Charlotte Kutyla
email: charlotte.kutyla@c2n.upsaclay.fr

Sylvain Lemettre
email: sylvain.lemettre@c2n.upsaclay.fr

Clement Bessouet
email: clement.bessouet@c2n.upsaclay.fr

Alain Bosseboeuf
email: alain.bosseboeuf@c2n.upsaclay.fr

Philippe Coste
email: philippe.coste@c2n.upsaclay.fr

Thierry Sauvage
email: thierry.sauvage@cnrs-orleans.fr

Olivier Wendling
email: olivier.wendling@cnrs-orleans.fr

Aurelien Bellamy
email: aurelien.bellamy@cnrs-orleans.fr

Piyush Jagtap
email: piyush.jagtap@im2np.fr

Stephanie Escoubas
email: stephanie.escoubas@im2np.fr

Wendling Olivier

Cemhti

Stéphanie Escoubas

Bellamy Aurélien

Christophe Guichet
email: christophe.guichet@im2np.fr

Johan Moulin
email: johan.moulin@c2n.upsaclay.fr

Clément Bessouet

Olivier Thomas
email: olivier.thomas@im2np.fr

Oxygen and hydrogen sorption in thermally activated yttrium-based getter thin films for MEMS vacuum packaging

Keywords: Yttrium alloys, oxidation, getter, thin films, lowtemperature bonding

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Since the end of 20 th century, getter materials were mainly studied and used in large vacuum facilities like particle accelerators, to provide Ultra and Extreme High Vacuum (UHV and XHV) [START_REF] Benvenuti | Vacuum properties of TiZrV non-evaporable getter films[END_REF]. Getter film has also an application in the vacuum packaging of microsensors. For several kinds of these devices a medium vacuum (10 -4 mbar to 10 -1 mbar) is required to get optimal performances during their whole lifetime (10 years or more) [START_REF] Dragoi | Wafer bonding technology for new generation vacuum MEMS[END_REF].

After unloading from the deposition set-up, a getter film is passivated by air exposure. Consequently, it must be thermally activated to re-generate a reactive metallic surface able to trap gases with a gas-and time-dependent rate. The activation results from the in-depth diffusion of the atoms of the passivating surface layer (mainly composed of oxides).

The role of a getter film for MEMS packaging is to compensate the outgassing during its sealing and also the residual leaks during its lifetime. The activation time and temperature of the getter must be limited to avoid device degradation. We consider in this work that the allowed thermal budget is at maximum one hour at a temperature of 350 °C.

Getter films previously investigated were mainly based on single element or alloys of transition metals of column IV (Ti, Zr, Hf) and V (V, Nb, Ta), respectively because of their high oxygen solubility or diffusivity [START_REF] Bosseboeuf | Effect of Environment on Activation and Sorption of Getter Alloys and Multilayers for Hybrid Wafer-level Vacuum Packaging[END_REF]. Many of these films have already proved high performances for the sorption of oxygen. In this work, we investigate a new generation of getter materials for their application to MEMS vacuum packaging. We have chosen yttrium as a new base element because of its high affinity with oxygen, its large oxygen diffusivity as well as its high solubility limit (20 at. %) for hydrogen and its ability to easily form a highly stable YH2 hydride [START_REF] Huiberts | Synthesis of yttrium trihydride films for ex-situ measurements[END_REF] [START_REF] Curzon | Thin film studies of yttrium, yttrium hydrides and yttrium oxide[END_REF].

Here, we investigate the sorption capacity of oxygen and hydrogen for Y-based films when activated in an environment with relatively high partial pressures of oxidizing species, as found in MEMS packages.

II. EXPERIMENTAL DETAILS

Thin films with thickness between 200 and 400 nm were deposited by UHV e-beam co-evaporation at room temperature on 100 mm in-diameter (100) silicon wafers. The background pressure before deposition was lower than 10 -8 mbar. The pressure during evaporation did not exceed 10 -7 mbar and was almost exclusively composed of H2 gas (~90%) and of H2O (~10%) as measured by mass spectrometry. The electron beam powers were adjusted to obtain a deposition rate of 0.5 nm/s for the main constituent of the getter film while deposition rates of the other constituents were chosen according to the desired composition.

To simulate the activation step occurring in a packaging process with a similar partial pressure of oxidizing species, the samples were annealed in a furnace under 99.9999 % pure Ar with traces of oxidizing gases, at 1 atm for 1 h and at temperatures ranging from 225 °C to 400°C. The argon flow during annealing was adjusted to 2000 sccm. This duration and range of temperature were chosen to correspond to the bonding temperatures commonly used in wafer-level packaging processes. For each annealing temperature, all samples of different compositions were annealed simultaneously.

The films sheet resistance was measured before and after annealing by the 4-point probe method at room temperature. The Temperature Coefficient of Resistance (TCR) was determined by using a method described previously Surface microstructure of as-deposited and annealed films was observed by Scanning Electron Microscopy (SEM). The in-plane mean grain size was estimated by using the mean linear intercept method following ASTM E112 standard. The microstructure and phase identification were carried out using an X-ray diffractometer (Philips-Panalytical) with Cu Kα source. The samples were analyzed in standard Bragg-Brentano geometry by varying the angle 2θ from 25° to 45°. The diffracted peaks were identified using reference data from powder samples.

Ion Beam Analysis (IBA) techniques were implemented to measure the stoichiometry of metal and light elements in getter films. The metallic elements composition and their indepth homogeneity in films were controlled by Rutherford Backscattering Spectroscopy (RBS) with a 2 MeV 4He beam. The thin films characterization before and after annealing was completed by NRA (Nuclear Reaction Analysis) and ERDA (Elastic Recoil Detection Analysis) to determine oxygen and hydrogen atoms contents, respectively. The oxygen depth profile was measured by the 16 O(d, α0) 14 N nuclear reaction with 900 keV deuterons. The α0 particles are detected at 166° detection angle. ERDA was applied with a 2.8 MeV 4 He + beam at 30° angle to determine the hydrogen depth profile through the nuclear reaction 1 H( 4 He, 1 H) 4 He.

The depth profiles were extracted from IBA spectra by using SIMNRA software [1999 Mayer] with SigmaCalc cross section [6] for NRA and ERDA. The O and H depth resolutions are close to 30-40 nm, depending on the film metallic composition.

III. RESULTS AND DISCUSSION

A. Microstructure of as-deposited films

Y, Y-Al, Y-V, Y-V-Al, Y-Zr-Al, and Y-Ti with various compositions were deposited. The surface of all films,as observed by SEM, exhibits small grains in the 15-30 nm range. A columnar structure is observed in the cross-section of the pure yttrium film. It is highly crystallized and showed strong texture with an intense 002 Y peak observed by XRD. Y-Ti alloys have an hexagonal structure with a 002-peak shifting between the one of titanium and the one of yttrium, except for the Y32Ti68 which is in an amorphous state [START_REF] Bessouet | Electrical and ion beam analyses of Yttrium and Yttrium-Titanium thin getter films oxidation[END_REF]. Y-Zr-Al alloys showed also an hcp crystalline structure with a 002-peak shifting between the one of yttrium (in Y rich alloy composition) and the one of zirconium, characteristic of a solid solution [START_REF] Kutyla | Hydrogen sorption in yttrium-based getter thin films[END_REF]. Y-V-Al alloys showed a broad and low intensity peak indicating a nanocrystalline structure, close to the amorphous state. Y-Al revealed a 002 Y-peak with a very low and spread intensity indicating a structure close to the amorphous state with small grains in the 10 nm range.

Electrical properties of as-deposited films are given in Fig. 1, with single metal data (Ti, Zr, V) extracted from [2019 Lemettre]. Alloyed films have a higher resistivity and lower TCR than single metal films. This is consistent with the fact that pure metals are less disordered than alloys, as shown by XRD analyses. The Mooij rule is followed since the alloys with a resistivity larger than 150 µΩ.cm present a negative TCR [START_REF] Gantmakher | Mooij rule and weak localization[END_REF], at the noticeable exception of the single metal yttrium film. Despite its high resistivity of 172 µΩ.cm, its TCR is positive at 9 x 10 -4 °C-1 . This exception may be due to the rather high hydrogen contamination of the film during its evaporation compared to Ti, Zr, and V. Such a contamination during evaporation was already observed by Huiberts et al [START_REF] Huiberts | Synthesis of yttrium trihydride films for ex-situ measurements[END_REF]. A hydrogen over metal (H/M) atomic ratio of 0.28 was indeed measured in Y film by ERDA 14 days after deposition, while negligible amounts of H were measured in Ti, Zr, V single metal films. Hydrogen insertion in solid solution may greatly affect the resistivity of the Y film. This would also explain the large difference of resistivity between Yttrium thin film value and bulk Yttrium (59.6 µΩ.cm [START_REF] Kittel | Physique de l'état solide[END_REF]). For Ti, Zr and V this difference is much lower. On the contrary, this incorporation of hydrogen in Y film might have less influence on the film TCR.

B. Oxygen and hydrogen sorption in yttrium

The oxygen and hydrogen depth profiles of yttrium film after different activation annealing are plotted in Fig. 2. As mentioned above, a high H/M ratio is observed in the asdeposited Y film. An unexpected significant hydrogen content is also detected inside the silicon substrate. The oxygen profiles after annealing show a diffusion of oxygen from the surface to the bulk and also a thickening of the surface oxide. At temperature higher than 300 °C the oxygen distribution tends to be uniform. No oxygen diffusion in the substrate is observed. The hydrogen contents both inside the film and inside the substrate largely increase after thermal annealing up to 300 °C. This is attributed to the enhanced diffusion during annealing activation. However, after 350 °C annealing, the total hydrogen content in the film decreases. It might be related to yttrium oxidation which became high enough to inhibit the diffusion of incorporated hydrogen atoms [START_REF] Huiberts | Synthesis of yttrium trihydride films for ex-situ measurements[END_REF].

C. Oxygen sorption in yttrium-based alloys

Whatever the film composition, a global increase of oxygen content determined by NRA with annealing temperature was observed. For comparison, electrical sheet resistance has been measured and an increase with temperature is also clearly visible [START_REF] Kutyla | Hydrogen sorption in yttrium-based getter thin films[END_REF]. At °C, for yttriumvanadium and yttrium-titanium based alloys, the sorption of oxygen is related to the amount of yttrium in the film: the quantity of sorbed oxygen atoms is higher in yttrium-rich alloys. As for pure yttrium, the sheet resistance of Y-alloys also increases with the amount of yttrium in the getter, until insulating thin films are obtained. This confirms that electrical measurement provides a good indication of oxygen sorption ability. The ratio between the number of sorbed oxygen and the number of metals for Y64V21Al15 is twice larger than for Y20V75Al5. Alloying yttrium with aluminium reduces the oxygen sorption capacity of the film by half.

TEM has been performed on Y32Ti68 film annealed at a temperature of 225°C. A gradient of oxygen from the surface to the interface with silicon is clearly visible, the film is not saturated in oxygen. Concerning metal atoms, yttrium content is in-depth homogeneous in the films while titanium appears in the form of grains (Fig. 4).

D. Hydrogen sorption in yttrium-based alloys

Without annealing treatment, Y-Al, Y-Ti, Y-Zr-Al, Y-V and Y-V-Al alloys exhibit a lower amount of hydrogen than pure yttrium. As shown previously in [START_REF] Kutyla | Hydrogen sorption in yttrium-based getter thin films[END_REF], depth profiles show an hydrogen gradient from the surface to the interface and then a diffusion inside the substrate. After the annealing process, the most efficient alloys for oxygen sorption (i.e Y-Ti and Y-V-Al) show the lowest hydrogen/metal ratio. Getter materials containing aluminium and zirconium exhibit a larger ability to sorb hydrogen as demonstrated in Fig. 5. Fig. 6 shows depth profiles for Y-Al and reveals a very low amount of hydrogen and oxygen for the as-deposited film compared to pure yttrium. However a diffusion of hydrogen in the substrate is also clearly visible. At 300°C, hydrogen is almost homogeneous in-depth while in pure yttrium a concentration gradient was observed.

E. Summary and conclusion

Thin films of yttrium-based alloys containing titanium, zirconium, vanadium and/or aluminum were studied by using IBA and electrical characterization to evaluate their ability for oxygen and hydrogen sorption a thermal treatment. SEM and TEM observations combined with previous XRD analyses were used to characterize their microstructure. The measured in-depth concentration profiles shows that oxygen diffuses into the film from the surface while hydrogen diffuses also in the substrate and is mostly located at the interface with tsilicon. This could be related to the creation of a diffusion barrier for =hydrogen when oxidation becomes high enough. At 300°C, the largest amount of oxygen has been measured for alloys containing vanadium and titanium. The lowest oxygen content was found in alloys containing zirconium and aluminium. Concerning hydrogen sorption, only Y-Zr-Al and Y-Al are highly efficient. Those results show that when thermal activation is not performed in ultra-high vacuum condition, oxidation has a large influence on hydrogen sorption.

As a conclusion, alloys containing yttrium, zirconium and aluminium with a large amount of yttrium seem to be the most promising for both oxygen and hydrogen sorption. Further studies, for example with isotopic gases, will be performed to differentiate diffusion and sorption of species adsorbed on the surface during air exposure and of sorbed gases during annealing.

Fig. 1 .

 1 Fig. 1. TCR vs. resistivity of as-deposited films.

Fig. 3 .

 3 Fig. 3. a) Ratio of sorbed oxygen atoms divided by metal atoms and b) sheet resistance as a function of yttrium amount at 300°C.

Fig. 5 .

 5 Fig. 5. Ratio of sorbed hydrogen atoms by metal atoms as a function of yttrium amount at 300°C.

Fig. 4 .

 4 Fig. 4. TEM images of Y32Ti68 annealed at 225°C. Left : cartography of yttrium and titanium. Right: cartography of oxygen.

Fig. 2 .

 2 Fig. 2. a) Evolution of the oxygen and b) hydrogen peak obtained by NRA and ERDA as a function of temperature for yttrium. c) Oxygen and hydrogen depth profiles in yttrium as a function of temperature.

ACKNOWLEDGMENT

This work was partially funded by the "Agence Nationale de la Recherche" (project ANR-19-CE08-0011), "Nano 2022" program, and by the French RENATECH network. The authors wish to thank Abdelhanin Aassime, Laetitia Leroy, Fabien Bayle, Cédric Villebasse and Ludovic Largeau for their work allowing a good operation of deposition, annealing and characterization set-ups in C2N clean room facilities.


Lucas Hudeley
email: lucas.hudeley@onera.fr

Olivier Le Traon

Thomas Perrier

Raphaël Levy

Jean Guerard

Alain Bosseboeuf
email: alain.bosseboeuf@c2n.upsaclay.fr

Planar Foucault pendulum silicon gyro

Keywords: Gyro, Axisymmetric, whole angle, navigation grade, silicon, MEMS

This paper presents theoretical performances of a novel silicon MEMS Coriolis Vibratory Gyroscope (CVG). Since whole-angle (WA) operation is promising to reach navigation grade, in this work we designed a new cell to obtain isotropic in-plane movement of a central mass. We project a high Q thanks to the original design of link elements between the outer frame and the central mass. They allow the movement of the central mass without energy losses in the outer frame. A FEM calculation gives very low energy losses in the outer frame, of order of 10 -7 of the total energy contained in the resonator. So, for an under vacuum device only the thermoelastic losses will limit the Q-factor, it is estimated for our design above 8.5 × 10 5 . This allowing promising scale-factor and angular random walk (ARW). Numerical model gives scale factors of 3 aF/(°/s) for 100Hz mismatch in open-loop operation. The ARW model predicts 0.1 °/√h for a 100 Hz mismatch and 0.007 °/√h with degenerated modes for a device of 2.5 mm  2.5 mm  80 µm. The operating frequency is about 8 kHz and the first eigen mode is upper 4.7 kHz making the device robust to harsh vibratory environments.

I. INTRODUCTION

Several works reported MEMS tuning fork out of plane gyroscope with very promising results for near navigation grade. The well-known and historic Honeywell's silicon cell [START_REF] Johnson | Development of a Navigation-Grade MEMS IMU[END_REF] and the quartz crystal cell from Systron Donner / Emcore [START_REF] Zotov | In-Run Navigation Grade Quartz MEMS-Based IMU[END_REF] are one of them. University of California demonstrated WA operation capability of an axisymmetric in plane silicon MEMS gyroscope, the quad mass gyro [START_REF] Askari | Performance of quad mass gyroscope in the angular rate mode[END_REF]. Thales [START_REF] Vercier | A new Silicon axisymmetric Gyroscope for Aerospace Applications[END_REF] and Safran [START_REF] Jeanroy | Inertial angular sensor of balanced mems type and method for balancing such a sensor[END_REF] also develop axisymmetric silicon in-plan MEMS gyroscope for WA operation. Boeing also demonstrated a fused silica or silicon axisymmetric Disc Resonating gyro with WA operation [START_REF] Challoner | Boeing disc resonator gyroscope[END_REF]. In consequence, WA in-plane silicon MEMS gyro seems to be an ideal concept for low size, weight and power (SWaP) navigation grade angular rate sensors. This paper presents an original design of a gyro cell [START_REF] Le Traon & Al | Gyromètre vibrant à structure plane[END_REF] applied to a silicon MEMS. A section details the model of open-loop operation. We present the methods to limit bias instability. A following section describes FEM simulations of this cell and its theoretical performances. Lastly, the paper mentions the first realization of the gyroscope.

II. GYRO CELL

The cell presented here is a symmetric in-plane CVG that sense the angular rate around the normal axis of the device. Its design is as insulated as possible from external conditions, thanks to the new concept that composes the gyro cell. The idea is to obtain a similar operation to a Foucault pendulum. A proof mass has an isotropic in-plane vibration that depends on the rotation around the normal axis to the vibration plan. For a simple gyro such as in Fig. 1, when the outer frame is free in space, the momentum of the inertial mass naturally induces an outer frame movement in compensation. This leads to the relation: Mc vC = M vM and an energy loss through this movement, where M (kg) is the mass of the central mass, vM its velocity (m/s), MC is the outer frame mass and vC its velocity.

In order to avoid this loss, we need a way to compensate the momentum of the proof mass. In that purpose, we propose an original design of link element (equivalent of springs in Fig. 1) that fulfil this role.

A. An original link element

The original link element has a configuration where its arms momentums themselves balance momentum of the proof mass (see Fig. 2). In this figure, the angle α depend on the material used; this angle needs to be wisely chosen to ensure that the physical properties are the same in each arms. For silicon <100> in our case, α needs to be 90° (180° is an alternative) since its crystal has a cubic symmetry. This is necessary for both arms to have the same mechanical properties to allow the translation movement of the mass onto the bisector direction.

In a balanced configuration, the resonator permits the referred point M to translate onto the bisector direction In [START_REF] Johnson | Development of a Navigation-Grade MEMS IMU[END_REF], choosing the right ratio between masses balances the configuration, which cancels ||vc||. Four of these link elements compose the final silicon cell.

B. Full cell 1) Mechanical behaviour

The proposed design uses Silicon On Insulator (SOI) technology. A proof mass connects the link elements in an axisymmetric configuration, which forms the gyroscope. This allows two orthogonal in plane modes. Those are a requirement to obtain two matched modes and the whole angle operation of the gyro cell. One mode along the common bisector of two link elements and the second mode is identical with a 90° rotation around the normal axis of the device. The number of link elements depends on the material used. An equivalent cell in quartz is also in development and uses three link elements to fit with the trigonal symmetry of the quartz crystal [8].

In open-loop operation, a transduction mechanism (described in the following section) excites one of the two orthogonal modes of the CVG to its resonance frequency. The coupling between the drive mode and the angular rotation of the structure generate the movement of the sense mode. Fig. 3 depicts both modes of the structure. The drive amplitude x excited at its resonance pulsation x is linked to the drive Q-factor Qx and the exciting force fx by [START_REF] Lynch | Vibratory Gyro Analysis by the Method of Averaging[END_REF]:

 ẍ + ω x Q x ẋ + ω x 2 x = f x  
The sense amplitude y is expressed as a function of its pulsation y, Q-factor Qy and the Coriolis force. This force depends on the angular velocity  (°/s) and k the angular gain. In a perfect realization model:

 ÿ + ω y Q y ẏ + ω y 2 y = -2kΩẋ 
However, in reality, fabrication processes introduce diverse defects that affect the vibration behavior and generate undesired coupling. We then rewrite (3) as in [START_REF] Lynch | Vibratory Gyro Analysis by the Method of Averaging[END_REF]:

ÿ + ω y Q y ẏ + ω y 2 y = -∆( ω 0 Q 0 )ẋsin(2θ c ) + … ω 0 ∆ω x sin(2θ k ) -2kΩẋ (4)
Where:

ω 0 Q 0 = ω 1 Q 1 + ω 2 Q 2 ; ∆( ω 0 Q 0 ) = ω 1 Q 1 - ω 2 Q 2 ω 0 2 = ω 1 ² + ω 2 ² 2 ; ω 0 ∆ω = ω 1 ² -ω 2 ² 2

 

The real vibratory modes are indexed 1 for the drive and 2 for the sense. In ( 2) and ( 4), pulsation and quality factors also depends on θ c and θ k . These two angles represent an angular shift of the stiffness (k) or damping (c) of the vibratory modes. This is a consequence of fabrication defects. Different sources such as a temperature variation may affect both of these shifts. This generates bias and bias instability.

To improve bias stability over temperature each arms are in the same crystallographic direction. Thus, as long as the temperature variation is even over the entire cell, the properties of the four link elements stay strictly the same. It guarantees the isotropy of vibration whatever the temperature (x(T) = y(T)) for WA operation or the stability of scale factor versus temperature (x(T) -y(T) = Cste) for an open-loop configuration.. Another anticipated way of improvement for the bias instability is the use of highly doped silicon. A high doping value (over 10 18 cm -3 ) allows reducing or nullifying the temperature coefficient of frequency (TCF). It is the case for n-doped silicon [START_REF] Jaakkola | Design Rules for Temperature Compensated Degenerately n-Type-Doped Silicon MEMS Resonators[END_REF]. Hence, we chose a heavily doped substrate (over 10 18 cm -3 ) for our gyroscope.

2) Electromechanical transduction

The material used for this gyroscope being silicon the electrostatic driving and capacitive sensing is appropriate. To keep the symmetry of the device, both driving and sensing will use the same transducer. To maximize the sense signal it is important to increase the capacitive surface. The use of interdigitated combs allow maximizing the capacitive surface on a limited area of the substrate. We choose variable gap transduction to ensure higher sensitivity of the vibratory cell. To be able to generate force with interdigitated combs it is mandatory to have an asymmetric configuration between gaps.

With these configurations, we can express the variation of charges (in C) due to an angular rate as:

Q sense (Ω) = 4V DC εh y(Ω) ( 1 g 1 2 - 1 g 2 2 )(2N E L E R+N c L c )
Where y() is the sense displacement. In Fig. 3., amplitude of the counterbalance masses is larger than the central mass amplitude, the ratio between both amplitude is R. Fig. 4. presents last variables of [START_REF] Challoner | Boeing disc resonator gyroscope[END_REF]. N is the number of fingers on counterbalances mass combs (indexed E) or on the proof mass (indexed C). L is the overlap length of the fingers (with same indexation as N) and h their width. In a low range of polarization and displacement (below static pull-in) this relation could be used to compute the collected electric charges. We compute a static pull-in bias of about 15 V in our case. III. FEM SIMULATIONS The geometry of the design affects every parameters of the CVG. That is why we realized numerical simulations to optimize dimensions. The software used was Simcenter 3D with the OOFELIE::Multiphysics solver. The performance parameters evaluated to discriminate each device geometry were the Q-factor (Q thermoelastic or QTED and Qinsulation), the frequency and the scale factor.

The simulations take into account the doping of the substrate by using elastic parameters of the doped silicon referenced in [START_REF] Ng | Temperature dependence of the elastic constants of doped silicon[END_REF]. We measured the resistivity of three of the SOI wafers used for the realization with a four-pointprobe method. We computed a mean doping value of 1.7e 18 cm -3 over these wafers. Our SOI are antimony doped so the use of the values of elastic parameters given in [START_REF] Ng | Temperature dependence of the elastic constants of doped silicon[END_REF] for 1.3e 18 cm -3 antimony doped silicon is relevant.

A. Parameter evaluation methods

Our model only computes modal deformations, Eigen vectors and harmonic displacements. We need to evaluate each parameters from theses simulations results.

1) Q insulation:

We compute this parameter with a modal simulation of the free structure. We evaluate Qinsulation by the ratio between the kinetic energy of the whole structure over the kinetic energy of the anchor region. This represent the quantity of energy kept in the vibratory structure.

2) Angular gain:

Reference [START_REF] Bryan | On the Beats in the Vibrations of a Revolving Cylinder or Bell[END_REF] define the angular gain and we can evaluate it thanks to the expression:

 K ⃗⃗ = 1 V ∭ (k p ⃗⃗⃗⃗ ∧ k d ⃗⃗⃗⃗ ) dV V  
Where the angular gain is the component of K along the out of plane axis, kp is the Eigen vector of the drive displacement, kd the Eigen vector of the sense displacement and V the volume of the vibratory cell.

3) Q thermoelastic:

A harmonic simulation taking into account the thermoelastic properties of silicon allows access to the Qfactor. In harmonic simulation anchor are fixed. We impose a prescribed force F on the structure to excite it. We then extract the displacement of the proof mass. Thanks to a A least square method, we fit the resonance amplitude A with a Lorentzian expressed as:

 A=A s √ 1 Q TED 2 ((1 - ω 2 ω 0 2 ) 2 + ω 2 ω 0 2 Q TED 2 )
 

Where As is the static amplitude due to F force.

4) Scale factor and bias:

We perform two harmonic simulations with a structural damping to prescribe a quality factor as computed previously. One without prescribed angular rotation to access bias and one with a known prescribed rotation to access scale factor. We compute these harmonic responses in phase and in quadrature. From (4), we can extract the scale factor (m/(°/s)): 4) and ( 9), we can express the bias (°/s):

SF mecha = y(Ω) -y(Ω=0) Ω = -jω x 2k ω y 2 -ω x ² + jω x ω y Q y x(ω x )  From (
Ω Bias = y(ω x ,Ω=0) SF mecha

 

We then use [START_REF] Challoner | Boeing disc resonator gyroscope[END_REF] to convert this mechanical scale factor in an electro-mechanical scale (C/(°/s) or F/(°/s)):

 SF em = Q sense (SF mecha ) 

B. Theoretical performances

Table I reports the theoretical performances resulting from simulations based on the design shown in Fig. 3. This value of Qinsulation indicates an excellent insulation; this is encouraging for confining energy in the vibratory part. QTED is the limiting factor in this system. In our case, the cell is in the isothermal regime, thus Q depend on f/e² (f the frequency and e the width of the beam). Then, we may reduce beam's widths that are already small to increase the Q-factor. The perfect angular gain would be one, but this is only obtainable if all the mass of one mode is used to transfer the energy through the Coriolis effect. In our case, a part of this mass is used to compensate the momentum and insulating the vibratory part (m2 in Fig. 1). The QTED value is near the highest Q in gyro MEMS sensors [3][6]. Fig. 5 is a representation of the harmonic results. In this figure, we see that for a 0.34 µm drive amplitude, 100 Hz mismatch and 1 °/s of angular velocity, a sense displacement of 1.75 pm is generated. This leads to a mechanical scale factor of 1.75 pm/(°/s) computed with [START_REF] Lynch | Vibratory Gyro Analysis by the Method of Averaging[END_REF]. This represents 3 aF/(°/s) computed with [START_REF] Ng | Temperature dependence of the elastic constants of doped silicon[END_REF]. Nominally, for a perfect symmetric design, bias will be zero. However, bias and its stability are strongly fabrication dependent, which is why we do not anticipate their values.

We can improve the overall performances by several means. The first one is by increasing the mechanical scale factor by lowering the mismatch. We need to keep in mind that the bandwidth of the sensor decreased with the mismatch but the scale factor evolving with 1/f. Another way is to reduce the gaps, but this depends more on the fabrication feasibility and the comb stability.

These results allows us to compute a first approximation of the ARW using the model presented in [START_REF] Jaakkola | Design Rules for Temperature Compensated Degenerately n-Type-Doped Silicon MEMS Resonators[END_REF]:

 ARW mecha = √ k B Tω y A 2 M ω x 2 Q y 1 1 + ω d 2 γ 2

 

Where A is the drive amplitude, kb the Boltzmann constant, M the mass of the proof mass, T the temperature, Qy the quality factor of the sense mode. y the sense angular frequency of the sense mode and x the one of the drive mode. We also define d = (y² + x²) / (2x) and y / (2Qy). For electrical ARW we use the noise on the charge amplifier to express it:

 ARW elec = en (C f + C 0y ) 1 SF em

 

Where en is the noise of the amplifier (usually ~10 nV/√Hz), Cf the counter capacity of the amplifier and Coy the sense capacity without rotation.

The theoretical ARW computed in open loop is 0.1 °/√h for 100Hz mismatch (electrical noise dominant). In matched modes, we compute an ARW of 0.007 °/√h (mechanical noise dominant). These values are computed for 10 V polarization, a higher polarization voltage improve the ARW when modes are mismatched however increasing the voltage might increase the asymmetry of the excitation and so the mismatch. The match mode configuration seems promising to reach the navigation grade that require less than 0.05 °/√h. The close loop configuration is one of our best chance to attain our objective since we can lower f while keeping a non-zero bandwidth. The WA is also a way to improve performances because of the direct extraction of angle that remove integration errors.

IV. FIRST REALISATION

We now need to compare these theoretical results to the real cell to verify their veracity. Moreover, we do not talked about bias instability here because it substantially depends on fabrication.

The process used to fabricate the vibrating cell is common in the MEMS silicon field. We use 4 inches SOI wafer and as mentioned before these SOI are highly Sb-doped. First, we deposit 17 nm/200 nm thick Cr/Au electrodes by evaporation and liftoff process. Then we use inductively induced plasma deep reactive ion etching (ICP DRIE) to obtain large aspect ratio (20 in our case). A HF based etching allows the release the vibratory part. The nonreleased cell in presented in Fig. 4.

In the near future, we will electromechanically characterize the fabricated cells to evaluate the real performances.

V. CONCLUSION

This paper propose a FEM study of novel design of MEMS Silicon CVG oriented on the insulation of the vibration to the outer conditions. This gyro cell has promising results in term of Q factor (> 8  10 5 overall). The ARW in a match mode configuration of 0.007 °/√h might allow attaining navigation grade with this structure. The design aims whole angle operation, thus near future fabrication and characterization will address this possibility and will allow the evaluation of bias and its instability.

Fig. 2 .

 2 Fig. 2. One of the four link elements of the device

Fig. 3 .

 3 Fig. 3. Modal deformation of both gyro modes of the proposed structure. Electrostatic combs are not represented here.

Fig. 4 .

 4 Fig. 4. Comb configuration where y is the excited or sensed direction depending on VDC (excitation if alternative and detection if constant). Differences and positions of combs.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Optical microscopy image of the gyro cell after silicon deep reactive ion etching step

TABLE I

 I

	.	THEORETICAL PERFORMANCES OF THIS WORK
	f in Hz	Qinsulation	QTED	Angular gain
	7950	9 × 10 7	8.5× 10 5	0.19

ACKNOWLEDGMENT

This study is part of a PhD thesis under a part fundings from AID/DGA (Defense Innovation Agency -French MoD). The authors would like to thank AID/DGA for their support. The authors also would like to thank Samson Edmond from C2N for its help with the fabrication process development.