] Ruffieux | Electronic Structure of Atomically Precise Graphene Nanoribbons[END_REF] These results show it is possible to tune the ribbon band gap over a wide energy range of 1.2 eV through correct choice of ribbon edge-functionalization. Notably, halogenation results in reduction in band gap of over 0.5 eV as compared to equivalent hydrogenated ribbons. We note that larger band gaps are normally concomitant with lower mobilities at the band edges (larger effective carrier masses, see Figure 1d), showing that transport properties can be dramatically modified depending on the nature of edge-functionalization. [START_REF] Long | Theoretical Predictions of Size-Dependent Carrier Mobility and Polarity in Graphene[END_REF][START_REF] Wang | Inverse relationship between carrier mobility and bandgap in graphene[END_REF] This variation in band curvature around the band gap can be seen in the band structures for -H and -OH terminated ribbons (Figure 1b). It can be understood because the dispersed states around the gap converge toward similar energies at X, necessarily imposing more rapid dispersion when the band gap at Γ is smaller.

An alternative route to control the ribbon band gap is through asymmetric functionalization of the two ribbon edges. Different configurations of graphene edge hydrogenation (-H) have been studied in the literature, and doubly hydrogenated (-2H) armchair graphene edges have been identified as the most stable. [START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF][START_REF] Ivanovskaya | Low-Energy Termination of Graphene Edges via the Formation of Narrow Nanotubes[END_REF][START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] The three possible cases with constant hydrogen densities along each edge, "-H/-H", "-2H/-H", and "-2H/-2H", are shown in Figure 2a. The edge stability does not vary with ribbon width in the width range 6-15 (∼6.05-17.05 Å); thus opposing armchair edges can be considered as energetically decoupled down to very small widths (<10 Å). [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF] The band gap for ribbons with these different edge terminations is shown in Figure 2b. The band gaps for the most widely studied case of a symmetric "-H/-H" edge hydrogenation are in good agreement with previous studies, reflecting a characteristic 3N periodicity. [START_REF] Barone | Electronic Structure and Stability of Semiconducting Graphene Nanoribbons[END_REF][START_REF] Wagner | Ripple edge engineering of graphene nanoribbons[END_REF][START_REF] Baldoni | Electronic properties and stability of graphene nanoribbons: An interpretation based on Clar sextet theory[END_REF][START_REF] Wassmann | Clars Theory, π -Electron Distribution, and Geometry of Graphene Nanoribbons[END_REF] For the other two opposed edge combinations, a shift is visible, demonstrating that doubly hydrogenating an armchair edge reduces the effective ribbon width by 1. This principle has also been observed in the literature, [START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF][START_REF] Zheng | Band gap engineering in armchair-edged graphene nanoribbons by edge dihydrogenation[END_REF] and can be understood as a reduction of the sp 2 carbon network (see Figure 2a). Shifting the effective ribbon width in this way, that is, by modifying the edge-functionalization asymmetrically, results in significant band gap modification (for the width 7 AGNR, the gap varies over a 1.24 eV range; see Figure 2b). This demonstrates a general principle, that asymmetry in functionalization (resulting in a change in hybridization of edge carbon atoms) between opposed edges of a graphene nanoribbon is another route to control the ribbon band gap. This could be achieved by attaching different functional groups to the two opposed ribbon edges, [START_REF] Martin-Martinez | Tuning aromaticity patterns and electronic properties of armchair graphene nanoribbons with chemical edge functionalisation[END_REF] or in other ways, for example by attaching ribbons orthogonally to a substrate.

A similar period three variation is reflected in the electron and hole effective masses, presented in Figure 2c. Varying the edge-functionalization for the width 7 ribbon shifts this periodicity as for the band gap, and can result in a near 7fold decrease in the carrier effective masses, which is likely to have a significant effect on carrier mobility. Interestingly, some asymmetry is observed in the (3n + 1) width ribbons, which have slightly lower hole than electron effective masses.

Zigzag Graphene Nanoribbons (ZGNRs). The structural behavior and influence on electronic properties of edgefunctional groups attached to ZGNRs are next investigated for width 6 ZGNR (∼11.3 Å wide), as shown in Figure 3. As for AGNRs, in general the effect of interaction between neighboring functional groups needs to be taken into account. On the perfect zigzag edge, the carbon-carbon distance is ∼2.46 Å. Because this is more than twice the hydrogen van der Waals radius (1.10 Å 57 ), it suggests that steric hindrance at hydrogenated zigzag edges should be negligible. However, for species such as Cl and Br with van der Waals radii of 1.75 and 1.83 Å, respectively, 58 significant interactions are expected. Steric hindrance between F atoms (van der Waals radius of 1.47 Å) attached to zigzag edges is expected to be weaker, and to compete against the preference for fully conjugated planar sp 2 carbon network geometry of the nanoribbon.

These steric out-of-plane effects are visible in Figure 3a, where both chlorinated and brominated z 1 edges exhibit rippling, with significant out-of-plane displacement of the functional groups (0.84 Å for Cl, 0.97 Å for Br). The ripple periodicity is a systematic alternation of neighboring functional groups, maximizing the spacing between the edge-functional groups; longer period ripples are found to be less stable.

The most stable edge configuration for hydrogenated ZGNRs was previously shown to be the nonmagnetic z 211 structure, [START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF][START_REF] Kunstmann | Stability of edge states and edge magnetism in graphene nanoribbons[END_REF] where every third edge carbon atom is doubly hydrogenated. Similarly, we find here that of the various edge configurations considered, the z 211 is the most stable irrespective of functional group -H, -F, -Cl (see Table 1). The exception is bromination, where pairwise addition is unfavored. Additionally, the z 211 configuration provides an alternative mechanism for strain relief because the functional groups on a doubly functionalized sp [START_REF] Novoselov | A roadmap for graphene[END_REF] -carbon already lie out-ofplane. Hence, in the z 211 configuration, out-of-plane rippling is much less prominent. In any case, the presence of the sp 3 coordinated edge species means that such ribbon edges will not remain flat on a substrate.

All of the z 1 edge configurations are most stable in the antiferromagnetic ground state (except -S, which is metallic), similarly to hydrogenated ribbons. [START_REF] Son | Half-metallic graphene nanoribbons[END_REF][START_REF] Lee | Electronic structures of zigzag graphene nanoribbons with edge hydrogenation and oxidation[END_REF][START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF] The hydrogenated, fluorinated, chlorinated, and brominated zigzag ribbons all exhibit significant band gaps, which further increase in the more stable, nonmagnetic, z 211 configuration (see Figure 3b). We note that these are LDA band gaps, and as discussed above are about ∼20-30% underestimated as compared to calculations using HSE03. Nonetheless, these gaps are smaller than equivalent width armchair ribbons, and notably there is significantly less variation in band gap with functional group (less than 0.08 eV).

We also considered functionalization of the related 5-7 zigzag edge, formed by rotating alternate carbon bonds along the ribbon edge, [START_REF] Koskinen | Self-Passivating Edge Reconstructions of Graphene[END_REF][START_REF] Koskinen | Evidence for graphene edges beyond zigzag and armchair[END_REF] but found in test calculations always to be less stable than the functionalized zigzag edges presented above.

Reconstructed Klein Graphene Nanoribbons (RKGNRs). When cutting a graphene sheet along the ⟨21̅ 1̅ 0⟩ direction, two parallel cutting lines are possible, resulting in either zigzag or Klein edges. As reported previously, the most stable hydrogenated edge structures along this direction are not zigzag edges, but reconstructed variants of the Klein edge. [START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] Thus, we have also considered similar functionalized reconstructed Klein ribbon edges (see Figure 3 and Table 1). The effective ribbon width definition is taken from ref 32, and hence reconstructed Klein-based GNRs have more carbon atoms per unit cell than zigzag GNRs, because the reconstructed Klein sp 3 -carbon atoms are not included when determining the width. The similarity in calculated band gaps for the z 211 and rk 22 + z 2 terminated ribbons further supports this definition of effective ribbon width.

As can be seen in hydrogenated ribbons. In both cases, the most stable edge configuration is the rk 22 + z 2 , where all edge carbon atoms are fully sp 3 coordinated. However, for the chlorinated and brominated ribbons, the reconstructed Klein-based ribbons are less stable than their zigzag analogues due to the significant steric hindrance between the functional groups. Consistency in calculated band gaps has been found between the most stable functionalized zigzag and reconstructed Klein graphene nanoribbons, which are relatively insensitive to the choice of edge-functional group.

CONCLUSION

In summary, we have studied different edge-functionalized graphene nanoribbon types, armchair, zigzag, and reconstructed Klein GNRs. All edge-functional groups except hydrogen tested on armchair GNRs (-F, -Cl, -Br, -S, -OH, -SH) lead to characteristic edge rippling. Similar behavior has also been found for the bulkier -Cl and -Br edge-functional groups, when attached to zigzag z 1 GNRs, while for -H and -F the z 1 zigzag structures stay flat in-plane. However, for -H, -F, and -Cl functionalized zigzag ribbons, the nonmagnetic z 211 configuration is more energetically stable than the z 1 , and edge rippling is reduced. An even more stable ribbon configuration was found for -H and -Cl edge-functionalization, through changing the zigzag graphene edge structure to form a regular mix of zigzag and reconstructed Klein edges, called rk 22 + z 2 GNRs. For these edge structures, steric hindrance is less prominent, and thus rippling is expected to occur rarely.

Regarding electronic properties, in particular the ribbon band gap, two general conclusions can be drawn. First, for thin armchair GNRs, the band gap can be tuned over a wide range ΔE gap ≈ 1.2 eV, while for thin zigzag and reconstructed Klein GNR types, the band gap is largely insensitive to edgefunctionalization. Second, the difference in band gap between z 1 and z 211 zigzag GNRs and the new rk 22 + z 2 reconstructed Klein GNR is small (considering similar ribbon widths, not including sp 3 hybridized reconstructed Klein carbons), and all of the most stable structures exhibit nonmagnetic ground states. Carrier effective mass in AGNRs are also highly dependent on ribbon width and edge-functionalization, with smaller gaps imposing more rapid band dispersion and hence lower effective masses. These results clearly favor thin armchair GNRs as possible candidates to design electronic properties via control of edge-functionalization.

Figure 1 .

 1 Figure 1. (a) Structures of different functionalized width 7 AGNRs. (b) LDA-PW92 band structures showing band dispersion along the ribbon axis for -H and -OH termination. (c) Calculated band gaps using different exchange-correlation implementations (LDA-PW92, GGA-PBE, and HSE03) (see Table1). (d) LDA-PW92 effective electron (m e *) and hole (m h *) masses at the Γ-point of the AGNRs.

Figure 2 .

 2 Figure 2. (a) The three different hydrogenated edge symmetries "-H/-H", "-2H/-H", and "-2H/-2H". Below each structure, the effective width derived from the sp 2 network is given. (b) LDA-PW92 band gaps with characteristic 3N periodicity calculated for different H-terminations of opposed armchair edges. (c) LDA-PW92 effective electron (m e *) and hole (m h *) masses at the Γ-point of the AGNRs. The width in both graphs is given in respect of the simple carbon atom ribbon width, the "-H/-H" case.

Figure 3 .

 3 Figure 3. (a) Perspective and side view of the edge configurations z 1 , z 211 , and rk 22 + z 2 (from left to right column) with different edge-functional groups -H, -F, -Cl, and -Br (from top to bottom) for width 6 GNRs. Edge formation energies are given in Table 1. (b) Calculated band gaps (LDA-PW92) for the different functionalized zigzag and reconstructed Klein GNRs. NM, nonmagnetic; AFM, antiferromagnetic.

Table 1 ,

 1 fluorinated reconstructed Klein ribbons demonstrate the same trend in energetic stability as

Table 1 .

 1 Edge Formation Energy E edge[START_REF] Wassmann | Structure, Stability, Edge States, and Aromaticity of Graphene Ribbons[END_REF][START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF] and Calculated LDA-PW92 Band Gap for Different Functionalized Zigzag, Reconstructed Klein-type (Both Width 6), and Armchair (Width 7) Graphene Nanoribbons a

		zigzag			recon. Klein types		armchair
	LDA-PW92	z 1	z 211	rk 22 + z 2	rk 11	rk 22	a 11	a 22
	E edge (eV/Å)
	-H	+0.100	-0.011	-0.107	+0.303	-0.047	+0.019	-0.174
	-F	-0.580	-0.951	-1.512	-0.350	-1.436	-0.722	-1.690
	-Cl	+0.140	+0.057	+0.136	+0.246	+0.563	+0.041	-0.006
	-Br	+0.319	+0.325	+0.601	+0.411	unstable	+0.236	unstable
	-S	+0.164		+0.753	+0.445	+0.584	+0.068
	-SH	+0.154			+0.233		-0.005
	-OH	-0.985			-0.723		-0.617
	band gap (eV)
	-H	0.36	0.46	0.48	0.00	0.33	1.59	0.35
	-F	0.33	0.39	0.42	0.00	0.36	1.09	0.26
	-Cl	0.31	0.40	0.44	0.00		1.07	0.31
	-Br	0.32	0.38	0.14	0.00		1.09
	-S	0.00		0.42	0.00	0.30	0.90
	-SH	0.06			0.00		0.76
	-OH	0.16			0.00		0.76

a z, zigzag; rk, reconstructed Klein; a, armchair; subscripts indicate number of functional atoms per edge carbon atom.

[START_REF] Wagner | Stable hydrogenated graphene edge types: Normal and reconstructed Klein edges[END_REF]

ACKNOWLEDGMENTS P.W., C.P.E., L.M., P.P., S.R., and V.V.I. thank the NANOSIM-GRAPHENE project no. ANR-09-NANO-016-01 funded by the French National Research Agency (ANR) in the frame of its 2009 programme in Nanosciences, Nanotechnologies & Nanosystems (P3N2009). P.W., C.P.E., J.-J.A., and B.H. thank the SPRINT project ANR-10-BLAN-0819. S.R. acknowledges funding support from the Spanish Ministry of Economy and Competitiveness (MAT2012-33911). We thank the CCIPL, CLUSTUS (IMN), and GENCI (Grant 2013-097015) for computing time. We thank COST project MP0901 NanoTP for support.

AUTHOR INFORMATION

Corresponding Author *E-mail: chris.ewels@cnrs-imn.fr.

Notes

The authors declare no competing financial interest.


Arkadiusz Karpinski

Axel Ferrec

Mireille Richard-Plouet

Linda Cattin

Mohammed Abdou Djouadi

Luc Brohan

Pierre-Yves Jouan
email: pierre-yves.jouan@cnrs-imn.fr

P.-Y Jouan

Deposition of nickel oxide by direct current reactive sputtering Effect of oxygen partial pressure

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

NiO is a wide band gap p-type semiconductor and as a thin film it provides high transparency. Therefore, the deposition of such a material on transparent conductive glass substrate becomes interesting for development of electrochromic and photovoltaic devices where high transparency is needed.

In recent years, numerous methods have been used in order to deposit nickel oxide film such as metal organic chemical vapor deposition [1], thermal evaporation [2],s o l -gel deposition [3], pulsed-laserdeposition [4] and intermittent spray pyrolysis process [5].A m o n g them, reactive magnetron sputtering is the most widely used [6][7][8][9][10]. The film properties and structure growth are affected by various deposition process parameters such as O 2 partial pressure in the gas mixture, total pressure, sputtering power, substrate temperature, etc. The predominant parameters are probably the partial pressure of O 2 in the gas mixture [11] and substrate temperature [12]. In this paper, the impact of O 2 partial pressure on plasma characteristics has been investigated. Indeed, we show in this paper that the preferred crystalline orientation is directly related to the amount of oxygen and monitoring this amount should enable us to control the film properties: transparency, conductivity (p or n)…

Experimental methods

The NiO thin films were deposited on SnO 2 :F conducting glass ASAHI 120 by Direct Current reactive magnetron sputtering using a pure nickel target (99.99%) and an Ar-O 2 gas mixture. A 1.3 in. diameter (30 mm) target was used and substrates were placed at approximately 30 mm from this target. Thanks to mass flow controller, argon and oxygen gases could be adjusted. The resulting oxygen ratio in the fed gas was defined as: O 2 % = V(O 2 )/(V(O 2 ) + V(Ar)) (where V represents the mass flow of each gas) and varied from 0 to 25%. The base pressure was about 2.7.10 -4 Pa. In order to maintain the pumping speed constant, the turbo molecular pump valve was fully open. Then the discharge pressure in pure argon was 0.44 Pa and was modified from 0.45 to 0.51 Pa, depending on the oxygen content. Different discharge currents (110, 80 and 50 mA) were tested. Pre-sputtering process was done in two steps: first with argon to clean the target and second with the chosen working gases mixture in order to reach a steady state before deposition. The crystal structure of films, as a function of the thickness, was identified by X-Ray Diffraction (XRD) using a SIEMENS D 5000 diffractometer, in a grazing mode, for films close to 100 nm, (with a grazing angle 1°(θ), CuK α radiation). Thicker films were analyzed using a Brüker D8 Advance powder diffractometer operating in Bragg-Brentano geometry (Ge monochromator, CuK α1 λ = 1.540598 Å). The film morphology was examined by Scanning Electron Microscopy (SEM), using SEM Jeol JSM 7600F and JSM 6400F. The stoichiometry of thin films was evaluated by Energy-Dispersive Xray spectroscopy (EDX) equipment in a SEM Jeol 5800, operating at 7 kV. The film thickness was measured using a surface profilometer (DEKTAK 8). The electrical conductivity was measured, with the four-point probe technique, using copper electrode. The Hall effect measurements were 1 performed utilizing the Van der Paw arrangement. The majority carrier type has been checked using the hot probe technique.

Results and discussion

Estimation of the deposition conditions

There are three basic feedback signals for controlling the reactive process during sputter deposition: mass spectrometry [13],o p t i c a l emission spectroscopy [14] and cathode voltage (i.e. discharge voltage) [15]. In this study, we analyzed the electrical signal of the cathode because it is an easy handling technique giving fast updated information (potentially less than a few milliseconds). The signal strongly depends on the properties of the material which is formed on the cathode surface during the sputtering process [16]. The behavior of the cathode voltage was studied by increasing the oxygen content (Fig. 1a) with fixed values of current and flow of inert gas (10 sccm, standard cubic centimeter per minute, of Argon). In Fig. 1a, we can also observe that increasing the discharge current leads to a shift of the curve toward higher oxygen flux. Indeed, the discharge current is the sum of both electron and ion current (Ar + created by electron collisions near the cathode). Intensifying the ion current leads to an increase of the sputtering of the target, thus, as expected or as generally reported, the oxygen partial pressure required to observe the oxidation of the cathode is higher. In general, the influence of the reactive gas content on the discharge voltage, leads to 2 regions, related to the target state: metallic or poisoned [17,18]. In the case of the nickel target, we differentiated 4 regimes, on the characteristic of discharge voltage (Fig. 1b). Each region exhibits a typical X-ray diffraction pattern in which the peaks intensities are significantly varying. A similar behavior of discharge voltage has been already observed by Hotovy et al. [6],b u t it was not clearly correlated to the preferential growth of NiO films. The 4 regimes are as follows:

• In the first region, the target is in a metallic state, the oxygen species reaching the target are removed by the ion bombardment, allowing the target to remain metallic and therefore the discharge voltage does not change significantly. • In the second region, the target is in an intermediate state, the amount of oxygen reaching the target starts to become higher than the oxygen removed by sputtering. We observe an enhancement (absolute value) of the target potential which indicates a decrease in electrical conductivity of the cathode with a low increase of secondary electron emission coefficient. • In the third region, the target potential seems to stabilize, suggesting that the cathode is in a semi-conductive state, due to the formation of NiO layer on the target surface. However this behavior is difficult to understand, some authors attribute this stabilization at high potential, for a critical oxygen partial pressure, to the formation of negative oxygen ions [19,20]. In fact near the cathode we have two electron populations [21], high energy electrons which can collide to O 2 molecules leading to dissociation-ionization (maximum of the cross section at about 100 eV [22])o ft h eO 2 molecules with the creation of low energy electrons increasing the second population of electrons which are thermalyzed (4 eV) and which can get attached to O atoms or O 2 molecules but also collide to O 2 molecules to form 2O - (maximum of the cross section at about 6 eV [22]). These negative ions consume electrons and are not trapped by the magnetic field, hence we observe a diminution of the ionic density (n i =n e )i nt h e plasma and thus, in order to maintain the discharge current, the cathode voltage increases. • In the fourth region, the target is fully oxidized, the discharge voltage stabilized again at a lower (absolute value) discharge voltage. This is the so-called poisoned mode of the cathode. This low discharge voltage could be due to an increase in the secondary electron emission coefficient and in discharge pressure (oxygen is added).

During our experiments, the hysteresis loop (consumption by the walls) that normally occurs, when decreasing the reactive gas partial pressure, was not observed. This is probably due to the small surface of the cathode (1.3 in. in diameter) and the relatively high pumping speed (260 L.s -1 ).

Deposition rate

The deposition rate is directly proportional to the sputtering yield which depends on the discharge current (i.e.: power) and on the oxidation degree of the cathode. Thus, the deposition rate appears to be a consequence of the discharge conditions, it could not be considered as a parameter in this study. The experimental deposition rates of NiO as a function of oxygen content in the gas mixture for 3 discharge currents are presented in Fig. 2. The deposition rate augments with increasing discharge current as commonly observed. For each examined current, the deposition rate reaches a maximum in the semi-conductive region (zone 2) and then decreases strongly due to the low sputter yield of the oxide for zones 3 and 4. The same evolution of the deposition rate as a function of oxygen content has been observed by Liu et al. [23].

Crystallographic orientations

The structure of as-deposited NiO was found to be polycrystalline but with preferential orientations, strictly related to the oxygen content in the mixture of working gases (Fig. 3). The preferential growth of NiO was always correlated to the shape of the discharge current (Fig. 1b), and therefore to the oxygen partial pressure. The region defined as 1 was out of consideration because the deposited films were also composed of metallic nickel. The region 2 with relatively low oxygen content was accompanied by an increase in the ( 111 oxygen content in the mixture of working gases. Nevertheless, the nonstoichiometry in nickel, for example, may also produce the same effect, causing less X-rays diffraction from the heaviest Ni atoms. As plotted in Fig. 4, the O/Ni ratio of as deposited NiO films (supported by EDX analysis) suggests that some vacancies in nickel atoms occur leading to the formula: NiO 1.13 , for high flux of oxygen gas. This non-stoichiometry is associated to a shift of the diffraction peaks toward low 2θ angles, indicating an expansion of the cell parameters. This observation is coherent with previously reported data concerning NiO obtained by spray pyrolysis from nickel nitrate [24].

Reported XRD results for NiO thin films deposited by reactive sputtering, show that increasing the temperature and/or the discharge power for a fixed oxygen flow rate leads to [111] preferential orientation. Note that in both cases (increasing the temperature and/or the discharge power) the effect on the film is a reduction of oxygen content which is consistent with our results [25][26][27].

Crystallographic studies of the deposits at lower thicknesses indicated that preferential growth of the NiO structure is not only affected by oxygen content but also depends on the thickness of thin films. Fig. 5a shows crystallographic reflections of samples at about 80 nm thickness, for increasing oxygen contents.

The intensity of the (111) peak decreases with increasing oxygen content from 8.2 to 9.9% O 2 (Fig. 4a). For the deposits at 14.5% or 23.5% O 2 the intensities of the (111) and (200) reflections are comparable. This gives an idea that at high oxygen content the grain growth process is competitive for all the directions, for very thin layers (lower than 100 nm) at high oxygen values. The structural evolution is clearly observed for samples deposited at 23.5% O 2 when the thickness of thin film enhances (Fig. 5b). At the thickness of 96 nm, both (111) and (002) grains are present. Further growth leads to progressive predominance of the growth along the [200] direction when the thickness reaches 272 nm and 1148 nm. Predominance of the preferential orientation with the thickness can be associated to competitive growth and grains coarsening. The thin film, at the very beginning of the growth, is randomly oriented while overall microstructure evolves toward a final well crystallized state (intense and narrow diffraction peaks) driven by the appropriate deposition conditions [9,10].

Morphology and cross-section

Fig. 6 shows a typical surface of 80 nm thin films deposited for various oxygen contents at 110 mA current. The surface appears composed of round grains with size ranging from 50 to 100 nm. When the NiO thickness grows to about 1 μm, grain coarsening is observed (Fig. 7).

Initial round grains evolve to flakes like crystals close to 300 nm for low oxygen contents (Fig. 7a). For high oxygen values (Fig. 7d), the surface aspect is different with rectangular and triangular grains in the size range close to 100 nm. We observe that this size evolution depends on the orientation: the grain narrowing is associated to film growth along the [200] direction. Cross-section images (Fig. 8) reveal the columnar nature of the films in every case. The sample deposited at lowest oxygen value grows as columns with limited extension and different growth directions, leading to low ordering of the microstructure. Deposited NiO at 9.9 and 14.5% O 2 is also composed of columns, expanding during nucleation process, which is affected by competitive growth along both preferential orientations. Such a type of growth is often accompanied by the formation of intercolumnar voids (see Fig. 7b,c). Sample deposited at high oxygen flow (Fig. 8d) consists in highly ordered cylindrical grains having mainly the [200] orientation.

Conclusions

Nickel oxide thin films were deposited onto conductive transparent glass substrates. The optimization of deposition conditions allowed us to control the preferential structure growth along [111] or [200] direction in a continuous manner as a function of the oxygen content in the mixture of working gases. Whatever the experimental conditions, the discharge voltage characteristic presents an unusual shape with a local maximum. The position of this maximum is shifted toward higher oxygen percentages when the current increases. The deposition conditions allowed us to tune the polycrystalline microstructure and preferentially oriented growth could be observed. Finally, we have checked that the NiO semi-conductor films are p-type and their resistivity ρ,is around 2 Ω.cm. The UV-Visible measurements have already been reported. The optical band gap lies between 3.2 and 4 eV, as expected, with a transparency up to 80% [28].

Fig. 1 .

 1 Fig. 1. Discharge voltage vs oxygen content for different discharge currents: a) experimental data, 50 mA blue circle, 80 mA red square, 110 mA black triangle, b) differentiation of 4 zones in the discharge curve (see text).

Fig. 2 .

 2 Fig.2. Deposition rate for various evaluated discharge currents, 50 mA blue circle, 80 mA red square, 110 mA black triangle.

  ) diffraction peak; region 3, where discharge voltage reached absolute maximum was related to structure growth of NiO with peak intensities close to standard JCPDS card (Bunsenite, NaCl-type structure, 2003 No: 89-7130) NiO powder-like structure; region 4, rich oxygen content was characterized by the predominance of the (200) peak intensity.Films deposited in the oxygen range allowing transition of preferential growth from [111] to[200] directions exhibit intense and narrow diffraction peaks, however XRD diagrams of the deposits carried out in relatively high oxygen content show less intensity signal for films having similar thickness. This may indicate the presence of amorphous phase or a loss in the number of crystallite domains, due to a raise of

Fig. 3 .

 3 Fig. 3. XRD diagrams for 1 μm NiO thin films deposited at discharge currents: a) 110 mA, b) 80 mA and c) 50 mA. The peaks attributed to metallic nickel were marked by an asterisk (*).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Evolution of the stoichiometry for 1 μm NiO x thin films deposited at 110 mA for various oxygen contents, calculated from EDX measurements.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Representative surface of 80 nm thin films deposited at 110 mA: a) 9.9% O 2 , b) 23.5% O 2 .

Acknowledgments

This work was financially supported by the French Research Agency (ANR) through the SOLHYPIN project. The authors would like to thank Dr. M. Morsli from the University of Nantes for electrical measurements and N. Stephant from the CMC-University of Nantes for SEM observations.