Jean-Marc Kelif

Stephane Senecal

Marceau Coupechoux

Quality of Service and Performance Evaluation: A Fluid Approach for Poisson Wireless Networks

The evaluation of performance and quality of service needs the modeling of wireless networks. Among the different models usually considered, the hexagonal network is the most popular. However, it requires extensive numerical computations. The Poisson network model, for which the base stations (BS) locations form a stochastic spatial Poisson process, allows to consider a non constant distance between base stations. Therefore, it characterizes more realistically operational networks. The Fluid network model, for which the interfering BS are replaced by a continuum of infinitesimal interferers, allows to establish closed-form formula for the SINR (Signal on Interference plus Noise Ratio). This model was validated by comparison with an hexagonal network. The two models establish very close results. In this paper, we show that the Fluid network model can also be used to analyze Poisson networks. Therefore, the evaluation of quality of service and performance becomes very easy, whatever the type of model, by using the analytical expression of the SINR established by considering the fluid model.

I. INTRODUCTION

Performance and quality of service (QoS) evaluations of wireless networks can be analyzed by using simulations or analytical models. Several QoS parameters (like throughput, outage probability) can be derived from the SINR distribution. Analytical models thus try to derive simple SINR formula in order to quickly evaluate the performance of a cellular network. Due to the explosion of mobile services demand, the estimation of performance and QoS has to be more and more precise. Therefore, their analysis need tractable and accurate models of networks.

The most popular wireless network model is the hexagonal one: the transmitting base stations constitute a regular infinite hexagonal grid ([1]- [START_REF] Viterbi | CDMA -Principles of Spread Spectrum Communication[END_REF]- [START_REF] Lagrange | Principes et évolutions de l'UMTS[END_REF]). Although this model seems rather "reasonable" for regular deployments of base stations, it is intractable from an analytical point of view. Therefore, it implies extensive numerical computations. Several numerical techniques have been developed to perform such computations. For example, Monte Carlo simulations are widely used in conjunction with this model [START_REF] Gilhousen | On the Capacity of Cellular CDMA System[END_REF], [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF] or numerical computations in hexagonal networks [START_REF] Viterbi | Other-Cell Interference in Cellular Power-Controlled CDMA[END_REF], [START_REF] Viterbi | CDMA -Principles of Spread Spectrum Communication[END_REF].

Let us notice that tractability of a wireless network model allows to dramatically reduce the computation time. For example, several optimization problems that can be solved by 1 Jean-Marc Kelif is with Orange Labs, France Email: jeanmarc.kelif@orange.com 2 Stephane Senecal is with Orange Labs, France Email: stephane.senecal@orange.com 3 Constant Bridon is with ENS Cachan and Orange Labs, France Email: constant.bridon@ens-cachan.fr 4 Marceau Coupechoux is with Telecom ParisTech, France Email: marceau.coupechoux@telecom-paristech.fr metaheuristics like Tabu Search [START_REF] Kamal | Tabu Search for Dynamic Spectrum Allocation (DSA) in Cellular networks[END_REF] or Simulated Annealing [START_REF] Sambale | Decode-and-Forward Relay Placement for Maximum Cell Spectral Efficiency[END_REF], require extensive SINR calculations. Moreover, a network model highlights the parameters of the system, and thus allows to better understand their impact, especially on the quality of service and performance.

Another wireless network model consists in the Poisson model: the base stations are randomly distributed on the considered area according to a spatial Poisson process [START_REF] Baccelli | Tournois Downlink admission/congestion control and maximal load in CDMA networks[END_REF]. Although this model is less popular than the hexagonal one, it allows to take into account a more realistic environment since the distances between base stations are not constant. Measurements from operational networks have shown that hexagonal model is rather optimistic and Poisson model is rather pessimistic.

Another model of network is the fluid model [11] [12]. This model considers the interfering base stations as a continuum of infinitesimal interferers distributed in space. The main interest of this model consists in its tractability, in the possibility to establish closed form formula of the SINR, whatever the location of a UE, and to establish the SINR distribution [START_REF] Kelif | Impact of Topology and Shadowing on the Outage Probability of Wireless Networks[END_REF] [14], too. Furthermore, the fluid model was shown to be very close to the intractable hexagonal model in terms of evaluation of the SINR. As a consequence the evaluation of the quality of service and performance can be done by using the simple expression given by the fluid model when the spatial distribution of base stations is regular hexagonal.

Our contribution: In this article, we show that the evaluation of the quality of service and performance, based of the distribution of the SINR, can be done by using the simple expression given by the fluid model, when the spatial distribution of base stations is random Poisson, and whatever the density of BS. Therefore, the determination of the quality of service and performance of a wireless network can be done in a simple way, whatever the network model considered.

The organization of the paper is as follows. In Section II, we present the system model. We introduce the different types of network models used to analyze wireless networks in Section III. We show, in Sections IV and V , that performance and QoS of Poisson model can be fitted by using a fluid model. A conclusion is given in Section VI.

II. SYSTEM MODEL

Let us consider a wireless network. We focus on the downlink transmission part. Our aim is to evaluate the performance and the quality of service of a single user. We consider an access technology in which the radio resources of a base station (BS) are divided in a number of parallel, orthogonal, non-interfering channels (subcarriers), i.e. OFDMA. Therefore, only inter-cell interference is considered, no intra-cell interference.

A. SINR of a user

We consider a single frequency network composed of N base stations, transmitting at power P on each subcarrier. We define g i (u) the path gain between BS i and user u on a given subcarrier. The SINR u of user u served by BS i on the considered subcarrier is given by:

u = P g i (u) P j6 =i P g j (u) + N th . ( 1
)
with N th the thermal noise on a subcarrier. We consider a urban environment, where the thermal noise can be neglected. The SINR can then be expressed as:

u = P g i (u) P j6 =i P g j (u) , (2)

B. Performance and quality of service

The knowledge of the SINR allows to calculate the throughput that may be reached by a user. Indeed, considering any subcarrier as an AWGN (Additive White Gaussian Noise) channel, the SINR received by a mobile enables the determination of the spectral efficiency D u (in bits/s/Hz) by using the Shannon formula:

D u = log 2 (1 + u ). (3)
Let us notice that there are alternative approaches like using a modified upper bounded Shannon formula or throughput-SINR tables coming from physical layer simulations. Moreover, expressions (2) and ( 3), calculated at any location of the network, allow an evaluation of the CDF (Cumulative Distribution Function) of the SINR (or the throughput). The SINR CDF also provides the outage probability, i.e. the probability that a user cannot be accepted in the network since he cannot have a sufficient throughput. It is therefore important to develop a method which allows to determine these characteristics with a high accuracy, for a user at any distance r from his serving BS.

Moreover, since the throughput allows to know the quality of service that can be offered to a user, these methods make it possible to determine this characteristic with a high accuracy in a simple way. In particular, the minimum throughput, obtained at cell edge, can be derived. By doing an integration all over the cell range, the average throughput of the cell can be calculated, too. Dynamical analysis also need the knowledge of the SINR [START_REF] Rong | Performance Evaluation of Cellular Networks Offering TV Services[END_REF] as input.

III. WIRELESS NETWORKS MODELS

In this section, we review three cellular network models: the hexagonal one, the Poisson one and the Fluid one. We also recall how to calculate the SINR thanks to the fluid model.

A. Hexagonal Network

In a hexagonal configuration, the base stations are regularly distributed in the area (Fig. 1). The model is characterized by a single parameter, which is the Inter Site Distance (ISD). Since all the zones covered by any BS are equivalent, it is sufficicent to analyze any unique cell in the whole system.

B. Poisson Network

In a real network, the inter site distance is variable. The Poisson model network, characterized by the density of BS, allows to take it into account by considering a Poisson distribution of base stations in a given area (Fig. 2). In this configuration, the cells of the network form a Voronoï diagram. Therefore, it becomes necessary to analyze a wide zone, with a great number of base stations, to determine the statistical characteristics of the network in terms of performance and quality of service. We consider a path gain g j (u) = Kr ⌘ j (u), where K is a constant, r j (u) is the distance between user u and BS j and ⌘ the path loss exponent. Let us consider that a BS (eNode-B) transmits at power P on each subcarrier. Denoting r = r i , we can express (2) as (dropping u):

= r ⌘ P j6 =i r ⌘ j , (4)
By considering expression (4), we can see that the calculation of the SINR of a user depends on its location, on the location of its serving BS, and on the location of each interfering BS.

Considering the fluid model, the SINR only depends on the distance r of the user to its serving base station [START_REF] Maqbool | Analytical Performance Evaluation of Various Frequency Reuse and Scheduling Schemes in Cellular OFDMA Networks[END_REF]:

(r) = ⌘ 2 2⇡⇢ BS r ⌘ (2R c r) 2 ⌘ (5) If ⇢
BS is known, this analytical model allows to significantly simplify the calculation of the SINR: the only required variable is the distance of the mobile to its serving BS. This model has been proven to be reliable and close to the reality for homogeneous networks [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF], as well as for heterogeneous networks [START_REF] Kelif | Impact of Transmitting Power on Femto Cells Performance and Coverage in Heterogeneous Wireless Networks[END_REF].

The cell edge throughput can particularly be calculated by setting r = R c in [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF]. Therefore, the minimum performance and quality of service offered to UE is evaluated in a simple way. Moreover, a simple integration over the cell range allows to calculate the average throughput of the cell.

These results on the fluid model are valid for a constant inter site distance 2R c , and the purpose of the next section will be to find a correspondence between a completely stochastic network and the fluid model.

IV. FLUID MODEL OF POISSON NETWORK

In a real network, as mentioned in the introduction, the inter site distance is evidently variable. However, the fluid model relies on the hypothesis of a regular network. This section shows that it is possible to adapt the fluid model in order to find an equivalence between a Poisson random network and a regular network associated to the fluid model in terms of CDF of the SINR.

The first studied network is a Poisson network. We consider a given area of surface S A . A. Poisson network 1) Base stations distribution: In order to place the base stations, we set the expected half inter site distance as R c . This hypothesis fixes the density of base stations ⇢ BS . The values of ⇢ BS and of the studied surface S A give the Poissonian characteristic of the network: the number of BS is drawn according to a Poisson distribution of parameter ⇢ BS S A . The surface is chosen to obtain in average 50 stations in the area. It allows to have a significant number of cells, representative of a realistic zone covered by BS, and a significant number of interfering BS for the computation of the SINR. Those BS are then placed in the network, with no pairwise constraint (Fig. 2): distances between neighboring base stations may be very low.

2) SINR computation: The users are uniformly distributed on the whole area S A . Then, the SINR of a UE is computed from its definition: the best serving BS gives the power of the received signal, and all the other stations generate interference. Several Monte Carlo simulations are run. At each run, the number and locations of the BS change, whereas the set of studied points (UE) is fixed. As a result, for the set of studied points, we obtain the corresponding SINR with different configurations of BS. Therefore, it becomes easy to compute the CDF of the SINR received by UE in this zone. Considering a toroidal shape of the network allows to consider it as virtually infinite with no "edge effect" for the computation of the SINR.

B. Fluid network SINR calculation

As a comparison of this result, we calculate the SINR in a cell of radius R c with the fluid model by using [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF]. Let us recall that the fluid model has been proven to be a good approximation for the computation of the SINR for hexagonal networks [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF].

V. NUMERICAL EXPERIMENTS A. Assumptions

Various parameters may have an impact on the SINR, and therefore the reachable throughput and the coverage of BSs. We present hereinafter the parameters we chose in our analysis:

• downlink OFDM LTE, • carrier frequency 2.6 GHz,

• channel bandwidth 10MHz,

• BS transmitting power: we set it at 43 dBm, as in a realistic transmission environment, • pathloss parameter : varies between 2.2 and 4.2,

• density of base station ⇢ BS = 1 BS/km 2 . The simulations consider that UEs are uniformly distributed all over the network.

B. SINR CDF for the Poisson and Fluid models

Since the CDF of the SINR characterizes the performance and the quality of service of wireless systems, we establish the CDF of the SINR, considering the Fluid network and the Poisson network for a wide range of values of ⌘, comprised between 2.2 and 4.2. Fig. 4 shows an example of the curves established for ⌘ = 2.8 and 3.6. We first notice that a Poisson network gives always lower values of SINR than a regular hexagonal network: let us recall that the fluid model was validated by comparison to a hexagonal network. The difference between the two curves is about 2 dB for ⌘ = 2.8 and 4 dB for ⌘ = 3.6. These results mean that the Poisson network is rather "pessimistic" and the Fluid network "optimistic". It can be explained by the fact that there is a probability to have interfering base stations very close to a user in a Poisson network. Another noticeable aspect is the difference of those two curves along the abscissa axis. In fact, observations point out that the CDF of the SINR of the Poisson network is similar to the CDF of the SINR established by using the Fluid network model, translated along the abscissa axis. Moreover, this translation seems to depend on the value of the propagation parameter ⌘. The simulations take into account 11 different values of ⌘ with an increment of 0.2, ranging from 2.2 to 4.2 (usual range for the path-loss exponent is comprised between 2.8 and 3.6).

C. SINR CDF Fitting

As mentioned, the difference between the two SINR CDF curves along the abscissa axis seems to be constant, and increases with ⌘. Therefore, we apply a curve fitting with a polynomial approximation in order to model this phenomenon. In this aim, we compute the mean value of that difference for each ⌘. Then, the comparison of this value with a polynome of degree 1 in ⌘ is sufficient to achieve an excellent approximation. The result turns out to be interesting: the linear expression in ⌘ fits the curve with the same coefficients, whatever the value of ⌘. The SINR calculated is corrected by the linear expression, and in this case, we establish the formula:

SIN R f itted F luid = SIN R F luid (a⌘ + b) (6)
which yields

CDF P oisson ⇡ CDF f itted F luid (7)
where a = 3 and b = -6.

As observed on some examples (Fig. 5 and6), for ⌘ = 2.8, 3, 3.6, 3.8, the CDF of the SINR established by the fitted Fluid model and by the Poisson model are very close: the differences between them are less than 0.4 dB (let us notice that, for an outage probability of 5%, this difference may reach 0.9 dB for some values of ⌘).

D. Impact of the density of BS

We analyzed [START_REF] Kelif | Coupechoux A Fluid Approach for Poisson Wireless Networks[END_REF] the impact of the density of BS by considering two other densities of BS : ⇢ BS /10 = 0.1 BS/km 2 and 10⇢ BS = 10 BS/km 2 . It is interesting to observe that the fitting does not depend on the density of BS. Indeed [START_REF] Kelif | Coupechoux A Fluid Approach for Poisson Wireless Networks[END_REF] shows that when the density is multiplied by a factor 10 or divided by a factor 10, the CDF are identical, and the fitting is the same. As a consequence, these results can be used whatever the density of the network, i.e. whatever the intersites distance between base stations. It is easy to understand this property when we realize that, since ⇢

BS = p 3 6R 2 c
, the expression of the SINR [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF] given by the fluid model, can be expressed as:

(x) = 6 p 3 ⌘ 2 2⇡ x ⌘ (2 x) 2 ⌘ (8)
where x = r Rc represents the relative distance of a UE to its serving base station. This expression does not explicitely depend on the ISD (distance between neighboring BS). This expression holds whatever the ISD. Moreover, it does not depend on the density of BS.

E. Correlation coefficient

We compare the correlation coefficient (9) computed between the CDF curves of the fitted fluid model and of the Poisson model. We recall that, given X and Y , two samples of length n with respective means X and Ȳ , the correlation coefficient ⇣ is given by:

⇣ = P n i=1 (X i X)(Y i Ȳ ) q P n i=1 (X i X) 2 q P n i=1 (Y i Ȳ ) 2 (9)
This comparison is synthetized in Table I illustrating that these correlation coefficients are excellent. The linear approximation becomes less accurate for the minimum and the maximum values of ⌘: 2.4 and 4. Indeed, when this coefficient is not better than 0.99, the correlation is usually considered as not good.

F. Model limitations

The fluid model with a simple linear fitting in ⌘ with known parameters gives the same CDF of the SINR as a Poisson network computing the SINR by considering ( 5) and ( 6). The fluid model is thus also reliable with random networks. For values of ⌘ lower than 2.6 or higher than 3.8 (i.e. out of the range of usual values of ⌘), it can be observed that the proposed fitting is less accurate.

In conclusion, in this validity field, we can say that there is an equivalence between the Fluid network and a Poisson network, in terms of CDF of SINR, therefore in terms of outage probability and throughput. The fluid model can thus be used to evaluate the performance and quality of service of any kind of network, hexagonal and Poisson.

VI. CONCLUSION

In this paper, we established that the Fluid network model can be used to analyze Poisson networks as well as hexagonal networks. As a consequence we showed that the evaluation of performance and quality of service can be done in a simple way, whatever the type of wireless network model, by using the analytical expression of the SINR established by the Fluid network model.

Fig. 1 .

 1 Fig. 1. Hexagonal Network

Fig. 2 .

 2 Fig. 2. Poisson Network

Fig. 3 .

 3 Fig. 3. Fluid model: Network and cell of interest

Fig. 4 .

 4 Fig. 4. CDF of the SINR with ⌘ = 2.8 (left) and ⌘ = 3.6 (right), for a Poisson model network and a Fluid model network.

Fig. 5 .

 5 Fig. 5. CDF of the fitted SINR with ⌘ = 2.8 (left) and ⌘ = 3 (right), for a Poisson model network and a Fluid model network.

Fig. 6 .

 6 Fig. 6. CDF of the fitted SINR with ⌘ = 3.6 (left) and ⌘ = 3.8 (right), for a Poisson model network and a Fluid model network.

TABLE I CORRELATION

 I COEFFICIENT FOR THE COMPARISON OF THE CDF OF THE SINR FROM THE FITTED FLUID NETWORK AND THE POISSON NETWORK WITH RESPECT TO ⌘

	⌘	Correlation coefficient ⇣
	2.2	0.9755
	2.4	0.9961
	2.6	0.99909
	2.8	0.99966
	3	0.99952
	3.2	0.99941
	3.4	0.99934
	3.6	0.99897
	3.8	0.99861
	4	0.99824
	4.2	0.99683

ACKNOWLEDGMENT This work was partially supported by the Seventh Framework Programme for Research of the European Commission under grant number HARP-318489 and by the ANR France NETLEARN project ANR-13-INFR-004.


Denis Dezest

Fabrice Mathieu

Laure Liviu Nicu

PZT Nanofilm-B

Keywords: NEMS, actuators, piezoelectric a I

In this work, we present an unp integration of piezoelectric actuation mea functional nanoresonators at the wafer scale. W lead titanate zirconate (PZT) as piezoelectri because of its excellent actuation prope geometrically constrained at extreme scale. promising ways for NEMS to be used in con transduction capabilities are integrated at th providing effective fabrication process flow at t

INTRODUCTION

For nanoelectromechanical systems (NEM a viable alternative to their microscale microelectromechanical systems or ME integrated transduction at the wafer sca ultimate goal to be achieved. By integrated means both actuation and sensing of movem freestanding nanostructures. So far, releva sense have been obtained at the chip level eit traditional transduction schemes at the n delivering brand new schemes exclusively NEMS realm [START_REF] Unterreithmeier | Universal Transduction Scheme for Nanomechanical Systems Based on Dielectric Forces[END_REF]. Recent efforts to demo NEMS with integrated transduction means h 5] but remain rather seldom which demons the gap between chip and wafer scale is.

In this work, we demonstrate the piezoelectric actuation means on arrays of the wafer scale. We use lead titanate zi piezoelectric material mainly because of its e properties even when geometrically constr scale [START_REF] Guillon | Lead Zirconate Titanate Nanoscale Patterning by UV-based Lithography Lift-Off Technique for Nano-Electromechanical Systems Applications[END_REF].

II. FABRICATION

A. Design

Chips (16 mm x 16 mm) of 256 silicon have been fabricated from a 4 inches siliconwafer. Resonators with different sizes and been implemented. The beams' width has be µm to 5 µm and the length from 6 µm to 12 stack composed of a 150-nm thick PZT (Pb film sandwiched between a 100-nm thick la (LaNiO 3 or LNO) bottom electrode and a titanium/platinum (Ti/Pt) top electrode serve

B. Fabrication

The fabrication method was performed (100) wafer (340nm/1µm/525µm). Arrays o of different lengths (from 2µm to 10.5µm) have been placed on the wafer. The width 1µm to 2.8µm. A piezoelectric stack includ 120nm thick) and bottom LNO (100nm th well as the PZT material (150nm thick) has each nanocantilever in two covering configur or half-length of the nanocantilever. Th beneath the piezoelectric stack was made o silicon layer covered by a 30nm thick therm The whole fabrication process was perform scale, the lithography steps being realized u photo repeater (CANON FPA 3000i4/i5).

First, a 30 nm-thick layer of SiO2 is th serve as an electrical insulation layer. Both silicon device layers are then successively e Alcatel Inc.) by reactive ion etching (RIE) to of the structures. Collective integration of stack onto individual structures starts with th bottom electrode. A 100 nm-thick layer of and patterned on the structures by lift-o resolution bi-layers positive photoresist (purchased from MicroChemicals, Germany diffusion barrier against lead atoms migratio crystallization step. It prevents lead atoms to with any underlying Si-based material. A low magnetron sputtering deposition compati patterning is then used to define a 150 nm-th PbZr0.54Ti0.46O3 (PZT 54/46) [START_REF] Guillon | Lead Zirconate Titanate Nanoscale Patterning by UV-based Lithography Lift-Off Technique for Nano-Electromechanical Systems Applications[END_REF]. This co to the morphotropic phase boundary (M exhibits optimal piezoelectric and dielectric p patterning by lift-off is an alternative approa dry and wet etching techniques. The former to deteriorate the PZT material properties due of both physical damages and chemical con while the latter requires using HF solutio incompatible with other etchant-sensitive ma from undesirable undercut of masked deposition, a mandatory annealing at 625°p erformed to crystallize the PZT into the pe 12/120 nm-thick layer of Ti/Pt is then depos by lift-off as the top electrode. Each array of individually addressed with 100/700 n interconnections. The structures are released the sacrificial layer after a photoresist encap other sensitive layers. We use a 2.6 µm positive photoresist (purchased from Germany) with an additional baking at 115°C etching to enhance the photoresist adhesion. etchant (BOE) composed of 7-1 ratio of am (NH4F) -hydrofluoric acid (HF) mixture etch rate of around 77 nm/min is used at roo release the structures. Two subsequent photoresist encapsulations are performed to adhesion defects due to long etching time order to release the largest structures up on SOI 4" P-type of four cantilevers and similar width h was varied from ding top Ti-Pt (12hick) electrodes as been patterned on rations: full-length he elastic support of the SOI device mal silicon dioxide. med at the wafer sing a UV stepper hermally grown to thermal oxide and etched (AMS4200 o pattern the shape f the piezoelectric e deposition of the LNO is sputtered off using a high-LOR3A/ECI3012 y). LNO acts as a on during the PZT o diffuse and react w-temperature RFible with lift-off hick active layer of omposition is close MPB) where PZT properties [START_REF] Chen | Electrical properties' maxima in thin films of the lead zirconate-lead titanate solid solution system[END_REF]. PZT ch to conventional has been reported e to a combination ntaminations [START_REF] Chung | Etching Effects on Ferroelectric Capacitors with Multilayered Electrodes[END_REF][START_REF] Kim | Study on the etching damage characteristics of PZT thin films after etching in Cl-based plasma[END_REF] on which can be aterials and suffers features. After °C for 30 min is erovskite phase. A sited and patterned f structures is then nm-thick Ti /Au by wet etching of psulation to protect m-thick ECI 3012 MicroChemicals, C for 1 min before . A buffered oxide mmonium fluoride with an isotropic om temperature to wet etchingreduce photoresist while being able p to 5 µm wide.

Nevertheless, this critical step 33 %) of the implemented fabri

C. Experimental set-up

The actuation of the resonato different configurations: exte integrated actuation (on-chip) mounted onto a piezodisk ele analyzer, as shown on Fig.

collectively actuated through th read-out. In the latter configu metalized polyimide support interconnections. Devices of a s by biasing the top and bottom network analyzer.

For both actuation configurati individual cantilever is optica Fabry-Perot interferometer be clamped on a motorized piez translations) inside a vacuum secondary vacuum (around 10-30 mW helium-neon (of wav focused through a beam expan resonator and a long working (20X, N.A. 0.28). The fine posi by the piezo-stage displacem monitored with a CMOS cam ( 470nm). The air gap be and the substrate underneath Mechanical oscillations of the size of the optical cavity and interferences between the m intensity of the reflected opti driving frequency and its am resonator displacement. Det interferometry principle used as have been described by Carr a signal is reflected back throug diverted through a beamspl photodetector (New Focus mod output is connected to the ne 4395A) to track the mechanical frequency of the excitation amplifier detection. ors is performed according two ernal actuation (off-chip) and ). In the former the chip is ectrically driven by a network 2(a). Resonators are therefore he piezoshaker and individually uration, the chip is glued to a t with dedicated gold-wiring same array are then self-actuated electrodes (see Fig.

III. Q-FACTOR VERSUS AIR PRE

Performance of resonating structures gene many dissipative mechanisms [START_REF] Imboden | Dissipation in nanoelectromechanical systems[END_REF] which amplitude and the sharpness of the Dissipation, defined as the inverse of t D Q ), is a measure of the mechanica cycle of the oscillation (it is the ratio of thi the total energy stored in the resonator). mechanism can be described as an additiona linear force proportional to the resonator ve motion. Since no correlation exists betwee sources, the total dissipation within a system sum of all the dissipative contributions:

Q Q
For biosensing [START_REF] Roupioz | Individual blood-cell capture and 2D organization on microarrays[END_REF] applications, reson in gaseous or liquid media for which vis usually the dominant source of dissip investigated the effect of pressure on the structures with a multilayer stack. Measureme according to the following manner. The ch down to a pressure of 1.1 10 m measurements of the quality factor are c atmospheric pressure. A leak valve is used inside the chamber while monitoring the va pressure gauge (Inverted magnetron Pirani g 700, Agilent Inc.). The pressure gaug combination of two measurement systems cathode) with a measuring range from 5 atmospheric pressure and an accuracy Measurements of the quality factor are pe pressure is stabilized while the pumps are pumps are switched off during measurement starts to slightly increase toward atmospheric obtained for the fundamental resonant mod with geometry C8 are shown in Fig. 3. The Kn defined as the ratio between the mean fr molecules and the cantilever width w is plotte ESSURE erally suffer from h affect both the peak resonance. the quality factor al energy lost per is lost energy over . Any dissipation al and uncorrelated elocity that resists en the dissipation m is defined as the [START_REF] Villanueva | A Nanoscale Parametric Feedback Amplifier[END_REF] nators are operated scous damping is pation. We have quality factor for ents are performed hamber is pumped mbar after which carried out up to d to flow nitrogen cuum level with a gauge model FRGge is actually a (Pirani and cold 5 10 mbar to y of ± 30 %. erformed after the still on. Then the ts and the pressure c pressure. Results de of a cantilever nudsen number K free path λ of gas ed on the top axis. Extrinsinc damping due generally described by three pressures (K 10) the is mainly domina independent mechanisms (loss value of 837 for the quality fac at the lowest pressure and is se pressures (0.1 10) co regime in which energy los between individual gas molecu becomes the dominant dissipat structure an additional dissipati damping has to be added to ta an air gap between the resona total dissipation (1) becomes in

Q FM Q Q
As shown in Fig. 3, data a prediction regarding the accura spite of the complexity of the deviation from the model occu mbar (K 0.1) due to the cr and the continuum regimes. T this demarcation is in the sa reported elsewhere [START_REF] Li | Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications[END_REF][START_REF] Li | Bottom-up assembly of large-area nanowire resonator arrays[END_REF]. I molecules have to be consider fluid with a mean free path sma However none of the existing [START_REF] Newell | Miniaturization of Tuning Forks[END_REF][START_REF] Bergaud | Multi-mode air damping analysis of composite cantilever beams[END_REF][START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] are suitable for the the fabricated cantilevers and ar quality factor for a multilayer stack C8 for the fundamental flexural mode. ols) are fitted with a theoretical model olecular regime taking into account the rtical dashed line drawn at K 10 e intrinsic and the molecular regime. 0.1 pinpoints the crossover between the he relative variations of the resonant frequency in vacuum f plotted versus in the inset.

to a surrounding medium is different regimes [START_REF] Newell | Miniaturization of Tuning Forks[END_REF]. At low e so-called intrinsic regime, ated by intrinsic and pressurees within the material itself). A ctor is experimentally measured et as the Q value. Higher rrespond to the free molecular ss due to momentum transfer ules and the resonating structure tion mechanism. For suspended ion term relative to squeeze-film ake into account the presence of ator and the substrate [START_REF] Bergaud | Multi-mode air damping analysis of composite cantilever beams[END_REF]. The n this regime:

Q (2)
are in a fair agreement with the acy of the pressure gauge and in e multilayer stack resonator. A urs at a pressure of around 265 rossover between the molecular The Knudsen number relative to ame range of values [0.1-0.7] In the continuous regime gas red as a viscous incompressible aller than the cantilever's width. models for the viscous regime e non-homogenous geometry of re therefore not presented here.

The resonant frequency of the cantil expected with the increasing gas pressure. vibrates in a gaseous environment, it pushes of air around it that is generally described b However the total shift of the resonant freque to air is relatively small (0.12 %). It worth no increase is observed in the crossover reg molecular and viscous regimes prior to the g the resonant frequency. Such phenomenon elsewhere [START_REF] Gavan | Effect of pressure on the Q factor and the resonance frequency of SiN microcantilevers[END_REF] but is not yet fully understood IV. ESTIMATION OF THE LIMIT OF D For applications such as mass sensing, the (LOD) of the system is one of the key featu performance. For a beam loaded with a sma uniformly distributed over its surface, the s frequency is given by [START_REF] Ekinci | Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems[END_REF]:

δf δM 2M f
where δf is the resonant frequency shift due δM and f is the natural frequency. In this mass M is defined as follows:

M L m x Φ x dx L Φ x dx L
In the case of a uniform beam, the effectiv equal to the total mass of the beam. Howev case of a laminated beam with a non-uniform found to be equal to a fraction of the total and is computed by the FE model. The defined as the smallest added mass δM smallest measurable shift δf in the n LOD δM .

Experimentally the LOD of a resonato by monitoring the nominal resonant frequ variations) over a certain period of time a Allan deviation σ A , square root of the is an estimator of the true mathematic corresponds to an IEEE standard [START_REF] Barnes | Characterization of Frequency Stability[END_REF] co characterize the frequency stability of an osc domain. For a finite data set of N values frequencies f t measured with a sample ra frequency deviations can be defined accordin ONCLUSION e wafer-scale of self-actuated ith an integrated 150 nm-thick reported. Standard surface such as UV lithography appear en the degree of miniaturization and high-throughput manufacturing of resonators with integrated transduction means. Dynamic mechanical characterization of the fabricated cantilevers with different geometries has shown a preserved mechanical integrity and a dynamic behavior in a fair agreement with the finite element analysis. Quality factor values of ~900 in a vacuum and ~130 in air have been obtained at room temperature for fundamental resonant responses in the megahertz range. Piezoelectric actuation capabilities were validated in the linear regime for the first time at this integration level in spite of a low piezoelectric constant probably due to the lateral confinement. Discussion on the piezoelectric sensing shows how critical the transition from the micro-scale to the nano-scale is. This work shows promise and paves the way for further research dedicated to the NEMS realm to address the challenge of the transduction integration at the nano-scale to provide fabrication at the wafer-scale.

y f t f f A different

Fig. 1 .

 1 Fig. 1. (a) Wafer-scale fabrication of wafer. (b) Each chip is composed o scanning electron microscope (SEM) cantilevers with geometries A and B The color code use is the same for the two configurations for the multilayer st

Fig. 2 :

 2 Fig.2: The two actuation configuration chip) are respectively shown in (a) and

  2(b)) with a ions, the mechanical motion of ally read-out by a home-made ench. In this setup, a chip is oelectric precision stage (XYZ m chamber pumped down to a -6 mbar) at room temperature. A velength 633 nm) laser is nder (X5) onto the surface of a g distance microscope objective itioning of the laser is controlled ment while the operation is mera under a LED blue light etween the suspended resonator acts as a Fabry-Perot cavity. driven resonator affect the gap d modulate the intensity of the multiple reflected beams. The ical signal is modulated at the mplitude is proportional to the ails of the the Fabry-Perot s a sensing technique for MEMS and Craighead [10]. The optical gh the microscope objective and itter toward an AC coupled del 1601). The photodetector AC etwork analyzer (Agilent model l response of the resonator at the signal similarly to a lock-in ns external (off-chip) and integrated (ond (b).

Fig. 3 :

 3 Fig.3: Pressure dependence of the suspended cantilever with geometry C Experimental data (blue square symb (pink solid line) based on the free mo squeeze-film damping FM ). A ver indicates the separation between the Another vertical dashed line at K 0 molecular and viscous regimes. Th frequency f compared to the resonant the increasing gas pressure are shown i

  data set of normalized frequency obtained by changing the sampling time to τ positive integer) and averaging m adjacent obtain a new estimate [21]: clearly ver, in the present m mass density, it is mass of the beam LOD is therefore producing the natural frequency: or can be obtained uency f (or phase and computing the e Allan variance. It cal variance and ommonly used to cillator in the time s of instantaneous ate τ , normalized ng to: (5) y deviations can be τ mτ (with m a t values of y to (6)The Allan variance with o to the standard Allan variance variance) since more sample da calculation, leading to a b therefore a more reliable esti estimator of the Allan varianc[START_REF] Allan | Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators[END_REF]: y are adj between the measurements) no averaged over the sample rate τThe resonant frequency fo was monitored over a total measurement sample rate of τ (10 mbar) at room temper amplitude of 100 mV. The com in Fig.4.

Fig. 4 :

 4 Fig.4: Allan deviation with overlap geometry C8 computed for integration resonant frequency was monitored o temperature for a driving amplitude of The minimum measured equal to σ A τ 2.74 τ 9.29 s which yields accor δM 106 ag. It is commo express the LOD relatively to t so-called surface mass limi therefore: SMLOD 5 ag. μm

ACKNOWLEDGMENT

The French General Delegation for Armament and the French National Agency for Research (grant ANR/PNANO 2008, project NEMSPIEZO 'ANR-08-NANO-015') are gratefully acknowledged for financial support.