Syed Ali Ajwad
email: syed.ali.ajwad@univ-poitiers.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Michael Defoort
email: michael.defoort@uphf.fr

Tomas Ménard
email: tomas.menard@unicaen.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Output-feedback formation tracking of second-order multi-agent systems with asynchronous variable sampled data

Keywords: Formation tracking, directed topology, timevarying formation, sampled-data control, asynchronous sampling, continuous-discrete time observer

This paper deals with the problem of time-varying formation tracking for second-order multi-agent systems under directed topology, where the follower states form a desired formation while tracking the state of the leader. It is considered that each agent, including the leader, has second-order dynamics and can only transmit its position to its neighbors. The velocities and inputs are not exchanged between neighboring agents. In this work, it should be mentioned that contrary to many existing schemes, asynchronous and aperiodic sampling is considered. For each agent, an observer is proposed to estimate its state and the state of its neighbors from the available local asynchronous and aperiodic sampled position data. Using these estimates, a time-varying formation tracking protocol is developed. The stability of the closed-loop system which combines the continuous-discrete time observer and the formation tracking controller is analysed using an appropriate Lyapunov function. The effectiveness of the proposed outputfeedback controller is illustrated for various formations through simulation results.

I. INTRODUCTION

Multi-Agent Systems (MAS) have gained much attention in the last decades due to their application in a vast range of areas such as exploration, defence and rescue to name a few. MAS have also attracted the attention of researchers because of the theoretical challenges that arise in their control and coordination. Mostly, such challenges crop up while controlling the MAS in the absence of any centralized coordination mechanism. Distributed control, based on locally available information, provides much better performance for MAS in terms of scalability, efficiency and reliability.

Formation control has been one of the most attractive/appealing topics within the domain of MAS. It has a broad potential of applications in various areas like surveillance, heavy payload transportation, telecommunication relay, cooperative localization and environmental monitoring [START_REF] Wang | Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots[END_REF]. For the formation tracking problem, the followers have to form a desired geometric shape which could be time-varying while the whole formation tracks the trajectory generated by a leader (virtual or not).

Different formation control algorithms have been proposed and can be generally categorized as virtual structure method, leader-follower approach and behaviour based approach [START_REF] Dong | Time-varying formation control for general linear multi-agent systems with switching directed topologies[END_REF], [START_REF] Ren | Consensus based formation control strategies for multivehicle systems[END_REF]. Furthermore, many results have been proposed to derive consensus algorithms for MAS which provide distributed cooperation among the agents to reach some common state using only local information interactions [START_REF] Fu | Finite-time consensus for secondorder multi-agent systems with input saturation[END_REF], [START_REF] Yu | Finite-time consensus for second-order multiagent systems with disturbances by integral sliding mode[END_REF], [START_REF] Lin | Distributed consensus of second-order multiagent systems with nonconvex velocity and control input constraints[END_REF]. Consensus protocols can be applied to achieve formation tracking by choosing appropriate states on which the consensus is reached. Such a control technique is considered more reliable and robust as compared to other approaches especially in the case of failure of some agents [START_REF] Dierks | Neural network control of mobile robot formations using rise feedback[END_REF]. Furthermore, in [START_REF] Ren | Consensus strategies for cooperative control of vehicle formations[END_REF], it has been shown that the earlier mentioned approaches can be incorporated in the consensus based ones.

In [START_REF] Du | Finite-time formation control of multiagent systems via dynamic output feedback[END_REF], a consensus protocol has been proposed to solve the finite-time formation tracking problem, with stationary and moving formations. A distributed formation control law for second-order multi-agent systems has been discussed in [START_REF] Gu | Distributed network-based formation control[END_REF]. Although there exist many approaches to solve the formation tracking problem, only a few works consider the timevarying formation tracking problem. In [START_REF] Lu | Formation control for secondorder multi-agent systems with time-varying delays under directed topology[END_REF], a formation control algorithm has been proposed for second-order MAS with directed topology and time varying delays. In [START_REF] Xiao | Finite-time formation control for multi-agent systems[END_REF], the case of time-invariant and time-varying finite-time formation tracking problem has been studied. Other examples of timevarying formation protocols can been found in [START_REF] Dong | Time-varying formation tracking for second-order multi-agent systems subjected to switching topologies with application to quadrotor formation flying[END_REF], [START_REF] Dong | Time-varying formation tracking for linear multiagent systems with multiple leaders[END_REF], [START_REF] Ge | Time-varying formation tracking of multiple manipulators via distributed finite-time control[END_REF].

In the above discussed works, it is generally assumed that each agent's information state is continuously available to its neighbors. However, in practice, continuous measurement and transmission of the whole state is a difficult and costly procedure. Furthermore, due to the digital nature of the onboard equipment, each agent only receives asynchronous and aperiodic sampled position data from its neighbors. Among the different methods to reduce the microprocessor and network loads while guaranteeing the desired control objectives, event-triggered control (ETC) schemes, (where sensors send information to the controller when specific events occur), have been widely studied [START_REF] Zhu | Event-based consensus of multiagent systems with general linear models[END_REF]. This technique involves sensors sending information to the controller when only specific events occur. However, most of these schemes require that each agent's information state is continuously available to its neighbors. Recently, the conditions of continuous state availability are relaxed in [START_REF] Garcia | Decentralized event-triggered consensus with general linear dynamics[END_REF], [START_REF] Dolk | Event-triggered consensus seeking under non-uniform time-varying delays[END_REF] whereas the agents' states are measurable. This condition has been relaxed in [START_REF] Yang | Decentralized eventtriggered consensus for linear multi-agent systems under general directed graphs[END_REF] where an appropriate triggering function has been designed to update the output.

In the current paper, the objective is not to design an appropriate triggering function such that the formation tracking problem is achieved, but to design an outputfeedback formation tracking controller. Motivated by [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] where the consensus problem for second-order MAS with arbitrary asynchronous and aperiodic sampling periods has been investigated, an observer-based algorithm has been proposed to solve the distributed formation tracking problem of second-order MAS. Compared to the existing works, the contributions are as follows. Firstly, this paper is a first attempt to deal with time-varying formation tracking problems under aperiodic and asynchronous sampling, where the followers form a desired time-varying geometric shape while the whole formation tracks the trajectory generated by a leader. Secondly, a novel output-feedback controller is proposed. Indeed, it is clear that the results in [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF] cannot be directly applied due to the presence of the position and velocity offsets from the desired time-varying geometric shape in the tracking errors. Lastly, a new stability proof of the closed-loop system which combines the continuousdiscrete time observer and the formation tracking controller is derived. Simulations have been carried out for different scenarios (including time-invariant and time-varying formation tracking). The corresponding simulation results have demonstrated the efficiency of the proposed scheme.

The rest of the paper is as follows. In section II, some preliminaries are provided. Section III introduces the formation tracking algorithm along with its stability analysis. Simulation results are presented in section IV while a conclusion is given in section V.

II. PRELIMINARIES ON GRAPH THEORY

The communication topology among the agents in a MAS can be described by a directed graph. A directed graph G is a pair (V, E), where V represents a finite nonempty set of nodes and E ⊆ V × V is a set containing all the edges. An edge is an ordered pair of distinct nodes. For a given edge (i, j) in a directed graph, node j can receive information from node i. Therefore, node j is called the child node whereas i is the parent node. A sequence of edges in a directed graph is known as a directed path. A graph has a directed spanning tree if each node has at least one parent node except for one node, which is called the root and has a directed path to all other nodes in the graph. The adjacency matrix

A = (a ij ) ∈ R N ×N of G with N nodes is defined by a ij = 1 if (j, i) ∈ E and a ij = 0 otherwise. The Laplacian matrix L ∈ R N ×N is defined as l ii = j =i a ij , l ij = -a ij for i = j.
In this paper, the communication connection between the N followers is described by a directed graph G. This is more suitable for various practical applications especially when some agents are equipped with receivers only. The matrix H is a nonsingular M-matrix if and only if Assumption 1 holds [START_REF] Song | Pinning-controllability analysis of complex networks: an m-matrix approach[END_REF]. Furthermore, there exists a diagonal matrix Ω = diag(ω 1 , . . . , ω N ) such that H T Ω + ΩH > 0 [START_REF] Zhang | On constructing lyapunov functions for multi-agent systems[END_REF]. Let us denote the smallest eigenvalue of H T Ω + ΩH as ρ and maximum and minimum eigenvalues of Ω as ω max and ω min respectively.

III. MAIN RESULTS

A. Problem statement

Consider a group of N followers labeled from 1 to N and one leader labeled 0 (which could be virtual).

The followers have the following second-order dynamics:

ṙi (t) = v i (t), i = 1, . . . , N vi (t) = u i (t) (1)
where r i , v i ∈ R m represent respectively the position and the velocity of the i-th agent and m ∈ N. The dynamics of the leader is given by:

ṙ0 (t) = v 0 (t) v0 (t) = u 0 (t) (2)
where r 0 , v 0 ∈ R m represent respectively the position and the velocity of the leader and m ∈ N. Assumption 2: The leader input u 0 (t) is bounded, that is, there exists a constant δ 0 ≥ 0 such that u 0 (t) ≤ δ 0 for all t ≥ 0.

In this paper, it is assumed that each agent only measures its position r i (not its velocity v i nor its control input u i ). Furthermore, it is also considered that the agent position is transmitted asynchronously with nonuniform sampling periods. The sampling instant at which the position information is transmitted from agent j to agent i (if there is an edge (j, i) between these two agents) is denoted by t i,j k where k ∈ N, i = 1, . . . N and j = 0, . . . N . Each agent in the network has different sampling instants which are completely independent from all other agents.

Assumption 3: There exists a maximum sampling period τ M > 0 such that 0 < t i,j k+1 -t i,j k < τ M , for all k ∈ N, i = 1, . . . N and j = 0, . . . N .

The desired time-varying geometric shape that the followers have to form is defined by

f (t) = [f 1 (t) T , . . . , f N (t) T ] T , where the formation vector f i (t) = [f i,r (t) T , f i,v (t) T ] T for follower i ∈ {1, . . . , N } satisfies ḟi,r (t) = f i,v (t) (3)
f i,r (t), f i,v (t) ∈ R m corresponds to the position and velocity offset, respectively. It is worth noting that f i (t) does not represent global formation coordinates but relative offset vectors with respect to the leader. Figure 1 illustrates the notation of formation vector. In this example, the desired geometric shape is a square around the leader labeled 0, in an xy-plane i.e. m = 2. The corresponding formation vectors are

f 1 = [1, 1, 0, 0] T , f 2 = [1, -1, 0, 0] T , f 3 = [-1, 1, 0, 0] T and f 4 = [-1, -1, 0, 0] T .
Since it is a time invariant formation, the corresponding velocity offset components in the formation vectors are zero. Figure 1 provides a graphical representation of the desired formation in the leader frame (x 0 , y 0 ). Assumption 4: It is considered that each agent already has the knowledge of the formation geometry f (t).

x y y 0 x 0 0 1 2 3 4 f 1,r f 2,r f 3,r f 4,r
Remark 1: The information of formation geometry is provided beforehand to the followers and is not shared during the tracking process.

Definition 1: The formation tracking problem is said to be practically solved if there exists ¯ > 0 such that

lim sup t→+∞ (x i (t) -f i (t) -x 0 (t)) ≤ ¯ (4)
where

x i (t) = [r i (t) T , v i (t) T ] T and x 0 (t) = [r 0 (t) T , v 0 (t) T ] T .
The objective of this paper is to design the protocol u i (t), i ∈ {1, . . . , N } such that the MAS (1)-( 2) achieves the practical time-varying formation specified in Definition 1 using asynchronous and aperiodic sampled position data.

B. Output-feedback formation tracking controller

Here, a time-varying formation tracking controller is designed and analyzed. First, for each follower, a continuousdiscrete time observer is proposed to estimate its state and the state of its neighbors from the available local asynchronous and aperiodic sampled position data. Using these estimates, an output-feedback formation tracking protocol is developed.

Similarly to [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], for each follower i ∈ {1, . . . , N }, the following continuous-discrete time observer is used: ṙi,j (t) = vi,j (t) -2θe -2θ(t-t i,j k ) ri,j (t i,j k ) -r j (t i,j k ) (5) vi,j (t) = ḟj,v (t) -θ 2 e -2θ(t-t i,j k ) ri,j (t i,j k ) -r j (t i,j k ) [START_REF] Lin | Distributed consensus of second-order multiagent systems with nonconvex velocity and control input constraints[END_REF] where j = 0, 1, . . . , N , t ∈ t i,j k , t i,j k+1 , k ∈ N and θ > 0 is the observer tuning parameter. Moreover, ḟ0,v = 0 as leader does not have any offset. Using the available local asynchronous and aperiodic sampled position data, this observer guarantees the estimation of the state of agent j by agent i. ri,j and vi,j are the estimated position and speed of the agent j by the agent i. The initial conditions ri,j (0), vi,j (0) ∈ R m of the observers can be chosen arbitrarily.

Using these estimates, for each follower i ∈ {1, . . . , N }, the formation tracking algorithm is proposed as

u i (t) = ḟi,v (t) (7)
-cλ 2 N j=1 a ij [r i,i (t) -f i,r (t) -ri,j (t) + f j,r (t)] -c2λ N j=1 a ij [v i,i (t) -f i,v (t) -vi,j (t) + f j,v (t)] -cλ 2 b i [r i,i (t) -f i,r (t) -ri,0 (t)] -c2λb i [v i,i (t) -f i,v (t) -vi,0 (t)]
where c > 0 is the coupling strength and λ > 0 is the controller tuning parameter. Remark 2: From Definition 1, the tracking error can be defined as e i = x i -f i (t) -x 0 for i ∈ {1, . . . , N }. It should be noted that due to the presence of the offset f i , the controller ( 7) is different from the one given in [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF]. Therefore, it makes the proof of the closed-loop stability which combines the continuous-discrete time observer and the formation tracking controller relatively complicated.

Theorem 1: Assuming that Assumptions 1-4 are satisfied. There exist constants ¯ > 0 and ∈ (0, 1) such that if the observer and controller gains verify

θ ≤ ¯ τ M (8) c ≥ ω max ρ (9) λ = θ ( 10
)
red such that λθ > 1, then the time-varying formation tracking problem is solved in the sense of Definition 1 using the output-feedback controller ( 5)- [START_REF] Dierks | Neural network control of mobile robot formations using rise feedback[END_REF].

Proof: Let us provide a sketch of proof for Theorem 1 inspired from [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF]. The agent dynamics can be re-written as

ẋi = Ax i + Bu i i = 0, . . . , N r i = Cx i with A = 0 m I m 0 m 0 m , B = 0 m I m and C = I m 0 m .
Denoting xi,j = (r T i,j , vT i,j ) T , the observer ( 5)-( 6) can be written as ẋi,j (t) =Ax i,j (t) -θ∆ -1 θ K o e -2θ(t-κi,j (t)) (r i,j (κ i,j (t)) -r j (κ i,j (t))) for i = 1, . . . , N and j = 0, . . . , N and where κ i,j (t) = max t i,j k | t i,j k ≤ t, k ∈ N is the last instant when the position of agent j has been received by agent i,

∆ θ = I m 0 m 0 m 1 θ I m , K o = 2I m I m T .
Similarly, the formation dynamics can be written as

ḟi = Af i + B ḟi,v
Denoting the estimation error xi,j = xi,j -x j for j = 0 . . . N and i = 1 . . . N , the input u i can be written as

u i = -cK c Γ λ N k=1 H ik e k -cK c Γ λ N k=1 H ik xi,k +b i cK c Γ λ xi,0 + B ḟi,v (t) with K c = I m 2I m and Γ λ = λ 2 I m 0 m 0 m λI m
. Therefore, the formation tracking error dynamics is

ėi = Ae i + Bu i -Bu 0 -B ḟi,v
while the estimation error dynamics is

ẋi,j (t) = (A -θ∆ -1 θ K o C)x i,j (t) -θ∆ -1 θ K o z i,j (t) -Bu j (t) + B ḟj,v (t)
where z i,j (t) = e -2θ(t-κi,j (t)) C xi,j (κ i,j (t)) -C xi,j (t) .

Similarly to [START_REF] Ménard | Observerbased consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods[END_REF], and using

η c = [E T 1 . . . E T N ] T , η o i = [( Xi,1 ) T . . . ( Xi,N ) T ] T , i = 1, . . . , N and η o 0 = [ X1,0 . . . XN,0 ] with E i = Γ λ e i
and Xi,j = ∆ θ xi,j , the following Lyapunov functions are selected:

V c (η c ) = (η c ) T [Ω ⊗ Q]η c (11) V o ( Xi,j ) = ( Xi,j ) T P ( Xi,j ) (12)
Vo ( Xi,j ) = N i=1 N j=0 s ij V o ( Xi,j ) (13)
where s ij = 1 if agent i receives information from agent j and 0 otherwise for i = 1, . . . , N , j = 0, . . . , N , P (resp. Q) is the symmetric positive definite matrix solution of the equation

P + A T P + P A = C T C (resp. Q + QA + A T Q = QBB T Q).
Using these Lyapunov functions, based on conditions ( 8)- [START_REF] Gu | Distributed network-based formation control[END_REF], one can show that

N i=1 e i (t) ≤ αe -λ γ t + βδ 0 λ , ∀t ≥ 0 ( 14
)
where α, β, γ > 0 and δ 0 is the upper bound of the leader input. Furthermore, β does not depend on θ, λ, c, τ M and the initial conditions of the agents and observers. Details are omitted due to lack of space. Remark 3: The conditions ( 8)- [START_REF] Gu | Distributed network-based formation control[END_REF] for the selection of the controller and observer gains, obtained from Lyapunov analysis, are only sufficient. It is clear from inequality (14) that the system achieves practical consensus where the tracking error is gradually reduced and enters in a ball centered at the origin. The radius of the ball depends on δ which means that in case of static leader and time-invariant formation, the tracking errors converge exponentially to zero. Furthermore, the radius of the convergence ball of the tracking errors can be reduced by increasing the controller gain λ. This will also increase the speed of the controller dynamics.

IV. SIMULATION RESULTS

Let us consider a MAS consisting of one leader, labeled 0, and four followers, labeled 1, . . . , 4, described by ( 1)- [START_REF] Dong | Time-varying formation control for general linear multi-agent systems with switching directed topologies[END_REF]. The communication topology between agents is directed and is shown in Fig. 2. The leader can send its position information only to agent 1. One can also note that agent 4 can receive data from both agent 1 and agent 3. The

A. Time-invariant formation

A square geometric shape is considered where four followers track the leader while maintaining a constant square shape around the leader. The formation vector is

f 1 = [-1, 1, 0, 0] T , f 2 = [1, 1, 0, 0] T , f 3 = [1, -1, 0, 0] T and f 4 = [-1, -1, 0, 0] T .
Case 1: First, it is considered that the leader moves at a constant speed, i.e. u o = [0, 0] T . The tracking result for this case is shown in Fig. 4. Both x and y position errors are depicted in Fig. 5. It is clear from these results that the formation pattern is successfully obtained and all the followers keep the formation while tracking the trajectory of the leader. Moreover, exponential consensus is obtained. Case 2: In this case, the leader moves with a constant acceleration, i.e. u 0 = [0.1, 0.05] T . The formation tracking result for this case is illustrated in Fig. 6 while Fig. 7 shows the position errors. Practical stability is achieved due to the presence of u 0 as explained in Remark 3.

B. Time-varying formation

The time-varying formation vector is

f i (t) =    
10 cos(0.1t + 2π(i -1)/4) 10 sin(0.1t + 2π(i -1)/4) -sin(0.1t + 2π(i -1)/4) cos(0.1t + 2π(i -1)/4)     with i = 1, 2, 3, 4. If this formation is achieved, the followers will turn in a circle around the leader. Two cases will be considered. Case 1: In this first case, the leader is static while the followers turn in a circle around the leader. Fig. 8 shows the time-varying formation tracking while Fig. 9 represents the position error.

  Let the diagonal matrix B = diag(b 1 , b 2 , . . . , b N ) be the interconnection relationship between the leader and the followers. If follower i can receive information from the leader then b i = 1 and 0 otherwise. Communication graph including the followers as well as the leader is denoted by Ḡ. Also H = L + B Assumption 1: The pinning joint communication topology Ḡ has a directed spanning tree.

Fig. 1 .

 1 Fig. 1. Example of square geometric shape.

Fig. 2 .

 2 Fig. 2. Communication topology.

Fig. 3 .

 3 Fig. 3. Sampling periods for data transmission among the agents.

Fig. 4 .

 4 Fig. 4. Time-invariant formation tracking with constant leader velocity

Fig. 6 .Fig. 7 . 2 :

 672 Fig. 6. Time-invariant formation tracking with leader input

Fig. 9 .

 9 Fig. 9. Position error for time-varying formation with static leader


Lorenzo Poggini
email: lorenzo.poggini@icmcb.cnrs.fr

Mathieu Gonidec
email: mathieu.gonidec@icmcb.cnrs.fr

Juan H González-Estefan

Gilles Pecastaings

Benoît Gobaut

Patrick Rosa

Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film

Keywords: Eutectic gallium indium alloy (EGaln), Spin crossover (SCO), Thin films, Tunnel junctions

Thin films of a molecular spin crossover (SCO) Iron(II) complex featuring a high transition temperature are grown by sublimation in high vacuum on TS Au and investigated by X-ray and UV photoelectron spectroscopies. Temperature-dependent studies demonstrate that the thermally induced spin crossover behavior is preserved in thin films. A large-area ultrathin switchable spin crossover molecular vertical tunnel junction with top electrodes of the liquid eutectic of gallium and indium, for which the spin-state switching of the films induces a two orders of magnitude change in the tunneling current density flowing through the junction, is reported here. The results on large-area junctions, rationalized by density functional theory calculations, demonstrate the high potential of SCO-based switchable molecular junctions as functional devices.

Introduction

From the technological point of view spin cross-over (SCO) molecular materials have been proposed for numerous applications ranging from memory devices1 to gas, pressure, and temperature sensors.2, 3 SCO materials are proposed to be good candidates as active components in prospective spintronics and molecule-based data storage devices4, 5 as well as in innovative devices taking advantages from additional features, combining SCO with other functionalities.6, 7 Considering all those possible applications, the study of the electrical properties of SCO compounds is fundamental, to understand how they behave under electronic stimuli, especially when prepared as thin films.8, 9 Indeed, these molecules can be switched between two electronic states-termed high-spin (HS) and low-spin (LS)with different magnetic, optical, and structural properties by the action of external stimuli (pressure, temperature, light-irradiation)10-14 making them promising systems for new functional materials.7 This is particularly important since the use of electrical stimuli to control (read/write) the spin-state of the system would provide a great advantage toward technological applications, compared to other conventional addressing methods such as light irradiation, and changes in temperature or pressure, that are less easily implemented. In this paper, we show that it is possible to design large area switchable molecular tunnel junctions, in which the switchable tunneling barrier is made of a thin film of a SCO compound. Those thin films, made by evaporation on TS Au (template-stripped gold), were thoroughly characterized using highly-sensitive and specific surface tools. Electrical switching has been studied as a function of temperature in a tunnel junction configuration, and the experimental results have been rationalized thanks to a theoretical model based on energy levels and electronic densities obtained at density functional theory (DFT) level. The good correlation we establish between experimental measurements and modeling proves the feasibility to design, manipulate, and read such ultrathin film devices, an important prerequisite for the development of future active multistable devices. The most critical issues toward the development of large-area spin-crossover based molecular junctions are i) to obtain high quality SCO thin films over large areas and ii) to have a measurement methodology that allows to measure their properties in an efficient and reliable manner. Moreover, to facilitate further developments and applications of those systems, it is highly desirable to have a transition temperature (and thus, possibly, a switching temperature) close to room temperature. For those reasons, we have chosen the [Fe(HB(trz) 3 ) 2 ] SCO complex, hereafter called 1 (HB(trz) 3 = tris(1H-1,2,4-triazol-1-yl)borohydride), 15-18 for which it was shown recently that it can be deposited as continuous thin films on surfaces by thermal evaporation. 19,20 Some recent efforts have focused on vertical large area SCO junctions with film thicknesses in the 10-200 nm range by evaporation of metallic top contacts over molecular films on two closely related SCO complexes.5, 21 This methodology works well for rather thick films (a few tens or hundreds of nm), but for thinner films-of ≈1-5 nm, as may be required to obtain good quality tunnel junctions-it is likely to yield an unacceptably high amount of shorted devices. Altogether, large area junctions incorporating very thin films of those two closely related compounds showed no significant variation of conductance with temperature for 10 and 30 nm films of [Fe(HB(trz) 3 ) 2 ],21 and only a small conductance switching in the case of 8.1 nm films of [Fe(H 2 B(pz) 2 ) 2 (phen)].5 Conversely, the same authors showed that junctions incorporating thicker films of 100-200 nm of [Fe(HB(trz) 3 ) 2 ], for which charge transport likely occurs in the hopping regime, exhibited a jump in conductance switching of about one order of magnitude upon spincrossover.21 This may be, in part, due to interpenetration of the molecular thin film and the evaporated top electrode, or (as mentioned by the authors themselves) to mechanical aspects linked to the nature of this sandwich metal-insulator-metal junction. Indeed the sensitivity of SCO compounds to external pressure leading to inhibition of spin state switching is known, and one possible hypothesis would be to think that this should be exacerbated for thin films.22-24

Results and Discussion

In the present study, we measured vertical SCO molecular junctions by using a soft conformal EGaIn (for Eutectic Gallium Indium alloy, 75% Ga and 25% In in mass, see schematic representation in Figure 1a) top electrode that specifically addresses the main challenges one can expect to encounter when measuring vertical SCO junctions. In particular, this top electrode provides a soft, deformable top contact which should ensure that the molecular film is intact (i.e., not breached by the top electrode) even at very low thicknesses; it should also provide mechanical compliance of the electrode with the deformation of the molecular film, while ensuring a small footprint which should minimize the occurrence of short-circuits. This type of soft top electrode has been used by us and others to study molecular tunnel junctions with great efficiency25-29 and has recently been employed to study inorganic complex systems.30-32 In our hands using an EGaIn top electrode afforded tunnel junctions with a two orders of magnitude abrupt change in conductivity at the crossover temperature. mm 100 nm thick strip of complex sublimated on Kapton foil (red dots).

Bulk and Thick Film Magnetic Characterization

Magnetic characterization of the thermally driven SCO in a 100 nm thick film on a Kapton substrate has been carried out with a SQUID magnetometer and compared with a bulk powder sample measured with a vibrating sample magnetometer (VSM) (see Figure 1b).

The measurement on the bulk powder shows a transition temperature T 1/2 of 326 K with a small 3 K hysteresis, in line with the values of T 1/2 recorded previously for the dehydrated complex (magnetic measurement gives 333 K18, 19 and Raman spectroscopy about 325 K18). The 390 K χ M T value of 3.08 cm 3 K mol -1 is higher than that previously reported, which was slightly below the expected spin-only value (3.00 cm 3 K mol -1 ), contrary to what is expected for HS Fe(II) (values typically in the range 3.1-3.8 cm 3 K mol -1 ). The microcrystalline powder sample of complex 1 is therefore almost fully high-spin at K, and becomes almost fully LS at low temperature, reaching a χ M T value of 0.1 cm 3 K mol -1 at 200 K.

The thin film on Kapton shows a very similar behavior, with an almost fully diamagnetic state at low temperature with a χ M T value of 0.14 cm 3 K mol -1 at 200 K. The highest temperature that can be reached with the Kapton tape without damaging it was 350 K, and the spin crossover is still incomplete at that temperature (2.63 cm 3 K mol -1 at 350 K), illustrating a slightly more gradual character than in the bulk sample. The observed T 1/2 of 333 K is again in line with previously reported measurements for both hydrated15, 33 and dehydrated complexes, and the thermal hysteresis width (if significant) is reduced to less than 1 K. A comparison between the thermal switching of the pristine powder sample and the sublimated film reveals that the nanostructuration has an effect on the compound, in particular inducing a slightly more gradual SCO transition and increasing the residual LS fraction at high temperatures (incomplete SCO). Recently it has been reported for the very same complex evidence for an increase of the spin crossover temperature by reducing the thickness of films deposited on silica wafer.34 These findings may arise from packing inhomogeneity, similarly to what was previously found for doped SCO systems13, 35 for which the cooperativity was different than that of the pure compound, and may also arise from a relevant increase of surface energies when going from the low-spin to the high-spin phase.10, 22, 34, 36-39

Surface Characterization

To further confirm that 1 has been deposited intact on the TS Au surface in low coverage regime, X-ray photoelectron spectroscopy (XPS) experiments were performed on a 6.7 nm thin film, and the semiquantitative results are reported in Table S1 in the Supporting Information. We further checked the integrity of 1 after thermal deposition by in-house ultraviolet photoelectron spectroscopy (UPS) and synchrotron-based X-ray absorption spectroscopy (XAS).

We analyzed the N1s, B1s, and Fe2p XPS regions to evaluate the atomic compositions of the functionalized surfaces (see Figure S1 and Table S1 in the Supporting Information). For thin films of evaporated on TS Au, the N1s region features two main components with a 2:1 intensity ratio (Figure S1, middle, Supporting Information) at 401.1 and 399.7 eV (64% and 36%, respectively) assigned to the two coordinated nitrogen atoms of the triazole ring and to the unsubstituted nitrogen atom, respectively.40

The line shape of the B1s band (see Figure S1, right, Supporting Information) is also consistent with the chemical formula of complex 1. [START_REF] Chastain | Handbook of X-Ray Photoelectron Spectroscopy: A Reference Book of Standard Spectra for Identification and Interpretation of XPS Data[END_REF] The semiquantitative analysis (see Table S1 in the Supporting Information) thus confirms the chemical integrity of complex 1 after thermal deposition on TS Au.

To follow the SCO transition we acquired XPS and UPS spectra at 298 and 360 K, and XAS at 100 and 350 K. In the XPS spectra (see Figure 2a) the Fe2p region is clearly different at 300 and 360 K thus providing evidence for thermally induced SCO in 6.7 nm thin film. The occurrence of SCO is particularly clear when considering the Fe2p 3/2 peak that is centered at 710.5 eV in the HS state and at 709.5 eV in the LS configuration. In the LS state this peak also appears narrower, and is principally dominated by a component at lower binding energy (709.5 eV, highlighted in green and called A) which is known to be due to the coupling of the photoelectrons with the partially-filled metal shell. [START_REF] Borod'ko | [END_REF] The spin-orbit splitting (ΔE SO ) value also changes significantly upon conversion, with values of 13 and 12.5 eV for the HS and the LS components, respectively, which is in good agreement with previously reported values.43-46 Indeed, ΔE SO changes upon SCO due to different orbital populations in the two spin states: in the HS configuration the measured transition involves e g -like orbitals that are unoccupied in the LS state. We used a peak fitting procedure (Figure 2a) to deconvolute the spectra. The intensity of the components unequivocally attributable to the LS state (A and Aʹ in Figure 2a) allow, qualitatively, to follow the SCO transition. XPS binding energies (B.E.), percentage of the different component, and spin-orbit splitting values (ΔE SO ) are reported in Table S2 in the Supporting Information.46, 47 We tested the stability of the bulk phase of 1 under X-Ray in ultra high vacuum (UHV) conditions by cycling back to 100 K (see Figure S2b in the Supporting Information). In line with what was previously observed by Shalabaeva et al.,19 the presence of water inside the crystalline plane seems to be relevant for the occurrence of the SCO. Therefore, a reversible SCO could not be observed in the sublimated films under UHV conditions due to the loss of water. It is noteworthy that on the contrary, the SQUID measurement on a 100 nm thick film, does show a reversible SCO (Figure 1b). Nevertheless, XPS probes only the first few tens of nanometer of the sample, while magnetic measurements probe the whole of the sample, so the former is certainly more sensitive to vacuum-induced loss of water. Further evidence for the occurrence of thermally driven SCO in the thin films of 1 on TS Au was obtained from UPS by analyzing the film at 300 and 360 K. This characterization, performed on the pristine film at 300 and 360 K (corresponding to the LS and the HS states, respectively) was compared with a DFT-based modeling that helped to clarify the observed effects. The UPS spectrum acquired on a 6.7 nm film at K (Figure S3, blue line, Supporting Information) evidences a highest occupied molecular orbital (HOMO) contribution at -2.82 eV, in agreement with similar systems7, 45, 48, 49 and the total density of states (TDOS) obtained from the DFT calculation results starting from the LS crystal structure. The temperature effect on the UPS spectra is visible mainly in the valence-Fermi region (from 0 to 6 eV). Indeed, upon warming up from 300 to 360 K, we observed a shift to -2.35 eV (Figure S3, red line, Supporting Information) in line with what was previously reported for similar SCO compounds,7, 45, 46, 48, 49 and again in very good agreement with the TDOS obtained from the DFT calculations results starting from the HS crystal structure. Our in-house spectroscopic analysis points out that evaporated ultrathin films of 1 on TS Au, retain the switchability of the bulk complex 1 even after thermal sublimation.

A quantitative analysis of the population of the HS and LS states was also made by XAS measurements, since absorption techniques are unmatched tools to qualitatively and quantitatively analyses ultrathin layers of molecular systems, providing the required sensitivity to monitor the spin state of the investigated system.38, 50-53 Figure 2b displays the Fe(II) L 3 edge absorption spectrum of a 6.7 nm thin film of 1 on TS Au at 100 and 350 K. The 100 K spectrum clearly shows the coexistence of LS and HS forms, which is in line with what we observed by magnetometry for a 100 nm film evaporated on Kapton (Figure 1b). A series of consecutive spectra recorded at 100 K on the very same position show the same FeL 3 line shape within the noise error (Figure S4, Supporting Information) confirming the stability under X-Ray irradiation. Upon heating from 100 to 350 K the spectral features related to HS-Fe(II) increase from a contribution of about 23% to ≈95%, respectively (see the Experimental Section for details). Atomic force microscopy (AFM) imaging measured on thin films of 1 (see Figure S5 in the Supporting Information) showed a rather smooth surface to confirm a nice defect-free SCO deposit excluding the presence of defective areas or Volmer-Weber growth. The AFM estimated roughness in an area of µm 2 resulted at about 0.4 nm (Figure S5, Supporting Information).

Electrical Measurements

In the light of the quality of the films we obtained, we incorporated the ultrathin SCO deposits on TS Au in vertical tunnel junctions using an EGaIn top electrode as depicted in Figure 1a. The choice of the substrate is dictated by the necessity to use an ultraflat bottom electrode to obtain functional, nonshorted junctions, over hundreds of square micrometers. We contacted those thin films with EGaIn conical tips top electrodes as described elsewhere,29 and made contacts of ≈100 µm in diameter. We controlled the temperature of the system with a Linkam microscope stage, and the resulting J/V curves exhibited the typical nonlinear behavior expected for nonshort-circuited tunnel junctions. We started with a 9.5 nm thick film, but currents measured at room temperature were too close to our instrument's lowest range (Figure S6, Supporting Information), so we switched to a 6.7 nm film in order to have unambiguous results.

The 6.7 nm thick junctions in the LS state (20 junctions, 200 traces, on a single film) present log |J| values of -7.82 ± 0.45 and -7.55 ± 0.37 at -1.0 and +1.0 V, respectively (Figure S7c, Supporting Information). The standard deviations of those measurements are in line with those observed previously for EGaIn junctions incorporating SAMs of small organic molecules. 25,28,54 The high reproducibility of those measurements and the absence of short-circuits are a further confirmation of the high quality of the ultrathin SCO films we used in the junction, and of the reliability of EGaIn as the method of choice for measuring such systems.

With the 6.7 nm thick film, the J/V curves registered at 297 and 340 K (corresponding to the temperature of LS and the HS states determined by magnetometry, XPS and XAS, respectively) are very strikingly different, and they provide evidence for switchable junctions with a clearly different behavior depending on the spin state of 1 (Figure 3a). Above T 1/2 , with compound 1 in the HS state, the tunneling current density at 1 V applied bias potential is about two orders of magnitude higher than for the LS state (with tunneling current densities of ≈5 × 10 -6 A cm -2 to ≈3 × 10 -8 , respectively). We also noted that the HS junction presents a moderate negative rectifying behavior, with the current flowing under negative bias being slightly larger than the one flowing at positive bias, which can be rationalized by considering that the HOMO of the compounds are brought within the conducting potential window at sufficiently negative biases (see Figure S9 in the Supporting Information). ; b) log |J| at V = -1 V reported as a function of temperature acquired in sweeping mode using a rate of 1 K min -1 (warming up and cooling down, violet and cyan respectively); the curved arrows are a guide for the eyes to emphasize the jump in electrical conductivity.

To investigate the temperature-driven switching behavior of the junction more in depth, and to verify that the great difference in current density we observed between low and high-temperature data truly arises from the change of the spin state, we characterized the junctions at different temperatures. Figure S7b (Supporting Information) shows the obtained values of log |J| measured at an applied bias of ±1 V. To establish just how abrupt the switching phenomenon is, we repeated the measurements while increasing and decreasing continuously the temperature at a rate of 1 K min -1 (see Figure 3b) and found no significant temperature difference between the switching temperature in warming and cooling modes. Measurements in sweeping and stepping mode showed the same abrupt change in current density of approximately two orders of magnitude in the temperature window in which we expected the SCO to happen, at a critical temperature T* of 333 K (we report the complete series of log |J|/V curves in Figure S7a together with Fowler-Nordheim plots in Figure S8 in the Supporting Information). The transition voltage spectroscopy (TVS) analysis provides transition voltage values of the order of 0.1-0.6 V, but those values are likely related to some density of states in the oxide layer covering the EGaIn tip, as shown by Ricoeur et al. 55 Extending the variable temperature measurement to junctions made on the 9.5 nm film evidenced the same transition, though with a smaller change in current density upon the transition (see Figure S6 in the Supporting Information). As said above the lower current densities are very likely biased by our experimental setup and residual currents, and thus overestimated. Overall, it is noteworthy that our junctions all presented an abrupt SCO-induced conductance switching, contrary to what was reported by Shalabaeva et al. when measuring a 10 nm thick film of this SCO complex in an ITO/SCO/Al junction.21 This difference in behavior together with major differences in current densities between the two sets of experiments could have a variety of explanations. For instance, the differences in current densities could be ascribed to the differences in work functions due to the use of different electrode materials, to the fact that evaporated electrodes are known to display much higher current densities than EGaIn or mercury electrodes due to the improved effective contact with the underlying layer,29 or possibly to phenomena related to the interfaces with the electrodes (stress, electronic coupling). On the other hand, the discrepancy in conductance switching behavior could be also possibly linked to the soft nature of our top electrode that unlike rigid, evaporated top electrodes permits the required geometrical deformations of the molecular thin films at the transition temperature. It is difficult to ascertain which are the most relevant parameters, and rigorous testing of all of those hypotheses would require a separate, dedicated study. Preliminary data on another compound showing a SCO transition below room temperature, where the EGaIn ceases to be liquid and becomes much more rigid, could be ascribed to support the rigidity hypothesis and will be reported in due course.

Theoretical Model

To rationalize the experimental data and in particular to explain the magnitude of the change in current density caused by the SCO we modeled the behavior of the junctions with a single-orbital Simmons-like formalism that combines molecular orbital energies obtained from gas-phase density functional theory (DFT) calculations and experimentally determined temperature-dependent high-spin fractions. We thus sought to reproduce the trends in tunneling current density as a function of temperature. For tunnel junctions consisting in a monolayer of molecules sandwiched between two electrodes, it has been shown that the model developed by Simmons'56-58 provides a reasonably simple way to describe the current flowing through the system.59, 60 Nevertheless, this model applies to homogeneous monolayer thin films consisting of a single compound, and it is not directly applicable to mixed multilayer stacks made up of compounds with different electronic configurations. We therefore adapted the classical Simmons' model to describe such a system by taking into account a random distribution of HS and LS molecules to compute the temperature dependent tunneling current density by adjusting the statistical population of the two states to the one determined experimentally (see the details of the model in the Supporting Information). We used DFT at the B3LYP/6-31++G(d,p) level to estimate the gas-phase orbital energies of the HS and LS compounds. For both HS and LS states, the calculated orbital energies for the highest occupied molecular orbitals (-5.8 and -6.2 eV, respectively) are closer to the Fermi energy function of the electrodes (-5.1 eV for Au and -4.2 eV for EGaIn) than the energies of the unoccupied molecular orbitals (-2.2 and -1.1 eV, respectively). Based on these results, assuming that tunneling is responsible for charge transport in our thin films, one would expect hole tunneling to be the dominant charge transport mechanism, and we therefore chose to take into account only the effect of hole tunneling in our model.

For both HS and LS states, we computed an approximated TDOS by homogeneous broadening of the orbital energies combined with an arbitrary rigid shift of 3.5 eV which provided DOS spectra that matched reasonably well the experimental UPS data (see Figure S3 in the Supporting Information). Combining the uncorrected orbital energies obtained by DFT with our mixed multi-layer Simmons-like model, we could simulate the tunneling current density as a function of temperature, by taking the temperature-dependent HS fraction we determined experimentally. The curves generated by using magnetometry data on bulk samples and thin film samples (see Figure S10 in the Supporting Information) do not provide a good agreement in terms of absolute current density values but the empirical corrections we used were established for SAMs of organic molecules, and minor adjustments of those parameters (in particular the relative carrier mass m*) could change the absolute values of current densities in a major way. Nevertheless, our model does provide an acceptable estimation of the expected change in the log |J| values when transitioning between the HS and LS states, with a small discrepancy between T 1/2 and T* (8 and 1 K, respectively). They fail to reproduce the exceedingly abrupt transition observed experimentally (surprisingly for such a nanoscale SCO system), exhibiting instead a more gradual conversion. While our tunneling data differ significantly from earlier reports, the relatively good agreement between our experimental and theoretical values is a strong argument in favor of using soft, conformal contacts (like EGaIn) for the characterization of transport properties of soft, pressure-dependent molecular ultrathin films. We have no direct evidence on the exact nature of the transport mechanism in both films we measured. Nevertheless, given the lack of temperature dependence that would support a thermally activated multistep hopping mechanism, in spite of the rough assumptions underlying the Simmons model, its ability to predict the correct change of conductivity upon spin crossover is remarkable. Most importantly, it underlines that we have built a molecular junction where materials, thicknesses and electrodes materials allows for resistance switching to be governed by the change of the MO levels and not by the bulk conductance of the SCO material.

Conclusions

In conclusion, we have shown through a detailed spectroscopic analysis that it is possible to prepare high-quality continuous ultrathin films of the [Fe(HB(tz) 3 ) 2 ] spin-crossover complex on gold, and that such thin films can exhibit a SCO behavior similar to that of the bulk material. Building upon this fundamental preliminary finding, we studied vertical EGaIn tunnel junctions of 1, a nondamaging technique generally used for simpler molecular systems, to probe the electronic state of a 6.7 nm film. The resulting device presented a two orders of magnitude change in current density upon spin state switching. Strikingly, this change occurred in a very abrupt manner in less than 1 K, which is not at all what is usually expected for SCO complexes when shaped at that scale, where elastic cooperative effects are not enough to provide abrupt transitions or hysteretic effects. 13 Altogether, the strategy we proposed and used to follow the spin state switching of the compounds in a passive SCO junction demonstrates the possibility to design large area switchable SCO tunnel junctions of ultrathin films (where SCO is induced thermally). This represents an important milestone toward the use of such junctions in the near future for the development of novel active spintronic devices in which the SCO layer could be used as a molecular switch, triggered by more complex stimuli including light irradiation and voltage pulses. Most importantly, our study also emphasizes the crucial importance of the interplay between the molecular material and its electrodes, which requires in depth fundamental analysis.

Experimental Section

Synthesis: Reagents were purchased commercially and used without further purification. MeOH was distilled over Mg(OMe) 2 under nitrogen. KHB(trz) 3 (Htrz = 1H-1,2,4-triazole) was prepared according to the reported experimental procedure61 and recrystallized in MeOH. [Fe(HB(trz) 3 ) 2 ] was obtained by layering 15 mL of a degassed aqueous solution of KHB(trz) 3 (1.02 g, 4 mmol) over 5 mL of a solution of FeSO 4 .7H 2 O (0.56 mg, 2 mmol) in degassed water. The square-shaped purple crystals that formed were washed with water and Et 2 O, and then crushed in a mortar. The resulting powder was dried under vacuum heating around 50 °C, and then purified by sublimation at 280 °C onto a cold finger under 2.10 -6 mbar vacuum using a turbomolecular pump protected with a liquid nitrogen trap. Yield: 208 mg (42%). Calculated for C 12 H 14 B 2 FeN 18 (487.83 g mol -1 ) C 29.54, H 2.89, N 51.68%; found C 29.41, H 2.94, N 51.23%. X-ray diffraction of the powder obtained after purification by sublimation showed perfect matching with the reported LS structure of the anhydrous complex reported previously, with no alteration whatsoever over weeks (Figure S11, Supporting Information).18 Magnetic Measurements: Bulk magnetic measurements were performed on a Microsense EZ7 Vibrating Sample Magnetometer with the 100-1000 K EV1-LNA temperature control option. 5.866 mg of sample was weighed accurately in a silver capsule, cold-sealed, and glued to a ∅3 mm carbon fiber rod with double-faced scotch tape. DC measurements were performed under a 10 kOe magnetic field. Nitrogen gas was used for temperature control. Background subtraction was performed using a closely matched silver capsule. The diamagnetic contribution from the sample was estimated using Pascal's constants. For the thin film on Kapton, the compound was evaporated as a 100 nm thick 6 × 90 mm strip on to a 10 × 10 cm 50 µm thick Kapton foil. The calculated volume of 54 × 10 18 Å 3 corresponds to 0.174 µmol of the compound when considering the unit cell of the reported structure at room temperature. The foil was then wrapped tightly in a small cylinder perpendicularly to the strip, then placed into a straw to be introduced in a Quantum Design MPMS-7XL SQUID magnetometer. The DC measurement was performed using a RSO head under 20 kOe magnetic field. Raw data were treated as described in reference 62.

Sample Deposition:

The evaporations were performed in a Plassys thermal evaporator connected to a glove box system. An alumina crucible was filled with the powder of 1 and once the pressure reached the 10 -7 mbar range, the temperature was gently raised up to the evaporation temperature. The evaporation rate was followed by a quartz microbalance, and was stabilized in the 0.05 nm s -1 range at a temperature of 450-480 K. The tooling factor was determined by evaporating a thick film of 1 on an ultraflat silicon wafer through a soft porous membrane, and by checking the thickness of that film ex situ by AFM (see Figure S5 in the Supporting Information for details). For a target thickness of 20 nm, an actual film thickness of 27 ± 4.7 nm was measured, which corresponds to a tooling factor of 135%. This value was then used to correct the thickness of the thin films prepared for physical characterizations, and therefore obtained films with corrected thicknesses of 6.7 nm (for a target thickness of 5.0 nm) and 9.5 nm (for a target thickness of 7.0 nm).

XPS and UPS Characterizations: XPS experiments were carried out in a UHV apparatus with a base pressure in the 10 -10 mbar range. Monochromatized Al Kα radiation was used for XPS measurements (1486.6 eV, 100 W). The detector was a SPECS PHOIBOS 100 hemispherical analyzer mounting a 2D-DLD detector, the angle between the analyzer axis and the X-ray source was 54.44°. The XPS spectra were measured with a fixed pass energy of 40 eV. The binding energy (BE) scale was calibrated setting the Au4f signal of the substrate at 83.9 eV. In order to minimize air exposure and atmospheric contamination, samples were mounted on the sample holder inside a glovebox directly connected with the fast-entry lock system of the XPS chamber. UPS spectra were measured using the He II line (40.8 eV) from a non-monochromatized gas discharge lamp and the same analyzer used for XPS, yielding an energy resolution of 0.18 eV. The spectra were taken in normal emission, and they were energy calibrated using the Fermi level. The variable temperature experiment was performed by using an e-beam heater system connected to the XPS/UPS sample holder. Every spectrum represented herein results from averaging four spectra collected after 1 h of thermalization at a specific temperature. Spectral analysis consisted in a linear background subtraction and deconvolution using a mixed Gaussian and Lorentzian line-shape for each spectral component. The stoichiometry was calculated by peak integration, using theoretically estimated cross-section for each transition.63

Synchrotron Characterization:

The XAS experiments were performed on thin film samples at the DEIMOS beamline64 of the SOLEIL Synchrotron facility. The flux of photon was reduced to prevent any radiation damage. None was observed. A magnetic field up to 20 kOe was applied along the photon propagation direction with the normal to the surface. The reported XAS spectra were acquired at the L 2,3 edges of Fe (applied parallel to the X-ray propagation vector) and using the two circular polarizations (left, σ+ and right, σ-) at normal incidence (θ = 0°). All the spectra were normalized following the procedures described in earlier reports.65, 66 Absorption spectra were measured in total electron yield (TEY) detection mode to guarantee the optimal detection sensitivity. The estimation of the temperature dependence of the HS-Fe(II) molar fraction of the evaporated coverage was performed through least-squares interpolation of its normalized L 3 XAS spectra with the ones of the bulk compound, featuring temperature dependence as recorded in similar experimental conditions.67

Electrical Measurements: In order to measure the current through the molecular thin films, a liquid top-electrode of eutectic gallium indium alloy (EGaIn) was used. EGaIn oxidizes readily in ambient conditions to form a thin passivating oxide layer (mostly composed of Ga 2 O 3 ) that allows to fabricate tips with a controlled geometry. 27 The so-called "selected conical tips" were used as top-electrodes on SCO films. To form stable contacts, the tips were brought close to the substrates until the tip apex wrinkling or deforming on the substrate could be clearly observed (by optical microscopy). The data on Figure 3 were acquired after a systematic thermalization time of 5 min and are the average of 12 J/V traces acquired in two different junctions (i.e., different position on the surface). For each individual tip, one junction (i.e., contact) was formed and J-V traces were recorded for forward bias 0→-V→V and J-V traces for the reverse bias V→-V. This procedure was used to avoid using a tip contaminated with absorbed impurities. The temperature control was achieved using a Linkam cryostat, and the I/V curves were acquired under nitrogen flux to avoid any oxidation of the molecular film upon increasing temperature. Each temperature corresponds to a different junction on the surface of the nanostructured surface, and for each temperature were acquired 1 sequence as described above. The temperature scale was calibrated by using a Pt-100 thermocouple to measure the effective temperature of the Au TS substrate. Due to the low magnitude of the currents measured in the junctions (often of the order of 1 to a few pA), the built-in potential contributed to a non-negligible offset of the I/V curves and the data were thus systematically corrected by subtracting the 0 bias current for every junction.

Theoretical Calculations: The orbital energies were estimated by performing single-point DFT calculations using the crystal-structures as input geometries without performing any optimization step, with the B3LYP functional and the 6-31++G(d,p) basis set as implemented in the Gaussian 09 program. [START_REF] Frisch | Gaussian 09, Revis. A02[END_REF] The TDOS spectra were computed by Gaussian broadening of the molecular orbital energies (after a rigid shift of 3.5 eV) using the Multiwfn software. [START_REF] Lu | [END_REF]

Figure 1 :

 1 Figure 1 : a) schematic representation of the TS Au//SCO//Ga 2 O 3 /EGaIn junction with the structure of 1 represented in the inset and b) VSM measurement of the purified starting material (black dots) under 10 kOe, and SQUID measurement of a 6 × 90 mm 100 nm thick strip of complex sublimated on Kapton foil (red dots).

Figure 2 :

 2 Figure 2 : a) Temperature dependence of the Fe2p XPS spectra evidencing thermal SCO behavior for 6.7 nm thick films of 1 on TS Au and b) temperature evolution of the normalized Fe L 3 edge XAS spectra of a thermally evaporated 6.7 nm thin film of 1

Figure 3 :

 3 Figure 3 : a) The complete series of J/V curve from 297 to 345K acquired with a rate of 15 mV s -1

Acknowledgements

This work was supported by the French National Research Agency (ANR) Investment for the Future Programme IdEx Bordeaux (ANR-10-IDEX-03-02) and the Mexican government via the National Council of Sciences and Technology (CONACyT, program 073096, announcement 291060). Computer time for this study was provided by the computing facilities MCIA (Mésocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of the Université de Pau et des Pays de l'Adour. The authors acknowledge the ELORPrintTec platform, the LCPO laboratory, all the ICMCB and DEIMOS (SOLEIL) staff for their support during the measurements.

Conflict of Interest

The authors declare no conflict of interest.


Hamed Samie
email: hamed.samie@univ-poitiers.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Rodolphe Vauzelle
email: rodolphe.vauzelle@univ-poitiers.fr

Frédéric Launay
email: frederic.launay@univ-poitiers.fr

Potential Feedback Control for Data Scheduling in LTE Cellular Networks

This article deals with a new Quality of Service (QoS) aware Two-Level Scheduler (TLS) for real-time (RT) data flows in downlink transmission of Long Term Evolution (LTE) cellular networks. It consists of two distinct scheduler levels.

At the upper-level scheduler, the Potential Feedback Controller (PFC) determines the quota of data to transmit in order to meet the delay constraints of RT data flows while, at the lower-lever scheduler, the Proportional Fair (PF) scheduler is used to reach a high fairness among Users (UEs). The PFC is well suited for the upper-level scheduler since it maintains the queue length as close as possible to zero which leads to a low Packet Loss Ratio (PLR). A comparison, by using LTE-Sim simulator, with wellknown QoS aware scheduling methods Frame Level Scheduler (FLS), Exponential (EXP)-rule and Logarithmic (LOG)-rule has been provided which shows that our proposed TLS outperforms the EXP-rule and LOG-rule scheduling methods and it slightly decreases the PLR of RT data flows compared to the FLS scheduling method while it significantly reduces the computation time.

I. INTRODUCTION

Since 2009, LTE is represented as the next generation broadband technology with a promising framework (high data rate, scalable bandwidth and high mobility support) compared to the 3G networks [START_REF] Rumney | LTE and the Evolution to 4G Wireless: Design and Measurement Challenges[END_REF]. LTE supports all Internet Protocol (IP)-based communication services containing Voice-over-IP (VoIP), video streaming, online gaming, and transmission control protocol-based services such as emailing and data transferring [START_REF] Liu | Smart Downlink Scheduling for Multimedia Streaming Over LTE Networks With Hard Handoff[END_REF]. According to Ericsson Mobility, in 2015, LTE has exceeded 1 billion worldwide subscribers and it will reach 4.3 billion subscribers by 2021 [START_REF] Ericsson | Ericsson Mobility Report: On the pulse of the networked society[END_REF]. Therefore, optimizing LTE features is an essential subject of research in wireless communications.

In RT multimedia services, in order to meet the Quality of Experience (QoE) requirements, the end-to-end delay constraints should suit the human perception of interactivity requirements [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF]. For example, for the VoIP data flow, a maximum of 100ms delay should be considered for a good perceived quality [START_REF] Na | Allowable propagation delay for VoIP calls of acceptable quality[END_REF]. As soon as the RT data flow decoding starts with the chosen playout delay, it is compulsory for every encoded packet to respect this deadline. Therefore, if the encoded packet does not arrive within this deadline, it will be considered as a lost packet [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF]. As a result, in RT multimedia services, ensuring bounded delivery delays is equivalent to reducing packet losses.

LTE specifications inset a data bearer which identifies each data flow based on its QoS requirements [START_REF] Ekstrom | QoS control in the 3GPP evolved packet system[END_REF]. The packet scheduler sorts packets of a given data bearer using a packet filter based on the well-known fivetuple: source and destination IP addresses, source and destination ports, protocol identifier [START_REF] Capozzi | Downlink packet scheduling in LTE cellular networks: Key design issues and a survey[END_REF]. Therefore, packet scheduler can assign available radio resources per-flow basis. LTE standard does not inflict any specific scheduling method to vendors and gives them the freedom to design their own scheduling methods [START_REF] Capozzi | Downlink packet scheduling in LTE cellular networks: Key design issues and a survey[END_REF]. Consequently, the problem of defining a simple scheduling method for allocating available radio resources among UEs has extensively been studied. Classical scheduling methods such as Maximum Throughput (MT) [START_REF] Schwarz | Low complexity approximate maximum throughput scheduling for LTE[END_REF], PF [START_REF] Kim | A proportional fair scheduling for multicarrier transmission systems[END_REF] and Weighted Round Robin [START_REF] Luo | Quality-driven cross-layer optimized video delivery over LTE[END_REF] are not specified to meet the RT data flow QoS requirements. Therefore, several contributions propose QoS aware scheduling methods for RT data flows in downlink transmission of LTE cellular networks. A delay-prioritized scheduling method is proposed in [START_REF] Sandrasegaran | Delay-prioritized scheduling (DPS) for real time traffic in 3GPP LTE system[END_REF]. In [START_REF] Choi | MAC scheduling scheme for VoIP traffic service in 3G LTE[END_REF], a specific packet scheduler for VoIP data flow is proposed which firstly assigns all available radio resources to the VoIP data flows then, it assigns the remaining radio resources to the other data flows. In [START_REF] Lai | QoS-aware downlink packet scheduling for LTE networks[END_REF], Packet Prediction Mechanism (PPM) is proposed for downlink RT services which allocates available radio resources by predicting the behavior of future incoming packets. In [START_REF] Liu | Smart Downlink Scheduling for Multimedia Streaming Over LTE Networks With Hard Handoff[END_REF], a scheduling algorithm for downlink multimedia streaming is proposed in order to reduce the Hard Handoff (HO) procedure effect when a UE is moving from one cell to another. The famous delay-sensitive scheduling methods EXP-rule and LOG-rule have been presented in [START_REF] Sadiq | Delay-optimal opportunistic scheduling and approximations: The log rule[END_REF]. In [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], a QoS aware Two-Level Scheduler (TLS) has been proposed which determines a quota of data to transmit in frame time scale at the upper-level scheduler by exploiting a discrete-time feedback control called FLS and at the lower-lever scheduler, the TLS uses the PF scheduler method in order to reach a high level of fairness among UEs. Finally, in [START_REF] Skondras | A downlink scheduler supporting real time services in LTE cellular networks[END_REF], [START_REF]QoS-aware scheduling in LTE-A networks with SDN control[END_REF], based on the TLS proposed in [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], three level schedulers FLS-Advanced (FLSA) and FLSA -Cross Carrier (FLSA-CC) are proposed in order to have a better performance by prioritizing real-time traffics.

By applying the PFC first proposed in [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF], [START_REF] Phan | Potential feedback control for the power control in wireless sensor networks[END_REF], instead of the FLS at the upper-level scheduler, we develop a new QoS aware TLS for RT data flows in downlink transmission of LTE cellular networks. To demonstrate the efficiency of our proposed TLS, a realistic simulation has been provided by using the LTE-Sim simulator [START_REF] Piro | Simulating LTE cellular systems: an open-source framework[END_REF]. The performance of our proposed TLS is compared with respect to the EXP-rule, the LOG-rule and the FLS scheduling methods. The results confirm that our proposed TLS is able to respect the QoS requirements of RT data flows while it decreases significantly the computation time.

The paper is organized as follows: in Section (II), an overview on packet scheduling in LTE networks is provided. In Section (III), we develop our QoS aware TLS for RT data flows in downlink LTE systems. Section (IV) reports the simulation results. Finally, a conclusion is addressed in (V).

II. OVERVIEW OF PACKET SCHEDULING IN LTE

In LTE, radio resources are allotted into the time-frequency domain [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA)[END_REF]. In the time domain, they are distributed among UEs every Transmission Time Interval (TTI) that lasts 1ms. Conversely, in the frequency domain, the total bandwidth is divided into 180kHz sub-channels where each sub-channel contains 12 successive and equally spaced Orthogonal Frequency-Division Multiplexing (OFDM) sub-carriers. In LTE, time is split in frame scales where each frame is composed of 10 successive TTIs. Moreover, each TTI itself consists of two time slots with 0.5ms length where each slot contains either six or seven OFDM symbols for extended or normal Cyclic Prefix (CP) length respectively [START_REF] Sesia | LTE-the UMTS long term evolution[END_REF]. In LTE, Physical Resource Block (PRB), which is a time-frequency radio resource containing one time slot (0.5ms) and one frequency subchannel (180kHz), is the smallest unit of radio resource that can be assigned to a UE [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA)[END_REF].

Radio Resource Management (RRM) system has been designed to improve the usage of limited available radio resources. RRM exploits a mix of Media Access Control (MAC) and Physical (PHY) layers functions such as: Packet Scheduler (PS); Channel Quality Indicator (CQI) reporting; link adaptation through Adaptive Modulation and Coding (AMC) and Hybrid Automatic Re-transmission Request (HARQ) [START_REF] Capozzi | Downlink packet scheduling in LTE cellular networks: Key design issues and a survey[END_REF]. At the base station (eNodeB), the PS is the responsible for distributing radio resources between UEs. In our work, we are interested in the PS module performance in the downlink direction. Figure 1 represents the main RRM modules which interact with the PS in the downlink direction. The entire process of radio resource scheduling is represented by a sequence of operations which are repeated every TTI [START_REF] Capozzi | Downlink packet scheduling in LTE cellular networks: Key design issues and a survey[END_REF]:

1) Each active UE computes the CQI by decoding a pilot signal and send it back to the eNodeB on the Physical Downlink Control Channel (PDCCH). 2) Based on the CQI information and QoS constraints, the PS allocates available radio resources to active UEs.

3) The AMC module chooses the best modulation schemes (i.e., QPSK, 16-QAM or 64-QAM) for data transmission by scheduled UEs. 4) The scheduling information (scheduled UEs, allocated PRBs and selected modulation schemes) are sent to UEs on the PDCCH. 5) Each scheduled UE decodes its information data to the proper Physical Downlink Shared Channel (PDSCH). In [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], the TLS has been proposed in order to meet the packet delay constraints of the RT data flows in LTE downlink direction. The TLS is composed of two distinct levels that interact together to dynamically allocate available radio resources between the UEs. Figure 2 illustrates the TLS design for the RT data flows in the LTE downlink direction. At the upper-level scheduler, the FLS determines for each frame (10 TTIs) the amount of data of the i-th RT data flow u i (n) to transmit in order to meet its packet delay constraints. The non-RT data flows like best effort flows will be scheduled after transmitting the quota defined by the upper-level scheduler due to their longer packet delay. It should be mentioned that the upper-level scheduler does not take into account the quality of the transmission channel. On the other hand, the lowerlevel scheduler, every TTI, allocates the available PRBs to UEs by exploiting the PF scheduling algorithm in order to reach a high level of fairness between multimedia flows [START_REF] Dahlman | 3G evolution: HSPA and LTE for mobile broadband[END_REF]. The lower-level scheduler should take into account the quota of the data to transmit within the current frame defined by the upper-level scheduler for the RT data flows. Finally, if the lower-level scheduler meets this quota before starting a new frame, the best effort data flow is scheduled using the PF scheduling algorithm (see Figure 2). We refer the reader to [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF] for more information on the TLS design.

III. PFC FOR UPPER-LEVEL SCHEDULER

In this section, by exploiting the PFC first proposed in [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF], [START_REF] Phan | Potential feedback control for the power control in wireless sensor networks[END_REF], instead of the FLS at the upper-level scheduler, we develop a novel TLS for RT data flows in downlink transmission of LTE cellular networks in order to decrease the PLR of RT data flows and the computation time compared to the TLS given in [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF] (see Figure 3).

Let t n,i be the starting time of the n-th frame for the i-th RT data flow. Thus, the sampling interval Δt(n) = t n+1,i -t n,i corresponds to the LTE frame legnth. According to [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], evolution of the i-th RT data flow queue length can be represented by the following equation:

q i (n + 1) = q i (n) + d i (n) -u i (n) (1)
where q i (n) is the i-th queue length at t n,i ; q i (n + 1) is the i-th queue length at t n+1,i ; u i (n) is the amount of data to transmit during the n-th frame; 0 ≤ d i (n) ≤ d M is the amount of the data to be filled in the queue during the n-th frame.

The potential control has originally been applied in robotics to avoid collisions with obstacles [START_REF] Ogren | Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment[END_REF], [START_REF] Rimon | Exact robot navigation using artificial potential functions[END_REF]. The PFC is well suited to the upper-level scheduling problem because it maintains the state of the system (queue length) higher than a target value (queue min = 0) while keeping the state as close as possible to this target value in order to optimize the PLR for the RT data flows in downlink transmission. Beside the performance improvement, by applying the PFC at the upper-level scheduler, we can take advantage of its simple stabilization approach and its short computation time compared to the FLS proposed in [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF].

PFC

Queue length

d i (n) q i (n) u i (n) evolution system
Fig. 3: Feedback control loop for the upper-level scheduler Let us review the PFC for system (1). Theorem 3.1: System (1) under the initial condition q i (0) > c with c ≤ 0, with the feedback control

u i (n) = -k 1 (q i (n) -c) - k 2 q i (n) -c (2)
where u(n) = 0 for n < 0 and -1 < k 1 < 0 and k 2 > 0 is asymptotically stable with respect to the attractive and invariant set

S = ⎡ ⎢ ⎣2 k 2 (k 1 + 1); d M (1 -2k 1 ) + α + (d M (1 -2k 1 ) + α) 2 + 8k 2 1 k 2 4k 2 1 ⎤ ⎥ ⎦ where α = d 2 M -4k 1 k 2 - √ 4k 1 k 2 . Furthermore, we have q i (n) ≥ 2 k 2 (k 1 + 1) + c for all n ∈ N * = N \ {0}.
The proof of the Theorem 3.1 is provided in [17, Theorem 1].

IV. SIMULATIONS

In this section, we provide a realistic simulation of a multi-cell LTE Wireless Cellular Network (WCN) in order to compare the performance of our proposed TLS with respect to the one given in [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], and well-known scheduling strategies LOG-rule and Exp-rule [START_REF] Sadiq | Downlink Scheduling for Multiclass Traffic in LTE[END_REF]. For our simulations, we used the LTE-Sim simulator [START_REF] Piro | Simulating LTE cellular systems: an open-source framework[END_REF] which is an open-source simulator for LTE networks.

Our multi-cell WCN simulation scenario consists of 19 cells where we have one eNodeB and a variable number of UEs in each cell (see Figure 4). Mobility of each UE is described by the random way-point model [START_REF] Camp | A survey of mobility models for ad hoc network research[END_REF]. A 10MHz downlink bandwidth is considered for each base station. We have considered a variable mobility speed in order to analyze both pedestrian and vehicular UEs. Furthermore, the impact of the inter-cell interference is also considered. Table I provides a list of the simulation parameters. It should be mentioned that all simulation results are averaged over four simulations. In our simulations, each UE receives one video flow, one VoIP flow and one best effort flow at the same time. The video flow is encoded with H.264 standard compression at 128kbps average rate. The VoIP flow has adopted an ON/OFF Markov model where during the ON period, every 20ms, the resource sends 20 bytes sized packets (source data rate is 8kbps) while during the OFF period the rate is zero because of the presence of a voice activity detector. At last, we considered infinite buffer sources for the best effort flow [START_REF] Piro | Simulating LTE cellular systems: an open-source framework[END_REF]. At the Physical layer, we have supposed that the eNodeB exploits two antenna ports and a power transmission equal to 43dbm, uniformly diffused over all the 50 available sub-channels. In our simulations, the urban channel model is considered where the Path-Loss (PL) between the eNodeB and the UE is estimated according to [START_REF] Gpp | Technical Specification Group Radio Access Network; Physical layer aspects for evolved Universal Terrestrial Radio Access (UTRA) (Release 7)[END_REF] by

P L = 128.1 + 37.6log(d)
where d is the distance between the UE and the eNodeB in km. The large scale shadowing fading is modeled via a log-normal distribution and finally, time-frequency correlated signal multipath is modeled by using the Rayleigh fading channel model [START_REF] Piro | Simulating LTE cellular systems: an open-source framework[END_REF].

According to Theorem 3.1, in order to maintain the q i (n) ≥ 0, we must have -1 < k 1 < 0 and k 2 > 0. A numerical computation shows that by taking k 1 closer to -1 we have the q i (n) closer to 0. Therefore, we choose k 1 = -0.9 and k 2 = 0.5 for our simulations. Also, we choose c = 0.

In our simulations, we considered a variable set of delay targets in the interval [40ms -100ms]. It should be mentioned that we considered the same target delays for both video and VoIP flows to ensure that they are synchronously played out at the same UE, which is an important characteristic in video conferencing [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF]. It is important to remark that the enqueued packets are deleted from the transmission queue just when they expire or on the other word, they have not been transmitted before their target delay. In multimedia communications, packets received by the UE after their target delay have to be considered as lost packets because they are no longer usable by the decoding process.

Concerning multimedia data flows, PLR is a standard metric for evaluating QoS offered by the system at network layer. As the best effort data flows do not have strict QoS specifications, we consider the fairness index and the total goodput to compare the performance of different scheduling algorithms [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF].

Figures 5 illustrates the achieved PLR of video flows. It is easy to see that the achieved PLR increases with a higher network load (number of UEs). Furthermore, network with 120km/h user mobility speed achieves a higher PLR as the rate of useful bits which has been transmitted successfully by BE flows. It is easy to see that the goodput decreases with a higher network load. It should be mentioned that we have a higher goodput with respect to the PFC and FLS by using the LOG-rule and EXP-rule scheduling methods. This is due to the fact that in TLS (with the PFC or the FLS) the RT data flows have a higher priority compared to the non-RT data flows (i.e, BE flows).

Finally, Figure 9 provides the computation time of the PFC and the FLS for a 10s simulation for a video data flow with 0.08s target delay. Obviously, the FLS has a significant higher computation time with respect to the PFC. According to [START_REF] Piro | Twolevel downlink scheduling for real-time multimedia services in LTE networks[END_REF], for each flow, the FLS should execute (M -1) multiplications and 3(M -1) + 1 sums in order to compute u i (n) where M is the target delay of the RT data flow.

V. CONCLUSION

In this article, we developed a new QoS aware TLS for RT data flows in downlink transmission of LTE cellular networks. The proposed TLS exploits the PFC at the upper-level scheduler. The PFC is well suited to the upper-level scheduling problem because it maintains the state of the system (queue length) higher than a target value (queue min = 0) while keeping the state as close as possible to this target value in order to optimize the PLR for the RT data flows in downlink transmission. Simulation results show that by using our proposed TLS, UEs experience a lower PLR specially for video flows compared to the EXP-rule and the LOGrule scheduling methods. Moreover, our proposed TLS slightly improves the PLR for RT data flows compared to the FLS while it significantly reduces the computation time.

Fig. 1 :

 1 Fig. 1: A simple model of packet scheduler

Fig. 2 :

 2 Fig. 2: Two-Level scheduler for real-time data flow in downlink direction.

Fig. 4 :

 4 Fig. 4: Simulation scenario for a cell

Fig. 5 :

 5 Fig. 5: Packet Loss Ratio of video flows

Fig. 6 :

 6 Fig. 6: Packet Loss Ratio of VoIP flows

Fig. 9 :

 9 Fig. 9: Computation time of the FLS and the PFC.

TABLE I :

 I Simulation parameters

	Simulation time	100 s
	Number of cells	19
	Radius of each cell	0.5 km
	Bandwidth of each cell	10 MHz
	Number of UEs per cell	10, 15, 20
	Speed of UEs	3 km/h (pedestrian), 120 km/h (vehicular)

  VoIP flows is considerably lower compared to the achieved PLR of video flows. As we have the PF scheduling method at the lower-level scheduler, the VoIP flows have the higher priority due to their lower source bit rate. The PFC and FLS have better performances compared to the LOGrule and EXP-rule in maintaining the target delay of the video VoIP flows while the PFC has a slightly lower achieved PLR of the video and VoIP flows with respect to the FLS. Figures 7 shows the fairness index for different number of UEs with different mobility speeds. It is easy to see that all considered scheduling methods provide a high degree of fairness. Figures 8 provides the total goodput of BE flows for different number of UEs with different mobility speeds. The goodput is determined

		1						10
			PFC, 3km/h	EXP-rule, 3 km/h	LOG-rule, 120 km/h			PFC, 3km/h	EXP-rule, 3 km/h	LOG-rule, 120 km/h
	Fairness Index	0.8 0.85 0.9 0.95	FLS, 3 km/h LOG-rule, 3 km/h	PFC, 120 km/h FLS, 120 km/h		EXP-rule, 120 km/h	Total goodput of BE flows (Mbps)	2 4 6 8	LOG-rule, 3 km/h FLS, 3 km/h	FLS, 120 km/h PFC, 120 km/h	EXP-rule, 120 km/h
		0.75	0.04	0.06	0.08	0.1		0	0.04	0.06	0.08	0.1
	Target delay (s) (a) 10 UEs Target delay (s) 0.06 0.08 EXP-rule, 3 km/h PFC, 120 km/h FLS, 120 km/h (b) 15 UEs Target delay (s) 0.06 0.08 EXP-rule, 3 km/h PFC, 120 km/h FLS, 120 km/h (c) 20 UEs Fig. 7: Fairness Index best effort flows. 0.04 0.1 PFC, 3km/h FLS, 3 km/h LOG-rule, 3 km/h LOG-rule, 120 km/h EXP-rule, 120 km/h 0.04 0.1 PFC, 3km/h FLS, 3 km/h LOG-rule, 3 km/h LOG-rule, 120 km/h EXP-rule, 120 km/h compared to the one with 3km/h user mobility speed. This Fairness Index 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Fairness Index 0.75 0.8 0.85 0.9 0.95 1 is due to the fact that the transmission channel quality varies faster with a higher mobility speed which leads to a higher slip in MCS selection by AMC module. It is important to mention that the achieved PLR increases by taking a lower target delay due to a larger quota of packets passing the deadline determined by the QoS requirements. Consequently, we have the highest PLR for the simulation scenario of a network with 20 UEs per cell, 40ms the target delay and 120km/h user mobility speed. Figures 6 illustrates the achieved PLR of VoIP flows. The achieved PLR for Target delay (s) (a) 10 UEs Target delay (s) 0.04 0.06 0.08 Total goodput of BE flows (Mbps) 0 2 4 6 8 10 PFC, 3km/h FLS, 3 km/h LOG-rule, 3 km/h EXP-rule, 3 km/h PFC, 120 km/h FLS, 120 km/h (b) 15 UEs Target delay (s) 0.04 0.06 0.08 Total goodput of BE flows (Mbps) 0 2 4 6 8 10 PFC, 3km/h FLS, 3 km/h LOG-rule, 3 km/h EXP-rule, 3 km/h PFC, 120 km/h FLS, 120 km/h (c) 20 UEs Fig. 8: Total Goodput of best effort flows. 0.1 LOG-rule, 120 km/h EXP-rule, 120 km/h 0.1 LOG-rule, 120 km/h EXP-rule, 120 km/h

and

VI. ACKNOWLEDGEMENTS

This work is supported by the region Poitou-Charentes of France and The French National Center for Scientific Research (CNRS).