R Guglielmi
email: roberto.guglielmi@uni-bayreuth.de

Indirect stabilization of hyperbolic systems through resolvent estimates

Keywords: Indirect stabilization, systems of hyperbolic PDEs, resolvent estimates AMS subject classifications: 93D15, 35L53, 47D06, 47A10

We prove a sharp decay rate for the total energy of two classes of systems of weakly coupled hyperbolic equations. We show that we can stabilize the full system through a single damping term, acting on one component only of the system (indirect stabilization). The energy estimate is achieved by means of suitable estimates of the resolvent operator norm. We apply this technique to a system of wave-wave equation and to a wave-Petrovsky system.

Introduction

The issue of stabilizing a system of partial differential equations through suitable damping terms (possibly in feedback form) on each component of the system has its origin in the pioneering works of Lagnese and Lions [START_REF] Lagnese | Modelling analysis and control of thin plates[END_REF] and Russell [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF]. In their approach, the multiplier method is the main tool to reach the desired estimates on the energy of each component of the system. This techniques has been further developed in [START_REF] Komornik | Exact controllability and stabilization[END_REF] and later in [START_REF] Alabau | Stabilisation frontière indirecte de systèmes faiblement couplés[END_REF] for systems of hyperbolic equations.

In particular, Russell [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] addressed the indirect stabilization problem, that occurs when the damping (or the control) acts on a reduced number of equations of the system. In this situation the (uniformly) exponential decay rate usually cannot be achieved, but weaker decay rates might hold. This is the case in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] and [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF], where (uniform) polynomial stability for the whole system is showed, under a suitable compatibility condition on the operators involved in the system, by means of multipliers properly adapted to the peculiar structure of the system under investigation. Indeed, it turns out that different compatibility condition and multipliers are required to cope with systems with boundary conditions of similar type on each component [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] or mixed [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF].

In [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF] a different method has been proposed to prove polynomial stability for the solution of abstract first order Cauchy problems. When applied to the indirect stabilization of a weakly coupled system, this technique requires stronger compatibility conditions among the operators of each component of the system, in order to perform the needed spectral analysis, thus applying, so far, to fewer systems of hyperbolic equations.

An operator-theoretical approach has been recently proposed in [START_REF] Charles | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF] and [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], addressing the optimality issue for the established decay rate. Indeed, the decay rate of the system ruled by the dynamics A is related to the asymptotic behaviour of the norm of the resolvent operator of A on the imaginary axis (see, for example, [START_REF] Lebeau | Équation des ondes amorties[END_REF]). This technique has been successfully applied in [START_REF] Tebou | Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms[END_REF] to the indirect stabilization of systems of Euler-Bernoulli and wave equations with globally distributed coupling and damping. Let us point out that, however, multipliers cannot be completely avoided, since also this approach requires suitable multipliers (adapted to the operators of the system) in order to deal with elliptic estimates of the resolvent operator.

In this paper we study the indirect stabilization problem for two classes of systems of hyperbolic-type equations, by means of the general criterion given in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]. Since these systems fall into the general description given in [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF], polynomial stabilization is already ensured (see equation ( 8)). Here we succeed in improving the decay rate of the total energy of the systems, thanks to a sharp analysis of the behaviour of the resolvent operator along the imaginary axis.

More precisely, let Ω ⊂ R d , d ≥ 1, be open and bounded, with sufficiently smooth boundary Γ. We denote by ∆ = d i=1 ∂ 2 xixi the Laplacian operator; moreover, for sake of simplicity, we use the subscript u t to denote the partial derivative of u with respect of the variable t. Let λ ≥ 0 and α, β > 0. We will first consider the weakly coupled system of wave equations

u tt -∆u + λu + βu t + αv = 0 in Ω × (0, +∞) v tt -∆v + αu = 0 in Ω × (0, +∞) (1)
with boundary conditions

∂u ∂ν (•, t) + σu(•, t) = 0 = v(•, t) on Γ , t > 0 , σ ≥ 0 , (2)
and initial conditions

u(0) = u 0 , u t (0) = u 1 , v(0) = v 0 , v t (0) = v 1 in Ω , (3)
for functions u i , v i (i = 0, 1) in suitable spaces (see ( 14)). We remark that max(σ, λ) > 0, meaning that, under Neumann boundary condition on u, we require λ > 0, in order to ensure the coercivity of the operator in the first component. Also operators with different boundary conditions on separated portions of the boundary can be treated. Indeed, let Γ 0 and Γ 1 be open subsets of Γ such that

Γ = Γ 0 ∪ Γ 1 , Γ 0 ∩ Γ 1 = ∅ .
In this situation, we can consider system (1) with boundary conditions

u(•, t) = 0 on Γ 0 , ∂u ∂ν (•, t) + σu(•, t) = 0 on Γ 1 v(•, t) = 0 on Γ t > 0
and initial conditions [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF]. Finally, we will focus on the Petrowsky-wave system

u tt -∆u + βu t + αv = 0 in Ω × (0, +∞) v tt + ∆ 2 v + αu = 0 in Ω × (0, +∞) , (4)
with Robin boundary conditions

∂u ∂ν (•, t) + σu(•, t) = 0 on Γ , t > 0 , σ > 0 , (5)
on u and either hinged

v(•, t) = 0 = ∆v(•, t) on Γ , t > 0 (6)
or clamped boundary conditions

v(•, t) = 0 = ∂v ∂ν (•, t) on Γ , t > 0 (7)
on v, with initial conditions (3). We first point out that the system of two evolution equations associated with the systems above fulfills the compatibility condition introduced in [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF], ensuring the estimate on the total energy

E(U (t)) ≤ C (1 + t) 1/4 |U 0 | 2 H (8)
for every initial condition U 0 = (u 0 , u 1 , v 0 , v 1 ) ∈ D(A) and for some constant C > 0 (see the section below for precise definitions of the spaces D(A), H and the energy E(U )). The aim of the present paper is to derive the optimal decay rate of the total energy E(t); indeed, we will show that it decays polynomially in time with a decay rate 1/2 for initial condition in D(A), that is,

E(U (t)) ≤ C (1 + t) 1/2 |U 0 | 2 H
for every U 0 = (u 0 , u 1 , v 0 , v 1 ) ∈ D(A) and for some C > 0 (see also equation (31)). In this way, we succeed to improve the decay rate of an exponential factor 2. Moreover, we observe that in all the systems above the constants α, β, λ and σ can be replaced by bounded functions of the space variable, provided they are strictly positive.

The paper is organized as follows: in the next section we introduce the abstract setting that fits the previous PDEs systems into a first order Cauchy problem and, in this framework, we state our main result together with general stability and spectral properties. Finally, in Section 3, we prove the required estimates on the two classes of systems of hyperbolic equations introduced above.

Abstract setting

Let (H, , ) be a Hilbert space on the field C of complex numbers with associated norm | | H . We consider the abstract system of evolution equations

u (t) + A 1 u(t) + Bu (t) + αv = 0 in H v (t) + A 2 v(t) + αu = 0 in H (9)
where (H1) A i : D(A i ) ⊂ H → H (i = 1, 2) are densely defined closed linear selfadjoint operators such that

A i u, u ≥ ω i |u| 2 ∀u ∈ D(A i ) , i = 1, 2 , for some ω 1 , ω 2 > 0 . ( 10
)
(H2) B is a bounded linear selfadjoint operator on H such that Bu, u ≥ β|u| 2 ∀u ∈ H , for some β > 0.

(H3) α is a real number such that 0

< |α| < √ ω 1 ω 2 .
We associate to the operator A i , i = 1, 2, the energy

E i (u, p) = 1 2 |A 1/2 i u| 2 H + |p| 2 H ∀(u, p) ∈ D(A 1/2 i ) × H (i = 1, 2) , (12)
so that assumption (H1) yields

|u| 2 H ≤ 2 ω i E i (u, p) ∀(u, p) ∈ D(A 1/2 i ) × H (i = 1, 2) . (13)
System [START_REF] Lagnese | Modelling analysis and control of thin plates[END_REF], with initial conditions

u(0) = u 0 ∈ D(A 1/2 1 ) , u (0) = u 1 ∈ H , v(0) = v 0 ∈ D(A 1/2 2 ) , v (0) = v 1 ∈ H , (14)
can be formulated as a first order Cauchy problem in the space

H = D(A 1/2 1 ) × H × D(A 1/2 2 ) × H ,
that becomes a Hilbert space on C endowed with the scalar product

(U, Û ) := A 1/2 1 u, A 1/2 1 û + p, p + A 1/2 2 v, A 1/2 2 v + q, q +α u, v +α v, û (15)
for every U = (u, p, v, q), Û = (û, p, v, q) ∈ H, that also induces the norm |U | H := (U, U ) 1/2 on H. Indeed, introducing the operator

D(A) = D(A 1 ) × D(A 1/2 1 ) × D(A 2 ) × D(A 1/2 2 ) AU = (p, -A 1 u -Bp -αv, q, -A 2 v -αu) ∀U ∈ D(A) , (16)
problem ( 9) can be recast as

                 U (t) = AU (t) =      0 I 0 0 -A 1 -B -αI 0 0 0 0 I -αI 0 -A 2 0      •      u u v v      U (0) = U 0 = (u 0 , u 1 , v 0 , v 1 ) ∈ H , (17)
where I stands for the identity operator on H. Moreover, for every U ∈ H, we define the total energy of system (17) by

E(U (t)) := E 1 (u, p) + E 2 (v, q) + 2αRe u, v = |U | 2 H 2 , (18)
where Re(z) denotes the real part of z ∈ C. By assumption (H3), the total energy E is equivalent to the sum of the energy of each component, indeed it satisfies

ν 1 (α) [E 1 (u, p) + E 2 (v, q)] ≤ E(U (t)) ≤ ν 2 (α) [E 1 (u, p) + E 2 (v, q)] , (19)
where

ν 1 (α) = 1 -|α|(ω 1 ω 2 ) -1/2 > 0 and ν 2 (α) = 1 + |α|(ω 1 ω 2 ) -1/2 .
The operator A generates a C 0 -semigroup e tA on H (see Lemma 4.2 in [START_REF] Alabau-Boussouira | Indirect stabilization of weakly coupled systems with hybrid boundary conditions[END_REF]), that satisfies e tA U 0 = (u(t), p(t), v(t), q(t)), where (u, v) is the solution of problem ( 9) with initial conditions ( 14) and (p, q) = (u , v ).

Stability properties and spectral criteria

In [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] the authors prove that system ( 9)-( 14), or, equivalently, system (17), fails to be exponentially (uniformly) stable, at least when H is infinite dimensional and the operator A 1 has compact resolvent. This feature is a consequence of the compactness of the coupling operator

K u u = 0 αu in the energy space D(A 1/2 i ) × H (i = 1, 2)
. Thus, if any stability property holds for system ( 9)-( 14), then such a property must be weaker than exponential stability. A first step in this direction consists in checking that the total energy of the system fulfills a dissipation relation.

Proposition 1. For every U 0 ∈ D(A) we have that d dt E(U (t)) = -|B 1/2 p| 2 H = Re(AU, U ) H . (20)
Our approach strongly relies on the spectral properties of the operator A. The next result gives a first insight into the spectrum of A.

Proposition 2. Assume hypotheses (H1)-(H2)-(H3) hold. Then iR ⊂ ρ(A), that is, the resolvent set of A contains the imaginary axis.

Proof. Let b ∈ R and U = (u, p, v, q) ∈ D(A) such that AU = ibU . (21)
We claim that U = 0. We first address the case b = 0. Then, equation ( 21) can be recast as

         ibu = p ibp = -A 1 u -Bp -αv ibv = q ibq = -A 2 v -αu . (22)
We multiply both sides of (21) for U and take the real part. Thus, owing to equation (20), we deduce that |B 1/2 p| H = 0. Then, hypothesis (H2) implies p = 0; by the first equation in (22) we have u = 0 (since b = 0); from the second equation in ( 22) we find v = 0 (since α = 0); finally, the third equation in (22) ensures that q = 0, so we conclude that U = 0. On the other hand, if b = 0, system (22) reduces to p = q = 0 and

A 1 u + αv = 0 A 2 v + αu = 0 . ( 23
)
We now argue by contradiction. Suppose there exist u = 0 = v satisfying system (23). Thus, multiplying the first equation therein by u and the second by v, thanks to hypothesis (H1), we have

ω 1 |u| 2 H ≤ A 1 u, u = -α v, u , ω 2 |v| 2 H ≤ A 2 v, v = -α u, v . (24)
Since the left upper terms in (24) are positive, we multiply both sides together and we use the Cauchy-Schwarz inequality,

ω 1 ω 2 |u| 2 H |v| 2 H ≤ α 2 | u, v | 2 ≤ α 2 |u| 2 H |v| 2 H , (25)
infering ω 1 ω 2 ≤ α 2 , that negates hypothesis (H3). Therefore u = v = 0 and we conclude again that U = 0.

As a consequence, thanks to a characterization due to Benchimol [START_REF] Benchimol | A note on weak stabilizability of contraction semigroups[END_REF], we deduce that system (17) is strongly stable.

Corollary 1. Under hypotheses (H1)-(H2)-(H3), the semigroup e tA is strongly stable, that is, lim

t→+∞ |e tA U 0 | H = 0 ∀U 0 ∈ H . (26)
We now further analyze the asymptotic behaviour of the contraction semigroup e tA with generator (A, D(A)). For this purpose, we recall a result due to Borichev and Tomilov [7, Theorem 2.4], which gives a necessary and sufficient condition for the polynomial decay of the semigroup norm. Given two functions f and g, we use the notations f (t) = O(g(t)) as t → ∞ when the function |f (t)/g(t)| is bounded for large t; and f (t) = o(g(t)) as t → ∞ if the function f (t)/g(t) tends to 0 as t → ∞. Proposition 3. Let T (t) be a bounded C 0 -semigroup on a Hilbert space H with generator A such that the imaginary axis iR lies in the resolvent set ρ(A) of A. For every fixed γ > 0, the following conditions are equivalent:

R(ib, A) L(H) = O(|b| γ ) as b → +∞ ; (27)
T (t)A -1 L(H) = O(|t| -1/γ ) as t → +∞ ; (28)
T (t)A -1 x H = o(|t| -1/γ ) as t → +∞ ∀x ∈ H . (29)
In particular, we are interested in the case γ = 4, that is the decay rate we will show for all the systems addressed in Section 1. Indeed, in the following sections we will prove for each of those systems the next stabilization result. (30)

Then, for every integer m ∈ N there exists C m > 0 such that

|U (t)| H = |e tA U 0 | H ≤ C m (1 + t) m/4 |U 0 | D(A m ) ∀t ≥ 0 , U 0 ∈ D(A m ) . (31)
Once condition (30) has been proved for each system under consideration in Section 1, we can conclude that the total energy decays polynomially at infinity, with respect to the regularity of the initial condition U 0 . In particular, for every

U 0 = (u 0 , u 1 , v 0 , v 1 ) ∈ D(A), E(U (t)) ≤ C (1 + t) 1/2 |U 0 | 2 H ∀t ≥ 0 . ( 32
)
Remark 1. i) Whether the decay rate in (32) is sharp or not, is related to the optimality of the estimate we give in the relation (30). Since, along the computations of the proof below, we try to get the finest estimate on the exponent γ (see equations ( 42)-( 75)), we conjecture that the estimate (32) is optimal for the systems at hand. ii) We can consider systems with more general coupling operators such as

u (t) + A 1 u(t) + Bu (t) + α 1 P v = 0 in H v (t) + A 2 v(t) + α 2 P * u = 0 in H ( 33
)
where P is a bounded linear coercive operator on H, P * is its adjoint operator, and hypothesis (H3) is replaced by (H3)' α 1 and α 2 are two real numbers such that 0

< α 1 α 2 < ω 1 ω 2 P 2 L(H) .
In the case α i > 0, the total energy of system (33) is defined by

E(U (t)) := α 2 E 1 (u, p) + α 1 E 2 (v, q) + α 1 α 2 P u, v + α 1 α 2 P * v, u
and still verifies the estimate of Theorem 1.

Indirect stabilization by resolvent estimates

In the following, we will denote by H k (Ω), H k 0 (Ω) the usual Sobolev spaces with norm

|u| H k =   Ω |p|≤k |D p u| 2 dx   1/2
where we have set

D p = ∂ p1 x1 . . . ∂ p d
x d for any multi-index p = (p 1 , . . . , p d ). Moreover, we will refer to C Ω > 0 as the largest constant such that the Poincars inequality

C Ω |u| 2 L 2 (Ω) ≤ |∇u| 2 L 2 (Ω) (34)
holds true for any u ∈ H 1 0 (Ω). In the following sections we set H := L 2 (Ω) and we define the bounded operator B : H → H by Bu = βu for all u ∈ H, for some positive constant β > 0.

Indirect stabilization of a wave-wave system

Consider the weakly coupled system of wave equations

         u tt -∆u + λu + βu t + αv = 0 in Ω × (0, +∞) v tt -∆v + αu = 0 in Ω × (0, +∞) ∂u ∂ν + σu = 0 = v on Γ × (0, +∞) u(0) = u 0 , u t (0) = u 1 , v(0) = v 0 , v t (0) = v 1 in Ω , (35)
where α ∈ R and λ, σ ≥ 0 with max(λ, σ) > 0. We can rewrite system (35) as ( 9)-( 14) (or (17), equivalently) introducing the operators

D(A 1 ) := {u ∈ H 2 (Ω) : ∂u ∂ν + σu (•, t) = 0 on Γ, t > 0}, A 1 u = -∆u + λu , D(A 2 ) := H 2 (Ω) ∩ H 1 0 (Ω) , A 2 v = -∆v .
Let µ 2 0 be the least eigenvalue of -∆ + λI with Neumann/Robin boundary condition, and λ 2 0 be the least eigenvalue of -∆ on H 2 (Ω) ∩ H 1 0 (Ω). So the assumption (H3) on α requires

0 < |α| < λ 0 µ 0 . (36)
For every

U ∈ H = H 1 (Ω) × L 2 (Ω) × H 1 0 (Ω) × L 2 (Ω), the energy associated to system (35) is E(U (t)) := 1 2 Ω u 2 t + |∇u| 2 + λu 2 + v 2 t + |∇v| 2 + 2αRe(uv) dx + 1 2 Γ σu 2 dΣ = |U | 2 H 2 , (37)
which, for every U ∈ D(A), satisfies

d dt E(U (t)) = - Ω βu 2 t dx = Re(AU (t), U (t)) H . (38)
Corollary 1 ensures that system (35) is strongly stable. In the sequel we want to show that condition (30) holds for system (35). In this way, we prove that the total energy of the system decays polynomially at infinity, with respect to the regularity of the initial condition U 0 . In particular, for every

U 0 = (u 0 , u 1 , v 0 , v 1 ) ∈ D(A), E(U (t)) ≤ C (1 + t) 1/2 |u 0 | 2 H 2 + |u 1 | 2 H 1 + |v 0 | 2 H 2 + |v 1 | 2 H 1 0 . ( 39
)
Proof of Theorem 1 for system (35). Thanks to Proposition 2 we know that iR ⊂ ρ(A). Thus, we need to show that

(ibI -A) -1 L(H) = O(|b| 4 ) as b → +∞ . ( 40
) Let U ∈ H and b ∈ R such that |b| ≥ max(1, β, λ). Since Rank(ibI -A) = H, there exists Z ∈ D(A) such that ibZ -AZ = U in H . (41)
Thus, the estimate (40) will hold once provided that there exists C α > 0 (depending on Ω and α but not on b) such that

|Z| ≤ C α |b| 4 |U | , (42)
where C α blows up as |α| goes to 0 or to λ 0 µ 0 , and, for simplicity of notation, we set |Z| = |Z| H . Denoting Z = (u, p, v, q) ∈ D(A) and U = (f, g, h, k) ∈ H, equation (41) reads as

         ibu -p = f in H 1 (Ω) ibp -∆u + λu + βp + αv = g in L 2 (Ω) ibv -q = h in H 1 0 (Ω) ibq -∆v + αu = k in L 2 (Ω) . (43)
We will proceed in several steps, evaluating each term of the norm |Z|.

Step 1: Estimate of |p| H and |bu| H . We first multiply by Z both sides of equation ( 41) and then take the real part of it. Thanks to the right identity in (38), we deduce that β|p| 2

H = Re(U, Z), so β|p| 2 H ≤ |U | |Z| . (44)
Then, from the first equation of system (43), and using equation ( 19), we deduce that

|bu| 2 H ≤ 2|p| 2 H + 2|f | 2 H ≤ 2 β |U | |Z| + 2 µ 2 0 |f | 2 H 1 (Ω) ≤ 2 β |U | |Z| + 2µ -2 0 1 -|α|(λ 0 µ 0 ) -1 |U | 2 , so |bu| 2 H ≤ 2 β |U | |Z| + K α |U | 2 , ( 45
)
where here and in the following K α denotes a generic constant depending on α and Ω, which blows up as |α| λ 0 µ 0 .

Step 2: Estimate of |∇u| 2 H + λ|u| 2 H + σ|u| 2 L 2 (Γ) . Consider the scalar product in H of the second identity in system (43) with u

Ω (ibp -∆u + λu + βp + αv)udx = Ω gudx .

Integration by parts leads to

Ω (|∇u| 2 + λu 2 )dx + Γ σu 2 dΣ = Re Ω (g -(ib + β)p -αv)udx . ( 46
)
We now evaluate each terms in the right-hand side integral. First,

Ω gudx ≤ |g| H |u| H ≤ 1 µ 0 |g| H Ω (|∇u| 2 + λu 2 )dx + Γ σu 2 dΣ 1/2 ≤ 1 2µ 2 0 |g| 2 H + 1 2 Ω (|∇u| 2 + λu 2 )dx + 1 2 Γ σu 2 dΣ ≤ K α |U | 2 + 1 2 Ω (|∇u| 2 + λu 2 )dx + 1 2 Γ σu 2 dΣ . (ib + β)pudx ≤ b 2 + β 2 4 |u| 2 H + |p| 2 H ≤ 1 2 |bu| 2 H + |p| 2 H ≤ 2 β |U | |Z| + K α |U | 2 .
Finally, thanks to (45),

Ω αvudx ≤ |α||u| H |v| H ≤ |α| |b| K α |Z| |bu| H ≤ |α|K α |b| |Z| |U | 1/2 |Z| 1/2 + K α |U | ≤ |α|K α |b| |U | 1/2 |Z| 3/2 + K α |U | |Z| .
Plugging the last three inequalities in equation ( 46), we derive that

Ω (|∇u| 2 +λu 2 )dx+ Γ σu 2 dΣ ≤ K α |U | |Z| + |U | 2 + |α| |b| |U | 1/2 |Z| 3/2 , ( 47
)
or, for a partial estimate,

Ω (|∇u| 2 + λu 2 )dx + Γ σu 2 dΣ ≤ 4 β |U | |Z| + K α |U | 2 + 2|α||u| H |v| H . ( 48
)
Step 3: Partial estimate of |∇v| H . From the third identity in system (43) we derive that q = ibv -h, and using this relation in the fourth equation of ( 43) we obtain -b 2 v -∆v + αu = ibh + k.

The scalar product by v of both sides of this relation gives, after integration by parts,

Ω |∇v| 2 dx = Ω b 2 v 2 dx + Re Ω (ibh + k -αu)vdx . ( 49
)
Since

Ω (ibh + k -αu)vdx ≤ |h| H |bv| H + |k| H |b| |bv| H + |α| |u| H |b| |bv| H ≤ |bv| 2 H + 1 4 (|h| 2 H + |k| 2 H ) + |α| 2 2b 4 |bu| 2 H ≤ |bv| 2 H + |α| 2 βb 4 |U | |Z| + K α |U | 2 ,
from equation (49) we deduce that

Ω |∇v| 2 dx ≤ 2|bv| 2 H + |α| 2 βb 4 |U | |Z| + K α |U | 2 . ( 50
)
Step 4: Estimate of |bv| H . Taking the inner product between the second equation in (43) and b 2 v, we have

Ω {[(ib + β)p -∆u + λu + αv]b 2 v}dx = Ω b 2 gvdx , so we have that Ω b 2 v 2 dx = b 2 α Re Ω [(g -(ib + β)p -λu)v -∇u. ∇v]dx . (51)
First, note that, thanks to (44), (45) and

|b| ≥ max(1, β, λ), b 2 α Ω (g -(ib + β)p -λu)vdx ≤ Ω 3b 2 v 2 4 + b 2 g 2 α 2 + b 4 p 2 α 2 + λ 2 b 2 u 2 α 2 dx ≤ 3 4 |bv| 2 H + K α α 2 b 2 |U | 2 + C α 2 b 4 |U | |Z| , ( 52
) where here and in the following C denotes a generic positive constant, independent from α and b. Second, observe that, thanks to (48) and (50), we find out that

b 2 α Ω ∇u. ∇vdx ≤ b 2 |α| |∇u| H |∇v| H ≤ b 2 |α| √ 2|α| 1/2 |u| 1/2 H |v| 1/2 H + 2 √ β |U | 1/2 |Z| 1/2 + K α |U | • √ 2|bv| H + |α| √ βb 2 |U | 1/2 |Z| 1/2 + K α |U | ≤ C b |α| 1/2 |bv| 3/2 H |bu| 1/2 H + b 2 |α| |U | 1/2 |Z| 1/2 |bv| H + K α |α| b 2 |U | |bv| H + |α| 1/2 |u| 1/2 H |v| 1/2 H |U | 1/2 |Z| 1/2 + |U | |Z| + K α |U | 3/2 |Z| 1/2 + K α |α| 1/2 b 2 |u| 1/2 H |v| 1/2 H |U | + K α |α| b 2 |U | 3/2 |Z| 1/2 + K α |α| b 2 |U | 2 . ( 53
)
Let ε be a positive real number. We recall Young's inequality in the form |xy| ≤ |x| p pq 1/q ε p/q + ε|y| q (54) that holds for every real numbers x, y and for a suitable pair of conjugate exponents (p, q). For p = 4 and q = 4/3, we deduce that

b |α| 1/2 |bv| 3/2 H |bu| 1/2 H ≤ ε|bv| 2 H + C ε b 4 α 2 |bu| 2 H ≤ ε|bv| 2 H + C ε b 4 α 2 |U | |Z| + K α,ε α 2 b 4 |U | 2 , (55)
where C ε and K α,ε are positive constants that diverge as ε goes to 0 + . Similarly, thanks to standard Young's inequality with parameter ε, we deduce that

b 2 |α| |U | 1/2 |Z| 1/2 |bv| H ≤ ε|bv| 2 H + C ε b 4 α 2 |U | |Z| , (56)
K α |α| b 2 |U | |bv| H ≤ ε|bv| 2 H + K α,ε α 2 b 4 |U | 2 , ( 57
)
|α| 1/2 |u| 1/2 H |v| 1/2 H |U | 1/2 |Z| 1/2 = (|bv| 1/2 H |U | 1/4 |Z| 1/4 ) |α| 1/2 |b| |bu| 1/2 H |U | 1/4 |Z| 1/4 ≤ |bv| H |U | 1/2 |Z| 1/2 + |α| 4b 2 |bu| H |U | 1/2 |Z| 1/2 ≤ ε|bv| 2 H + C ε |U | |Z| + 1 b 4 |bu| 2 H ≤ ε|bv| 2 H + C ε |U | |Z| + K α b 4 |U | 2 , ( 58
) K α |α| 1/2 b 2 |u| 1/2 H |v| 1/2 H |U | ≤ K α |α| 1/2 |b||U | |bu| H 2 + |bv| H 2 ≤ ε|bv| 2 H + K α,ε |α| b 2 |U | 2 + |bu| 2 H + K α |α| b 2 |U | 2 ≤ ε|bv| 2 H + K α,ε |α| b 2 |U | 2 + 2 β |U | |Z| . ( 59
)
Gathering estimates (55)-. . . -(59) in relation ( 53), we end up with

b 2 α Ω ∇u. ∇vdx ≤ 5Cε|bv| 2 H + C ε b 4 α 2 |U | |Z| + K α,ε α 2 b 4 |U | 2 + K α |α| b 2 |U | 3/2 |Z| 1/2 .
(60) Back to relation (51), owing to (52) and (60), we have

(1 -20Cε)|bv| 2 H ≤ K α,ε α 2 b 4 |U | 2 + C ε α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 ,
that, for a sufficiently small ε > 0, ensures that

|bv| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 . (61)
Step 5: Estimate of |∇v| H and |q| H . Using estimate (61), relation (50) yields

|∇v| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 . ( 62
)
On the other hand, by the third equation in system (43), we conclude that

|q| 2 H ≤ 2|bv| 2 H + 2|h| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 . ( 63
)
Final

Step: Resolvent estimate. Thanks to equations ( 44)-( 47)-( 62)-(63), we deduce that

|Z| 2 ≤ ν 2 (α) Ω p 2 + |∇u| 2 + λu 2 + q 2 + |∇v| 2 dx ≤ C α b 4 |U | 2 + b 4 |U | |Z| + b 2 |U | 3/2 |Z| 1/2 + |U | 1/2 |Z| 3/2 ,
where C α is a positive constant depending only on Ω and α (but not on b) that blows up as |α| goes to 0 or to λ 0 µ 0 . Applying again Young's inequality with suitable choices of conjugate exponents (p, q), we conclude that

|Z| 2 ≤ C α |b| 8 |U | 2 ,
that completes the proof of relation (42).

Observe that similar calculations, with suitably different space D(A 1 ) and total energy, ensure the same indirect stabilization result also for a system with different boundary conditions on different portion of the boundary, as in the case of system (1) with boundary conditions u(•, t) = 0 on Γ 0 , ∂u ∂ν (•, t) + σu(•, t) = 0 on Γ 1 v(•, t) = 0 on Γ t > 0 , with max(λ, σ) > 0, and where Γ 0 , Γ 1 are open subsets of Γ such that

Γ = Γ 0 ∪ Γ 1 , Γ 0 ∩ Γ 1 = ∅ .

Stabilization for a wave-Petrowsky system

We now focus on the stabilization problem for the weakly coupled system

     u tt -∆u + βu t + αv = 0 in Ω × (0, +∞) v tt + ∆ 2 v + αu = 0 in Ω × (0, +∞) u(0) = u 0 , u t (0) = u 1 , v(0) = v 0 , v t (0) = v 1 in Ω , (64)
for some β > 0 and α ∈ R, with Robin boundary conditions and

D(A 2 ) := {v ∈ H 4 (Ω) : v(•, t) = 0 = ∂v ∂ν (•, t) on Γ , t > 0} , A 2 v = ∆ 2 v (69) or D(A 2 ) := {v ∈ H 4 (Ω) : v(•, t) = 0 = ∆v(•, t) on Γ , t > 0} , A 2 v = ∆ 2 v , (70)
and defining (A, D(A)) as in (16). Let λ 2 0 be the least eigenvalue of -∆ with Robin boundary conditions, and µ 2 0 be the least eigenvalue of ∆ 2 with either clamped or hinged boundary conditions. So the assumption (H3) on α yields 0 < |α| < λ 0 µ 0 . For every

U ∈ H = H 1 (Ω) × L 2 (Ω) × H 2 0 (Ω) × L 2 (Ω)
, the energy associated to systems (64)-( 65)-(66) or (64)-( 65)-( 67) is

E(U (t)) = 1 2 Ω u 2 t + |∇u| 2 + v 2 t + |∆v| 2 + 2αRe(uv) dx + 1 2 Γ σu 2 dΣ (71)
which, for every U ∈ D(A), satisfies

d dt E(U (t)) = - Ω βu 2 t dx = Re(AU (t), U (t)) H . (72)
We now analyze the asymptotic behaviour of the contraction semigroup e tA with generator (A, D(A)).

Proof of Theorem 1 for systems (64)-( 65)-( 66) and (64)-( 65)-(67). Proposition 2 ensures that iR ⊂ ρ(A). Thus, systems (64)-( 65)-( 66) and ( 64)-( 65)-( 67) are strongly stable and we are left to show that

(ibI -A) -1 L(H) = O(|b| 4 ) as s → +∞ . ( 73
) Let U ∈ H and b ∈ R such that |b| ≥ max(1, β). Since Rank(ibI -A) = H, there exists Z ∈ D(A) such that ibZ -AZ = U in H . (74)
Thus, the estimate (73) will hold once provided that there exists C α > 0 (depending on Ω and α but not on b) such that

|Z| ≤ C α |b| 4 |U | , (75)
where C blows up as |α| goes to 0 or to λ 0 µ 0 . Denoting Z = (u, p, v, q) ∈ D(A) and U = (f, g, h, k) ∈ H, equation (74) reads as

         ibu -p = f in H 1 (Ω) ibp -∆u + βp + αv = g in L 2 (Ω) ibv -q = h in H 2 0 (Ω) ibq + ∆ 2 v + αu = k in L 2 (Ω) . (76)
Step 1: Estimate of |p| H and |bu| H . We first multiply by Z both sides of equation ( 74) and then take the real part of it. Thanks to the right identity in (72), we deduce that |βp| 2 H = Re(U, Z), so

|p| 2 H ≤ 1 β |U | |Z| , (77)
and consequently, from the first equation of system (76), we deduce that

|bu| 2 H ≤ 2 β |U | |Z| + K α |U | 2 , (78)
where the constant K α blows up as |α| λ 0 µ 0 .

Step 2: Estimate of

Ω |∇u| 2 dx + Γ σu 2 dΣ.
We can estimate this term of the energy as in the previous example, proving that, for |b| ≥ max(1, β),

Ω |∇u| 2 dx + Γ σu 2 dΣ ≤ K α |U | |Z| + |U | 2 + |α| |b| |U | 1/2 |Z| 3/2 , (79)
or

Ω |∇u| 2 dx + Γ σu 2 dΣ ≤ 4 β |U | |Z| + K α |U | 2 + |α||u| H |v| H . (80)
Step 3: Partial estimate of |∆v| H . From the third identity in system (76) we derive that q = ibv -h, and using this relation in the fourth equation of (76) we obtain ∆ 2 v -b 2 v + αu = ibh + k.

The scalar product by v of both sides of this relation gives, after integrations by parts,

Ω |∆v| 2 dx = Ω b 2 v 2 dx + Re Ω (ibh + k -αu)vdx . (81)
Thanks to (78) and the assumption |b| ≥ 1,

Ω (ibh + k -αu)vdx ≤ |h| H |bv| H + |k| H |b| |bv| H + |α| |bu| H b 2 |bv| H ≤ |bv| 2 H + |h| 2 H + |k| 2 H b 2 + α 2 2b 4 |bu| 2 H ≤ |bv| 2 H + α 2 βb 4 |U | |Z| + K α |U | 2 ,
so from equation (81) we deduce that

Ω |∆v| 2 dx ≤ 2|bv| 2 H + α 2 βb 4 |U | |Z| + K α |U | 2 . ( 82
)
Step

|∇v| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 . ( 89
)
On the other hand, by the third equation in system (76), we conclude that where C α is a positive constant depending only on Ω and α (but not on b) that blows up as |α| goes to 0 or to λ 0 µ 0 . Applying again Young's inequality with suitable choices of conjugate exponents (p, q), we conclude that

|Z| 2 ≤ C α |b| 8 |U | 2 ,
completing the proof of relation (75).

Aknownledgement

The author wish to thank Fatiha Alabau-Boussouira for suggesting this problem and for fruitful discussions.

Theorem 1 .

 1 Suppose hypotheses (H1)-(H2)-(H3) hold, and let (A, D(A)) the operator defined in(16)-(17). Assume moreover that R(ib, A) L(H) = O(|b| 4 ) as b → +∞ .

  t) + σu(•, t) = 0 on Γ , t > 0 (65) on u (for some σ > 0) and either clampedv(•, t) = 0 = ∂v ∂ν (•, t) on Γ , t > 0(66)or hinged boundary conditions v(•, t) = 0 = ∆v(•, t) on Γ , t > 0 (67) on v. We can rewrite systems (64)-(65)-(66) and (64)-(65)-(67) as (9) (or (17), equivalently) introducing the operatorsD(A 1 ) := {u ∈ H 2 (Ω) : ∂u ∂ν + σu (•, t) = 0 on Γ , t > 0} , A 1 u = -∆u (68)

|q| 2 H ≤ 2|bv| 2 H + 2|h| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 p 2 +

 224422 |U | 3/2 |Z| 1/2 . (90) Thanks to equations (77)-(79)-(89)-(90), we deduce that|Z| 2 ≤ ν 2 (α) Ω |∇u| 2 + q 2 + |∇v| 2 dx + Γ σu 2 dΣ ≤ C α b 4 |U | 2 + b 4 |U | |Z| + b 2 |U | 3/2 |Z| 1/2 + |U | 1/2 |Z| 3/2 ,

  H |U | 1/2 |Z| 1/2 + |U | |Z| + K α |U | 3/2 |Z| 1/2 |U | 3/2 |Z| 1/2 + K α |α| b 2 |U | 2 . (86)Let ε be a positive real number. Applying Young's inequality together with estimates (55)-. . . -(59), from (86) we conclude that |U | 3/2 |Z| 1/2 . Estimate of |∇v| H and |q| H . Using estimate (88) in relation (82) ensures that

	(83) α 2 b 4 |U | |Z| . (84) α 2 dx b 4 p 2 α 2 b 2 |U | 2 + b 2 g 2 α 2 + H + + |bv| 2 3b 2 v 2 4 ≤ 3 4 K α C Afterwards, thanks to the well-known inequality First, note that b 2 α Ω (g -(ib + β)p)vdx ≤ Ω Ω |∆v| 2 dx ∀v ∈ H 2 (Ω) ∩ H 1 0 (Ω) , (85) and owing to (80) and (82), we find out that b 2 α Ω ∇u. ∇vdx ≤ b 2 |α| |∇u| H |∇v| H ≤ b 2 |α| |∇u| H |∆v| H ≤ Cb 2 |α| |α| 1/2 |u| 1/2 H |v| 1/2 H + 2 √ β |U | 1/2 |Z| 1/2 + K α |U | • √ 2|bv| H + |α| √ βb 2 |U | 1/2 |Z| 1/2 + K α |U | ≤ C b |α| 1/2 |bv| 3/2 H |bu| 1/2 H + b 2 |α| |U | 1/2 |Z| 1/2 |bv| H + K α |α| b 2 |U | |bv| H + |α| 1/2 |u| 1/2 H |v| 1/2 + K α |α| 1/2 b 2 |u| 1/2 H |v| 1/2 H |U | + K α |α| α Ω ∇u. ∇vdx ≤ 5Cε|bv| 2 H + C ε b 4 α 2 |U | |Z| + K α,ε α 2 b 4 |U | 2 + K α |α| b 2 |U | 3/2 |Z| 1/2 . (87) Back to relation (83), owing to (84) and (87), we have (1 -20Cε)|bv| 2 H ≤ K α,ε α 2 b 4 |U | 2 + C ε α 2 b 4 |U | |Z| + K α |α| b 2 So, for a sufficiently small ε > 0, we deduce that |bv| 2 H ≤ K α α 2 b 4 |U | 2 + C α 2 b 4 |U | |Z| + K α |α| b 2 |U | 3/2 |Z| 1/2 . (88) b 2 b 2 Step 5:

4: Estimate of |bv| H . The inner product between the second equation in (76) and b

2 v gives Ω b 2 v 2 dx = b 2 α Re Ω [(g -(ib + β)p)v -∇u. ∇v]dx . Ω (v 2 + |∇v| 2 )dx ≤

This research has been performed in the framework of the GDRE CONEDP. Supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN, Grant agreement number 264735-SADCO


Alain Audouard

Fabienne Duc

Loïc Drigo

Pierre Toulemonde

Sandra Karlsson

Pierre Strobel

André Sulpice

Quantum oscillations and upper critical magnetic field of the iron-based superconductor FeSe

Keywords: 74, 70, Xa -Pnictides and chalcogenides, 71, 18, +y -Fermi surface: calculations and measurements; effective mass, g factor, 74, 25, Dw -Superconductivity phase diagrams

Introduction. -The discovery of iron-based superconductors [1][2][3][4] has reactivated the questioning about the interplay, either competition or cooperation, between magnetism and superconductivity in correlated electron systems. Indeed, as the temperature decreases, parent phases of iron-pnictide superconductors, such as BaFe 2 As 2 , undergo a tetragonal-orthorhombic transition closely linked to the condensation of a spin-density wave (SDW), i.e. to the development of an antiferromagnetic long range order. In contrast, concomitant decrease of the Néel temperature and superconducting critical temperature rise is observed on doping in both Ba(Fe 1-x Co x ) 2 As 2 (electron doped) or Ba 1-x K x Fe 2 As 2 (hole doped) [5,6], suggesting competition between SDW and superconductivity [7].

Even though no clear quantitative consensus has been reached yet for the iron-chalcogenide FeTe 1-x Se x phase diagram, it can be stated that superconductivity emerges as x increases (from x in the range 0 ∼ 0.3) from a SDW state, similarly to the case of pnictide compounds [8] even though Te and Se are isovalent. However, at variance with parent phases of iron pnictide superconductors, no long range magnetic order has been detected in the FeSe superconductor although the tetragonal-orthorhombic phase transition is observed at ∼ 90 K. Nevertheless, according to NMR data, antiferromagnetic spin fluctuations are strongly enhanced at temperatures close to T c [9] which suggests that spin fluctuations may nonetheless play an important role in superconductivity. In line with this statement, itinerant SDW instability have been proposed [10].

Within this picture, nesting properties, hence Fermi surface (FS) topology, may play a major role for superconductivity of iron-based superconductors [11]. According to band structure calculations based on density functional theory, the FS of FeSe is composed of 2 concentric quasi-two dimensional electron and 3 hole tubes, located at the corner and center of the first Brillouin zone, respectively, with their axis parallel to the c * direction [10,[12][13][14]. Although these tubes are significantly corrugated, imperfect nesting can be considered [10], as in the case of ironpnictide superconductors. However, strong discrepancy between band structure calculations and ARPES data has been reported [13]. Namely, much smaller tube areas have been observed and interpreted on the basis of a strong band renormalization and Fermi energy shift. In that respect, the tetragonal-orthorhombic transition at 90 K, connected to a nematic state with an orbital character, plays a significant role [15,16]. Nevertheless, it can be remarked that SDW with imperfect nesting would also lead to small tube area due to FS reconstruction.

Quantum oscillations study is a powerful tool to obtain information on the FS [11,17]. As an example, the de Haas-van Alphen oscillations spectrum of LaFe 2 P 2 , which is a non-superconducting parent of iron-pnictide superconductors, is in agreement with band structure calculations [18]. This feature, which demonstrates the absence of any nesting of the FS at low temperature, is in conflict with the above mentioned picture of competition between SDW and superconductivity. Oppositely, Shubnikov-de Haas oscillations with frequencies in the range 60 T to 670 T, corresponding to orbits area from 0.2 to 2.3 % of the first Brillouin zone (FBZ) area, much smaller than predicted by band structure calculations, have been very recently observed in FeSe [14]. Even lower frequencies, in the range 45-230 T had been previously reported for thin non-superconducting FeTe 1-x Se x crystals [19], this latter compound having similar FS topology.

Besides quantum oscillations, temperature dependence of the upper critical magnetic field H c2 may provide information regarding both the superconducting gap topology and the either single or multiband nature of the superconductivity which are both related to the FS topology [7,17]. As it is the case for e.g. FeTe 0.5 Se 0.5 , discrepancies can be observed within the few H c2 measurements reported for FeSe [14,20,21].

The aim of this article is to report on Shubnikov-de Haas oscillations to get more insight on the Fourier spectrum, hence the FS topology and band renormalization, and upper critical magnetic field measured by contactless tunnel diode oscillator technique (TDO) on high quality FeSe single crystals.

Experimental. -Studied single crystals have been grown using the chemical vapor transport method in sealed quartz tube, starting from Fe and Se powders (with a 1.1:1 molar ratio) in an eutectic KCl+AlCl 3 chlorides mixture as detailed in Ref. [22]. The temperature profile used was inspired from the previous work of Chareev et al. [23] and Böhmer et al. [24] with a gradient temperature of 120 • C maintained during 6 weeks between the hot zone (440 • C) and the cold zone (320 • C) of the furnace. The average composition of the obtained crystals was de- termined to be Fe 1.02(1) Se by EDX micro-analysis of the surface of different crystals in a SEM.

As reported in Ref. [25], the device for radio frequency measurements is a LC-tank circuit powered by a TDO biased in the negative resistance region of the currentvoltage characteristic. This device is connected to a pair of compensated coils made with copper wire (40 µm in diameter). Each of these coils is wound around a Kapton tube of 2 mm in diameter. The studied crystal, which is a platelet with dimensions of roughly 1.4 × 1.4 × 0.04 mm 3 , is placed at the center of one of them. The fundamental resonant frequency f 0 of the whole set is ∼ 24 MHz. This signal is amplified, mixed with a frequency f about 1 MHz below the fundamental frequency and demodulated. Resultant frequency ∆f = f -f 0 has been measured in zero-field and in pulsed magnetic fields of up to 55 T with a pulse decay duration of 0.32 s in the temperature range 1.5 K to 9 K. It has been checked that the data collected during the raising and the falling part of the pulse coincide, indicating that no discernible temperature change occurred during the field sweep. It must be kept in mind that the TDO frequency is sensitive to the resistivity, yielding Shubnikov-de Haas oscillations [25], and the London penetration depth [26], in the normal and superconducting states, respectively.

Results and discussion. -As previously reported for other iron-based superconductors [27,28], the zero-field TDO frequency displayed in Fig. 1 evidences a large increase as the temperature decreases linked to the decrease of the London penetration depth [26]. The onset of this frequency rise coincides with the onset of the zero-field cooled magnetization decrease and is therefore regarded as the superconducting transition temperature (T c = 8.6 ± 0.1 K).

Upper critical magnetic field. Field-dependent TDO frequency is displayed in Fig. 2 for magnetic field applied perpendicular (H c) and parallel (H ab) to the conduct- ing ab plane, at various temperatures. Marked transitions are observed for the two considered field directions. These data allow to reliably determine the temperature dependence of the upper critical magnetic field H c2 , reported in Fig. 3, in a similar way to the T c determination (see Fig. 1 and Ref. [28]). These data yield dH ab c2 /dT | T =Tc = -9.2 T/K and dH c c2 /dT | T =Tc = -2.3 T/K. Despite these values are by a factor of 1.4 higher than those deduced from resistivity data of Terashima et al. [14], they yield an anisotropy parameter γ = 4.0 at T = T c , in agreement with the data of Ref. [14]. This value is higher than the reported value of Braithwaite et al. (γ = 1.4) [29] and Vedeneev et al. (γ ∼ 1.5-2) [21]. This discrepancy could be attributed to better crystal quality in the two former cases.

In that respect, it has been evidenced that e.g. columnar defects induced by heavy ion irradiation decrease γ [30]. Otherwise, the measured γ value is also higher than the anisotropy parameter of FeTe 0.5 Se 0.5 crystals, determined with the same measurement technique (γ = 1.6) [28], suggesting a stronger two-dimensional character in the former

case.

Dashed lines in Fig. 3 are the best fits of the Werthamer, Helfand, and Hohenberg (WHH) model [31] to the data, assuming weak coupling [7]. In this framework, the temperature-dependent upper critical field is given by ln(1/t) = ψ(1/2 + h/2t) -ψ(1/2) where ψ is the digamma function, t = T /T c and h

= 4µ 0 H c2 /[π 2 (-dH c2 /dt) | t=1 ].
Orbital fields deduced within this framework (µ 0 H c c2 (0) = -0.693T c dH c2 /dT | T =Tc ) are µ 0 H c c2 (0) = 14 T and µ 0 H ab c2 (0) = 55 T. These values are lower than those deduced from TDO data of FeTe 0.5 Se 0.5 (µ 0 H c c2 (0) = 49 T and µ 0 H ab c2 (0) = 78 T) [28]. Hence, the deduced coherence lengths (ξ c = (φ 0 H c c2 (0))/(2π)/H ab c2 (0) = 1.2 nm and ξ ab = φ 0 /2πH c c2 (0) = 4.9 nm), which are close to the values deduced from resistivity measurements [14], are larger to that deduced from TDO data for FeTe 0.5 Se 0.5 . Even smaller coherence lengths are deduced from specific heat data of FeTe 0.5 Se 0.5 [32] suggesting larger effective mass, hence stronger renormalization for FeTe 0.5 Se 0.5 . The orbital field for H c yields, according to the Clogston formula [33], a superconducting gap ∆ = 1.1 meV , i. e. 2∆=3.1k B T c which is very close to the weak coupling BCS value (2∆=3.5k B T c ). This gap value is in agreement with muon-spin rotation [34] and specific heat data [35]. However, larger gaps are deduced from ARPES data [13] which suggests strong coupling instead.

While orbital effects account alone for the temperature dependence of H c2 for H c, Pauli pair breaking contribution must be included for H ab. In this case, the orbital 3) yielding µ 0 H c2 (0) = 23 T. Due to the contribution of the Pauli effect for H parallel to the ab plane, the anisotropy parameter decreases down to γ = 1.7 as the temperature goes to zero. Nevertheless, the still unexplained anisotropy inversion (in which H ab c2 is lower than H c c2 below ∼ 4 K), reported for FeTe 0.5 Se 0.5 [28,36,37] is not observed for FeSe. Noticeably, multiband superconductivity observed in muon-spin rotation data [34] and inferred from the almost linear temperature dependence of H c c2 and H ab c2 upturn at low temperature reported by Terashima et al. [14] is not detected in the temperature dependence of H c2 . This result suggests that despite the reported multiband nature of the Fermi surface, superconducting properties are dominated by one band, possibly due to strongly different diffusivities of the various bands [38] as discussed for 1111 and 122 arsenides by Lei et al. [39]. The different behaviour reported in Ref. [14] could be ascribed to the influence of vortices dynamics on conductivity data below T c . Shubnikov-de Haas oscillations. Field dependence of the oscillatory part of the TDO frequency, obtained by removing a monotonically field-dependent background (i.e. a polynomial with a constantly positive second derivative in the studied field range), and corresponding Fourier analysis are displayed in Fig. 4 for two directions of the magnetic field with respect of the conducting plane (θ = 0 and 11 • , where θ is the angle between the field direction and the normal to the conducting ab plane). Five frequencies are observed as reported in Table 1.

These frequencies correspond to orbits area in the range 0.2-2.3 % of the first Brillouin zone area. Similar and slightly smaller frequencies are observed in magnetoresistance data of FeSe [14] and FeTe 0.65 Se 0.35 [19], respectively. It should be noticed that less than 3 oscillations with the frequency F 1 are involved in the window field considered in Fig. 4 preventing any reliable data analysis, such as effective mass determination, for this Fourier component. This statement also holds for the data of Ref. [14]. Nevertheless, F 1 is close to the frequency labeled F γ , observed for FeTe 0.65 Se 0.35 in the field range ∼ 4-7 T (which involves about 4 oscillations) [19]. Therefore this very small frequency might be actually present in the oscillatory spectrum. Assuming circular orbits, band structure calculations predict electron and hole tubes with Fermi wave vector values corresponding to Shubnikov-de Haas frequencies of few thousands of Tesla [10,12,13]. In contrast, ARPES data at low temperature [13] evidence only one electron and one hole orbit with Fermi wave vectors k F = 0.18 Å-1 and k F = 0.05 Å-1 , respectively. Still assuming circular cross sections, these latter values correspond to frequencies of 1000 T and 80 T, respectively, which is roughly within the range of the frequency values observed in Fig. 4. Low temperature ARPES data of Refs. [15,16] yield additional orbits, due to the orthorhombic distortion, with Fermi energies of few tens of a meV, in agreement with the data in Table 1 as well. Taking FS warping into account, the number of frequencies observed should be twice the number of orbits due to necks and bellies. Besides, eventual presence of harmonics should be taken into account. Therefore, the oscillation spectrum should be more complex than reported in Table 1. Nevertheless, additional frequencies such as those labeled F ′ and F " in Fig. 4 cannot be excluded. As for harmonics, F 2 and F 3 could be the second harmonics of F 1 and F 2 , respectively. This hypothesis can be checked through the effective mass determination. Indeed, in the framework of the Lifshitz-Kosevich and Falicov-Stachowiak models [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF][START_REF] Falicov | [END_REF], the effective mass m * pi of the p th harmonics of the frequency F i is given by m * pi = p × m * i where m * i is the effective mass of the i orbit. Temperature dependence of the oscillation amplitude is displayed in Fig. 5a for the frequencies F 2 , F 3 and F 5 , yielding, in m e units, m * 2 = 0.75 ± 0.20, m * 3 = 2.0 ± 0.4 and m * 5 = 3.2 ± 0.6. Owing to the error bars, it cannot be excluded that F 3 is the second harmonics of F 2 . It should be noticed that such statement is at variance with previous data [14] for which it is reported that F 2 is the second harmonics of F 1 , instead, even though the effective mass relevant to F 1 cannot be reliably, determined as above discussed.

Otherwise, effective mass values m * 2 and m * 3 are significantly smaller than those, corresponding to frequencies F 2α and F β , respectively in Ref. [14], measured in a lower temperature range which remains to be understood. Nevertheless, the measured values are close or even slightly larger than those deduced from quantum oscillations of underdoped cuprates with similar frequency. For example, the effective mass linked to F 5 = 660 ± 5 T (m * 5 = 3.2 ± 0.6) can be compared to the effective mass of YBa 2 Cu 3 O 6.5 (m * = 1.9 ± 0.1 for F = 530 ± 20 T) and YBa 2 Cu 4 O 8 (m * = 2.7 ± 0.3 for F = 660 ± 30 T), respectively [42]. This result is in line with the strong renormalization of the effective mass of FeSe observed by ARPES measurements [13,15,16]. Finally, the Dingle temperature deduced from the field dependence of the amplitude relevant to the frequency F 5 (see Fig. 5b) is T D5 = 3.7 ± 0.8 K, yielding mean free path λ 5 = 17 ± 4 nm. The WHH model accounts for the temperature dependence of the upper critical magnetic field for magnetic field applied both parallel (H ab) and perpendicular to the conducting ab plane (H c). While the orbital contribution accounts for the data with H c, a Pauli limiting contribution is evidenced for H ab. The good agreement of the data with the WHH model suggests that superconducting properties of FeSe in magnetic field are mainly controlled by one band, only, despite the multiband nature of the Fermi surface. Finally, the anisotropy of the critical magnetic field close to T c (γ = 4) is higher than for FeTe 0.5 Se 0.5 . Besides, γ stays above 1 as the temperature goes to zero, in contrast to FeTe 0.5 Se 0.5 . The larger anisotropy of FeSe suggests stronger two-dimensionality.

Shubnikov-de Haas oscillations with frequencies in the range 50 T to 660 T, corresponding to orbits area from 0.2 to 2.3 % of the first Brillouin zone area have been observed in agreement with ARPES data at low temperature. Some of the effective masses, measured in the temperature range 1.5 -4.2 K, are significantly smaller than the recently reported values measured in a lower temperature range which remains to be understood [14]. Nevertheless, they are at least as large as in the case of underdoped cuprates. Owing to the small frequency values, these data account for the strong band renormalization deduced from ARPES data [13,15,16].

Fig. 1 :

 1 Fig. 1: (Color online) Temperature dependence of the zero-field TDO frequency and zero-field cooled magnetization. Construction lines for the determination of Tc are displayed.

Fig. 2 :

 2 Fig. 2: (Color online) Field dependence of the TDO frequency for H perpendicular (a) and parallel (b) to the iron conducting plane, respectively.

Fig. 3 :

 3 Fig. 3: (Color online) Temperature dependence of the upper critical magnetic field for H c (black symbols) and H ab (blue symbols). Dashed lines are the best fits of the WHH model to the data in the whole temperature range and the high temperature range for H c and H ab, respectively. Solid line is the best fit to the data for H ab including a Pauli contribution (µ0HP = 36.5 T).

Fig. 4 :

 4 Fig. 4: (Color online) (a) Field-dependent oscillatory part of TDO signal at various temperatures. (b) corresponding Fourier analysis in the field range 15-55 T and 25-55 T below and above 0.4 kT, respectively. Salient frequency peaks are marked as thin lines (see text). The temperature and angle θ between the field direction and the normal to the conducting plane is indicated in (b).

Fig. 5 :

 5 Fig. 5: (Color online) (a) Temperature dependence of the Fourier amplitude of components with frequency F2, F3 and F5. (b) Field dependence of the Fourier amplitude A5 at various temperatures. Solid lines are best fits of the Lifshitz-Kosevich model yielding effective masses (m2 = 0.75, m3 = 2.0 and m5 = 3.2, in me units) and Dingle temperature (TD5 = 3.7 K) in (a) and (b), respectively.

  Summary and conclusion. -Shubnikov-de Haas oscillations and magnetic field-and temperaturedependent superconducting transition of single crystalline FeSe have been studied by contactless tunnel diode oscillator-based measurements. In zero-field, the temperature dependence of the TDO frequency yields a superconducting transition temperature T c = 8.6 ± 0.1 K in agreement with magnetization data.

Table 1 :

 1 Frequency (Fi), effective mass (m * i ) and Fermi energy (EF ), calculated assuming parabolic dispersion, of the Fourier components with index i observed in the data of Fig.4. Pauli field µ 0 H P = 37 T, i.e. α = 2.1 (see solid line in Fig.

	i	F i (T)	m * i /m e	E F (meV )
	1 2 3 200 ± 10 ∼ 50 96 ± 6 4 ∼ 580 5 660 ± 5	0.75 ± 0.20 2.0 ± 0.4 3.2 ± 0.6	15 ± 5 12 ± 3 24 ± 5
	critical field is reduced as µ 0 H P = µ 0 H orb c2 / the Maki parameter is given by α = √ 2H orb √ 1 + α 2 where c2 /H P . A
	very good agreement with experimental data is obtained
	with a

The authors acknowledge the support of the European Magnetic Field Laboratory (EMFL) and the financial support of UJF and Grenoble INP through the AGIR-2013 contract of S. Karlsson.


Samir F Matar
email: matar@icmcb-bordeaux.cnrs.fr

Jean Etourneau

Naïm Ouaini

Crystal and chemical anisotropy effects in AE 2 ZnN 2 , (AE = Ca, Sr, Ba) from ab initio

Keywords: Nitrides, DFT, Compressibility, Cohesive energy, Elastic properties, Bonding

In the tetragonal ternary nitrides AE 2 ZnN 2 (AE = Ca, Sr, Ba), Zn has a rare linear coordination with nitrogen (N-Zn-N) along the c axis and N is located in an AE 5 Zn-like octahedron. Such features lead to original anisotropic crystal and chemical bond effects addressed herein with DFT-GGA based methods accounting for the cohesive energies, the energy volume equations of state EOS, the elastic constants and the properties of chemical bonding. Along AE = Ca, Sr, Ba an increasing overall compressibility under hydrostatic pressure is inferred from the respective EOS's, but the elastic constants C ij exhibit highly anisotropic features with large magnitudes of C 33 along the tetragonal c axis versus smaller C 11 = C 22 along a, b. Decreasing C 33 -C 11 along the title compounds signals decreasing anisotropy. The three ternaries are calculated as small gap insulators in agreement with experimental results of conductivity. The chemical bonding characterizes strong Zn-N bonds versus weaker AE-N thus reflecting the crystal anisotropy.

Introduction and context.

Nitrides offer a broad range of chemical compounds with interesting physical properties mainly related with the covalent character of the host elements with nitrogen. This is illustrated for instance by the extreme hardness of light element nitrides within the B-C-N diagram such as the cubic boron nitride c-BN which replaces diamond in many tooling machines either as bulky materials or more recently as thin films [1], the ternary BC2N [2] and the theoretically devised C11N4 [3]. Also the refractory character of transition metal mononitrides [4] and of SiCN ceramics [5] is of large interest in industry.

AE2ZnN2 (AE = Ca, Sr, Ba) ternaries were synthesized by DiSalvo group in the 1990's [6,7].

They form the first nitrides with the A2HgO2 (A= alkaline element) archetype prepared earlier by Hoppe and Röhrborn in 1964 [8]. The structure is body centered tetragonal with large c/a ratio of 3.1 -3.5. It is characterized by a rare linear coordination of Zn to nitrogen: N...Zn...N and an AE5Zn irregular octahedron surrounding nitrogen with five long dAE-N and one short dZn-N (cf. Fig. 1). This leads to ionocovalent compounds with chemical generic formulation: 2AE 2+ {N...Zn...N} 4-where ZnN2 4-is a complex linear anion.

The directional N...Zn...N bonds along the tetragonal c-axis are likely to exhibit original anisotropic mechanical as well as peculiar chemical bonding effects leading to specific physical properties which are examined herein by calculating the electronic structure and energy derived quantities of the three compounds within the quantum mechanical framework of the density functional theory DFT [9]. Two computational methods were used in a complementary manner. The Vienna ab initio simulation package (VASP-PAW) code [10,11] allows accurate protocols of geometry optimization, cohesive energies, energy volume equations of state (EOS) and the calculation of the complete set of elastic constants. Then a full account of the electronic structure and of the properties of chemical bonding (not implemented in VASP) is done with full potential all electrons augmented spherical wave (ASW) method [12,13]. The DFT exchange-correlation XC effects were considered following the generalized gradient approximation (GGA) according to the scheme of Perdew, Burke and Ernzerhof (PBE) [14]. Details on the methods can be found in a recent work [15] and therein references.

Geometry optimization and energy based results

Geometry optimization and cohesive energies.

Starting from the experimental structure parameters of AE2ZnN2, AE = Ca, Sr, Ba [6,7] given in Table 1, unconstrained geometry optimization runs were carried out. In all three ternary compounds the tetragonal symmetry in 𝐼4/𝑚𝑚𝑚 space group was preserved after successive calculations with increasing precision of the Brillouin zone integration.

As shown in Table 1, the fully relaxed structure parameters are found close to the starting ones and the shortest interatomic distances are in fair agreement with experiment. The distance d(Zn-N )= 1.84 Å is almost constant throughout the series and smaller than d(AE-N) which ranges from 2.4 Å (AE = Ca), to 2.54 Å (AE = Sr) and 2.60 Å (AE = Ba) whence the volume increase (Table 1). The calculated volumes present smaller magnitudes than experimental values but it will be shown in next paragraph that a better agreement is obtained from the energy-volume equations of states (EOS) fit values (cf. Fig.

2).

The last lines of Table 1 report the total and cohesive energies. The latter are obtained from:

Ecoh. = Etotal(compound) - E(constituents) for one FU.
The energies of the constituents were calculated in their ground state structures using available PAW-GGA potentials: FCC for AE with ECa (including p semi-core state) = -1.954 eV/atom, ESr (including s and p semi-core state) = -1.677 eV/atom EBa (including s and p semi-core state)= -1.911 eV/atom; hexagonal EZn = -1.250 eV/atom and N2 in a large cubic box, EN2. = -16.136 eV. The trend of decreasing cohesive energies along the series results not from the Zn-N constant separation but from the increasing AE-N distance as shown above.

Energy-volume equations of states

In view of the increasing lattice constants and subsequently of the volume from Ca to Sr then to Ba, it becomes relevant to examine the structure response to volume change by obtaining the energy-volume equations of state (EOS) for the three compounds. This proceeds from the E,V (energy-volume) set of calculations around the minima found from the geometry optimization (Table 1). The resulting

E = f(V) curves are shown in Figs. 2.
They have a quadratic variation which can be fitted with Birch EOS up to the 3 rd order [16]:

E(V) = Eo(Vo) + (9/8)VoBo[([(Vo)/V]) [2/3] B'-1] 2 + [9/16]Bo(B ' -4)Vo[([(Vo)/V]) [ 2/3] -1] 3 ,
where Eo, Vo, Bo and B ' are the equilibrium energy, the volume, the bulk modulus and its pressure derivative, respectively. In the inserts  2 values designating the goodness-of-fit are small enough to provide confidence to the fit results especially for the close values of equilibrium energies E0 to the ones obtained by energy optimization and for the volumes V0 which show a better agreement with the experiment than the geometry optimization results (Table 1). All three B' values are close to 4, i.e. within range of usually obtained values [17]. The respective Bo values have small magnitudes and they are found in the range of compressible metals and alloys [17] i.e., far from the magnitudes found in ultrahard nitrides as BC2N and C11N4 whose bulk modules are in the range of 300-400 GPa [2,3]. Also the B0 values decrease within the series: The compound with the largest volume (AE = Ba) becomes most compressible under hydrostatic pressure. However these observations do not provide sufficient information on the effects of the structure anisotropy reflected by the linear N…Zn…N entities along the c-axis . A detailed description can be provided through the calculation of the set of elastic constants Cij

Elastic constants

The elastic properties are determined by performing finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship. In tetragonal symmetry there are six independent elastic stiffness constants C11= C22, C33, C44, C66, C12, and C13. Most encountered compounds are polycrystalline where monocrystalline grains are randomly oriented so that on a large scale, such materials can be considered as statistically isotropic.

They are then completely described by the bulk modulus B and the shear modulus G, which may be obtained by averaging the single-crystal elastic constants. The most widely used averaging method of the elastic stiffness constants is the method of Voigt [START_REF] Matar | Matériaux ultra-durs: Concepts et Modélisations[END_REF] based on a uniform strain. The calculated elastic constants are given in Table 2. All Cij values are positive and their combinations: C11 > C12, C11C33 > C13 2 and (C11+C12)C33 > 2C13 2 obey the rules pertaining to the mechanical stability of all three title compounds. Focusing on C11, C22 and C33 translating stresses along the three orthogonal axes, it can be observed that while C11 = C22 (as expected in tetragonal symmetry where a = b) a much larger C33 (along the c axis) magnitude is found for all three compounds whereby the crystal lattice should be harder along the c-axis along which the N-Zn-N linear entities with short Zn-N connections are aligned, so that the Ca2ZnN2 compound presents an implicate hardness closer to the range of covalent nitrides [2,3,8]. Oppositely it is soft along the a, b directions.

However the difference between C11 and C33 decreases along the series. This can be interpreted in terms of largest anisotropy for Ca2ZnN2 and the lowest anisotropy for Ba2ZnN2.

The bulk (BV) and shear (GV) modules following Voigt are formulated as:

BV = 1/9 {2(C11 + C12) + 4C13 + C33} and GV = 1/30 {12C44 + 12C66 + C11 + C12 + 2C33 -4C13}
The numerical values (Table 2) show good agreement of BV with B0 obtained from EOS fits (Fig. 2), thus showing the complementary description of the compounds in the two approaches. The shear modulus which defines the rigidity of the material is significantly lower. The Pugh's GV/BV ratio [19] is an indicator of brittleness or ductility for GV/BV > 0.5 and GV/BV < 0.5, respectively. GV/BV ratio is found to decrease from 0.57 in Ca2ZnN2 to 0.43 in Ba2ZnN2, indicating reduced brittleness along the series. Note that for Ba2ZnN2 the GV/BV ratio approaches the value observed experimentally for the coinage metals Ag, Pt, or Au which possess GV/BV ratios in the range of 0.4-0.2 [20].

To conclude on this section, it has been shown that along AE = Ca, Sr, Ba, peculiar properties pertaining to decreasing cohesive energies and incompressibilities as well as decreasing anisotropy characterize the AE2ZnN2 compounds.

Electronic structure and chemical bonding

In as far as the calculated structure parameters are close to experiment for AE2ZnN2 (Table 1) we use the latter for the analysis of the electronic structure with all electrons calculations with the full potential ASW method [12,13]. In the minimal basis set used by the ASW method the valence states and the matrix elements were constructed using partial waves up to lmax+1 = 3 for AE and Zn and lmax+1 = 2 for N. Also the Zn(3d 10 ) filled subshell was not considered as part of the valence basis set of Zn and replaced by 4d 0 . The chemical bonding for pair interactions is analyzed qualitatively using the crystal orbital overlap populations COOP based on the overlap matrix elements Sij [21]. In the plots positive, negative and zero magnitude COOP indicate bonding, anti-bonding and non bonding interactions respectively. At self consistent energy and charge convergence the transfer of electrons follows the trend discussed above of positively charged AE and Zn and negatively charged N but does not translate the ionic of 2AE 2+ {N...Zn...N} 4-which is a formal picture rarely observed in the solid state. The site projected densities of states (PDOS) are shown in Fig. 3. The zero energy along the x-axis is with respect to the top of the valence band (VB) EV and the compounds are insulating -or small gap semi-conductors-with a ~2 eV band gap for Ca and Sr compounds and a reduced gap of ~1 eV for the barium compound. For Ca2ZnN2 the result agrees with the experimental observations of conductivity and magnetic measurements but no gap magnitude was provided [6].

As a general feature the lower part of the VB around -12 eV is occupied by N 2s states which show similar shape with the PDOS of AE and Zn. This is also observed for the pblock from -4 to EV. The large magnitude of AE and N PDOS as with respect to Zn arises from their twice larger number but the similar PDOS shapes are signatures of the quantum mixing between the valence states leading to the chemical bonding. The conduction band (CB) in the range of 2-6 eV is dominated by the empty AE states resulting from the positive ionization. In the case of barium, the broad band within the CB due to slike has the main effect of reducing the band gap as with respect to the Ca dn Sr compounds. Also the p-block PDOS show a progressive localization along the series especially for the intense peak at ~2eV relative to N-pz as inferred from a projection of Np PDOS.

In view of the coordination polyhedron of N in AE5Zn like octahedron (cf. Fig. 1) and of the short Zn-N distance versus AE-N ones, Figs. 4 show the COOP for the AE-N and Zn-N different interactions. In order to establish comparisons, one atom of each kind is considered in the plots. The whole VB is of bonding nature with positive COOP magnitude and the CB is antibonding thus illustrating the largely negative cohesive energies (Table 1).

Clearly the Zn-N bond is dominating over AE-N in the p-block just below EV and small bonding s-like COOP contribution, larger for AE-N is found at the lower part of the VB. The sharp peak at -2 eV corresponds to the directional N-pz states. Such anisotropic bonding Table 1. Calculated and experimental [4] (in parentheses) lattice and atomic parameters for AE2ZnN2, (AE = Ca, Sr, Ba). Space group I4/mmm with Zn at (2a): 0,0,0, AE and N at (4e): 0,0,z. Total electronic and cohesive energies are given as per formula unit (FU).

AE2ZnN2

Fig. 1 :Fig. 2 :

 12 Fig. 1: Crystal structure of AE 2 ZnN 2 (AE = Ca, Sr, Ba) family highlighting the linear N-Zn-N connections.

Fig. 3 :

 3 Fig. 3: AE 2 ZnN 2 : Site projected DOS for AE= Ca (top), AE= Sr (middle) and AE= Ba (bottom).

Fig. 4 :

 4 Fig. 4: AE 2 ZnN 2 , AE =Ca (top), Sr (middle), Ba (bottom): Atom-to-atom bonding magnitudes between AE-N and Zn-N.

Table 2

 2 AE2ZnN2, (AE = Ca, Sr, Ba: Elastic constants Cij in GPa units and the bulk and shear modules.

		AE = Ca		AE = Sr	AE = Ba
	a (Å)	3.56 (3.58)		3.83 (3.86)	4.12 (4.15)
	c (Å)	12.56 (12.66)	12.82 (12.93)	12.90 (13.05)
	V (Å 3 )	159.18 (162.26) 188.06 (192.65) 218.97 (224.75)
	z AE	0.303 (0.302)	0.342 (0.341)	0.348 (0.344)
	z N	0.147 (0.146)	0.144 (0.145)	0.143 (0.141)
	E Tot. (eV)/FU -25.47		-24.02	-23.75
	E coh. (eV)/FU -4.17		-3.27	-2.54
	AE2ZnN2 AE = Ca	AE = Sr	AE = Ba
	C11 ; C22	158.8	130.76	97.38
	C12	36.40	35.99		34.78
	C13	45.65	41.65		33.07
	C33	235.65	180.98	130.74
	C44	59.66	53.95		38.74
	C66	28.35	13.18		1.95
	C33-C11	76.85	50.20		33.36
	BV	89.85	76.57		58.59
	GV	51.33	39.96		24.99
	GV/BV	0.57	0.52		0.43

Acknowledgement.

Computations were conducted on the MCIA-University of Bordeaux computers. Support from Conseil Régional d'Aquitaine is gratefully acknowledged.

is at the origin of the large C33 elastic constant identified in the three compounds. They provide brittle mechanical properties which are reduced upon volume increase along the series.

To conclude on this section the results obtained from the geometry optimization are complemented by a description of the title compounds as insulating with ~2eV band gap.

The crystal anisotropy is well reflected by the larger Zn-N versus AE-N bonding.