G Chevallier
email: gael.chevallier@univ-fcomte.fr

E Sadoulet-Reboul
email: emeline.sadoulet-reboul@univ-fcomte.fr

Performances of an architectured composite plate with nonlinear stick-slip interactions for vibration reduction

Keywords: metamaterial, absorption, vibroacoustics, nonlinear, resonance

Metamaterials are architectured structures that have recently shown convincing performances for vibration reduction. Two main phenomena are commonly coupled to achieve these performances, local resonances and Bragg scattering that allow to develop interesting frequency bandgaps where waves cannot propagate. The performances for vibration reduction of a lattice metamaterial introduced as a core in a composite structure are here experimentally and numerically studied. This structure is composed of a series of slits with rotational symmetries and is placed between two aluminium plates : the planar configuration with periodic resonant patterns is propitious to generate bandgaps. Different parameters are adjusted to achieve good performances as the number of patterns, the width of the slots, or the length of the internal beams. Nonlinear effects consisting in stick-slip interactions inside patterns are evaluated as favorable aspects to increase the obtained global damping.

Introduction

Metamaterials are emerging lighweight strategies developed to reduce noise or vibrations. They are born from the extension of phononic crystals [START_REF] Vincent | Phononic Crystals, Artificial Crystals for Sonic, Acoustic, and Elastic Waves[END_REF] that consist in periodic composite media acting as a Bragg mirror, and exhibiting specific band frequencies where destructive interferences occur and wave do not propagate. The lowest frequency at which Bragg effects occur corresponds to twice the medium periodicity length, and it leads to solutions that are too large to be applicable at low frequencies. To overcome this limitation, local resonators have been introduced in the unit cells of the periodic media giving birth to the metamaterials [START_REF] Pierre | Acoustic metamaterials and phononic crystals[END_REF]. Stopbands are here linked to the resonance properties and not to the periodicity length, and it allows to reduce vibrations at smaller wavelength. Such a configuration has shown to be efficient for vibroacoustic applications for instance [START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF][START_REF] Claeys | A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation[END_REF]. A specific metamaterial is studied in this paper : its specificity it consists in an architectured structure introduced as a core between two aluminium plate.

Architectured mematerial with rotating beam elements

The designed metamaterial is intended to be used as a core between two aluminium plates. Tt consists in an architectured structure presenting resonators that are rotating beam elements with flexural behavior (Fig. 1a). One same unit geometry is repeated along two directions to create a 2D configuration that can be realized using a laser cutting facility (Fig. 1b). For sake of easy machining the material of the studied specimen is medium, and one interesting observed point is a consequent loss of rigidity after structure machining : it becomes a super flexible double curvature surface [START_REF]Super flexible double curvature surface -laser cut plywood[END_REF]. Thus the architectured plate presents a very lower static stiffness compared to the full medium plate, what is interesting for conformability in industrial applications. A numerical modal analysis has been performed on the metamaterial with material parameters E = 4.113 MPa and ν = 0.339 considering the medium as an isotropic homogeneous media. These date have been determined by correlation with experimental measurements of the three first modes by laser velocimetry (Fig. 1c). The plate is a square plate whose dimensions are 190x190x3 mm in which we insert the periodical pattern. The cutting diameter, which is the space between the beams, is 3 mm. When considering only one pattern in the two plate directions, the first local mode appears at around 145 Hz, which is less than the frequency of the the first global mode appearing around 231 Hz (Fig. 2a). A parametric study (Fig. 2b ) changing the number of patterns has shown that the eigenfrequency of the first global mode tends to decrease, in agreement with the drop in stiffness observed, unlike the eigenfrequency of the first local mode that The purpose of the proposed work is to study if this material whose flexibility properties are not common also presents an interesting behavior in terms of vibratory absorption. Indeed, the structure network consists in a periodic series of patterns exhibiting resonance effects to to the flexion of the ribs. Thus, when associated to a plate with tuned frequency, this metamaterial is assumed to absorb vibrations, leading to a reduction of the vibrations for the main plate. To check this point, the metamaterial is introduced as a core between two aluminium plates to compose a sandwich structure (Fig. 3). The aluminium skins are not in contact with the medium architectured plate. Overthicknesses are added to keep the space needed for the travel of the internal beams. The edges of the structure are all embedded. An harmonic force of 1 Newton in a range of frequencies going from 20 to 400 Hz with a step of 2 Hz is applied around a corner of the structure, and the displacement at the center of the plate is studied.This work is currently in the making but the first results with comparison to a configuration without metamaterial tend to show that the main plate displacement is decreased (Fig. 4). Ongoing study tends to tune with precision the frequency of the local material modes with those on the main skin plates. It also appears that the resonance effects are associated with non-linear stick-slip nonlinear effects that act in favor of reducing vibrations.

Figure 1 :Figure 2 :

 12 Figure 1: Architectured metamaterial with chiral hinge lattices

Figure 3 :

 3 Figure 3: Sandwich structure integrating the architectured metamaterial

Figure 4 :

 4 Figure 4: Central point displacement of a sandwich structure with and without the metamaterial

Acknowledgments

The authors want to thank the master's students Noaman Bachane, Théo Citerne and Julien Dessoly for their contribution to the presented work.


Pascal Briois

Alain Billard

J H Park

J W Son

Electrochemical Performances of SOFC as Function of the Sputtered Parameters of the YSZ Electrolyte

come

Introduction

Due to global warming, the depletion of fossil fuels and the growing demand for energy from the population, it seems necessary to develop clean, efficient and environmentally friendly technique to produce electricity (1)(2)(3). Solid Oxide Fuel Cell (SOFC) are one of the ways to meet this challenge because of their performance, their impact on the environment and the possibility to use a various fuel (1,(3)(4). To consider a mass development of SOFC, it is necessary to reduce their operating temperature (3). Indeed, a high operating temperature makes it possible to obtain good electrochemical performances, but this is done to reduce the lifetime of the cells (5). The reduction of the operating temperature without loss of electrochemical performance requires the development of SOFC thin film by the use of surface treatment techniques and reduce the temperature fabrication (6)(7). The required properties of an SOFC electrolyte are a good ionic conductivity of oxygen and no electrical conductivity, fine to limit the ohmic drop of the cell to reduce the temperature of use, dense to be gas-tight as well good chemical and thermomechanical compatibility with electrode materials (2,5).

The anode support cell developed in this work implements conventional SOFC materials namely: a Ni-YSZ cermet for the anode, a YSZ electrolyte, a GDC buffer layer and an LSCF cathode (3). The buffer layer in GDC is used in order to reduce the reactivity between the cathode and the electrolyte which leads to performance falls by the appearance of pyrochlore phase at the electrolyte / cathode interface (8). The technique used is magnetron sputtering because of its environmental friendliness and the fact that it is suitable for thin, dense and complex composition film synthesis (7). The work of J. A. Thornton (9), completed by A. Anders (10), , shows that the densification of a PVD film is obtained when the energy of the plasma species is important, therefore during a low working pressure during the deposition phase as well as the application of a polarization at the sample holder. This study focuses on the importance of electrolyte synthesis parameters on cell performance, in particular on the polarization power of the sample holder as well as on the thickness of the film. YSZ layers are developed under reactive conditions from a metal target.

Experimental details

Elaboration of the SOFC Anode layer: A 2 cm by 2 cm and 1-mm-thick NiO-YSZ (NiO/8YSZ = 56:44 wt%) anode support was fabricated by tape casting. 150-μm-thick NiO-YSZ tapes with poly(methyl methacrylate) (PMMA) as a pore-forming agent and 30-μm-thick NiO-YSZ tapes without PMMA were also fabricated by tape casting. Then, seven layers of the 150μm-thick tape and one layer of the 30-μm-thick tape were laminated at 75°C under a uniaxial pressure of 15 MPa. The laminated substrate was sintered at 1300°C for 4 h to fabricate a completely rigid substrate. For more details see our previous work (11).

Electrolyte layer: The experimental device is a 100-litre Alcatel SCM 650 sputtering chamber pumped down via a system combining XDS35i Dry Pump and a 5401CP turbo-molecular pump. The sputtering chamber is equipped with three 200 mm diameter magnetron targets and with a 620 mm diameter rotating substrate holder parallel to the targets at a distance of about 110 mm. The distance between the targets axis and that of the substrate holder is 170 mm. The Zr-16 at.% Y target is supplied thanks to a pulsed DC Advanced Energy dual generator allowing the control of the discharge current, power or voltage. In the present study, the discharge current is fixed at 2.5 A. The substrates are alumina pellets, glass slides as well as anode substrate positioned next to the target at 170 mm from the axis of the substrate holder. Argon and oxygen flow rates are controlled with MKS flowmeters and the pressure is measured using a MKS Baratron gauge. The deposition stage is monitored using a closed loop control PEM (Plasma Emission Monitoring) system using optical emission spectroscopy (OES). The technique is based on the measurement of the optical intensity of the zirconium emission line (360.12 nm) measured on the area close up to the target. The signal is sent via an optical fibre to a Ropper Scientific SpectraPro 500i spectrometer, with a 1200 groove mm-2 grating and a photomultiplier tube (Hamamatsu R 636). Then, the information is transferred to a computer where a program developed under Labview® monitors the oxygen flow rate to maintain the selected intensity of the optical signal of the zirconium line. For more details concerning the plasma emission monitoring techniques you can see our previous work on the elaboration of the electrolyte (12-13) and cathode material for SOFC (14).

Buffer layer: A 200-nm-thick gadolinia-doped ceria (GDC) buffer layers were deposited using PLD at a substrate temperature of 700°C and a Pamb of 6.67 Pa (11).

Cathode layer: A 3-μm-thick lanthanum strontium cobaltite (LSC) layer with a 1 cm × 1 cm area was deposited onto the GDC by PLD at room temperature and a Pamb of 13.3 Pa, followed by post-annealing at 650°C in air for 1 h to crystallize the cathode (11).

Structural and Morphological Characterization

The coatings morphology is characterized by Scanning Electron Microscopy (SEM) using a JEOL JSM 7800 F. The structural features of the coatings are identified in Bragg Brentano configuration X-ray diffraction using a BRUKER D8 focus diffractometer (CoKα1+α2 radiations) equipped with the LynxEye linear detector. X-Ray Diffraction (XRD) patterns are collected at room temperature during 10 min in the [20°-80°] scattering angle range by a step of 0.019°. The coating thickness is determined using the "step" method (a small surface of the substrate is covered with a tape that is removed after deposition leaving a step due to the coating thickness) with an Altysurf profilometer from Altimet allowing an accuracy of about 20 nm. Before each measurement, the calibration of the experimental device was realized with a reference sample number 787569 accredited by CETIM, France.

Electrical Measurements

Air and humidified H2 (3% H2O) were used as the oxidant and fuel, respectively. The cell operating temperature varied from 650 to 450°C at intervals of 50°C, and the electrochemical impedance spectra (EIS) and current-voltage-power (I-V-P) curves were obtained at each temperature. Each EIS was observed over a frequency range of 105 Hz to 10-1 Hz. The AC amplitude of the impedance measurements was 50 mV. An Iviumstat electrochemical analyzer (Iviumstat, Ivium Technologies) was used to obtain these EIS and I-V-P curves.

Result and Discussion

First Study

The sputtering parameter for producing the YSZ layer are summarized in Table I. Under the operating conditions, the deposition rate is 750 nm.h -1 , which is much greater without PEM system (13). Figure. 1 reveals that the film is crystallized under the cubic face-centered structure of the zirconia as-deposited and after a annealing treatment of 2 hours in air at 500°C. NiO diffraction peaks from the substrate are also observed. The annealing treatment leads to a displacement of the diffraction peaks to the right which reflects an evolution of the lattice parameter from 0.51411 nm to 0.51321 nm respectively for the asdeposited and annealed film. In agreement with JCPDS 30-4068 which gives a lattice parameter of 0.5139 nm, the as-deposited film is under extension constraints whereas after annealing it is under compression constraints. The appearance of stress during the deposition phase is induced by the application of the 20 W polarization at the porte substrate holder, and the modification of the nature of the stresses within the film during the heat treatment is due to the fact that the as-deposited coating is not saturated in oxygen (11). This oxygen deficiency is compensated during heat treatment under static air. The SEM observation of the fracture facies of the rough film shows good adhesion with quality interfaces (Figure 2a). The YSZ layer has a slightly columnar appearance that is typical of a PVD film. After the annealing treatment in static air for 2 hours at 500°C, the film is still adherent and denser (FIG. 2b) which reinforces the observations of the XRD measurement. Figure 3 shows the electrochemical performances of the cell for each temperature with figure 3a and 3b which respectively represent the I-V-P measurements and the Nyquist diagram for each temperature. Table II is obtained from these measurements in which a comparison of the performances of the cell with the theoretical magnitude for each temperature is shown. Before the measurement, the anode was reduced during10 h at 600°C and the measurements were recorded during decreasing temperature every 50°C from 600 to 450°C, then raised to 650°C. The value of the OCV is lower than the calculated theoretical value, respectively 1.038V and 1.122 V because of a problem of sealing of the YSZ layer. This problem of tightness of the electrolyte is caused by the too columnar appearance of the layer. In addition, the ASR of the cell obtained is greater than the theoretical value for all the temperatures. The increase in ASR results from the fact that the ionic conductivity of the YSZ film is lower than the conductivity of the bulk YSZ (16). The observation of the brittle cross section of the cell (figure 3c) as well as the cathode surface (figure 3d) after the electrochemical measurements reveal the good behavior of each layers as well as the integrity of the cell. Second Study

2Ɵ (°)

During this second series of experiments, we had to define new deposition parameters to increase the density of the films in order to produce films with different thicknesses from 2 to 14 μm. One of the possible parameters is to modify the power applied on the sample holder in order to increase the energy of incident species on the substrate holder. To do this, we used the results of a previous study conducted on gadolinia doped ceria coatings (GDC) (17), we observed that the power applied increase on the sample holder allows to densify the coating but from certain value (80W) it appears that films no adherent. This adhesion problem is mainly related to the presence of stress in the film, they relax during it cracking and / or its desquamation. The constraints are even more important that the atoms constituting the film are big and heavy, in our previous study the film of GDC is composed of Ce, Gd and O which are larger, and heavier than the elements Zr, Y and O from the YSZ for this study. We also had to minimize the constraints in the YSZ film because our goal was to make a fairly thick 14 μm film. The deposit conditions are reported in Table III As shown in figure 4, the increase of the polarization power makes it possible to densify the film and they are all adherent after an annealing of 2 hours in air at 500°C. We have not tried to put a bias voltage higher than 80 W for the reasons mentioned above. We have reduced the working pressure to 0.15 Pa which is lower than the first series (0.2Pa) because in agreement with the work of A. Anders (10), the reduction of the working pressure makes it possible to densify the film. However, when observing the effect of the polarization power on the porte substrate holder a working pressure of 0.1 Pa, but for such pressure the amount of gas is too low which leads to instability of the discharge. Such instability is not compatible with our goal of making a thick film of several micrometers. In addition, the set point has been increased to increase the oxygen deficiency of the film during processing which will lead to greater volume expansion during the annealing phase (13). Zirconia is a material that is transparent when it is saturated with oxygen. The measurement of the transmittance of the films produced under different set point shows that the films are more and more absorbent with the increase in the regulation set point, and thus with the oxygen deficiency. However, there is a compromise in the synthesis between putting enough oxygen to prevent cracking of the film during its crystallization and add a limited amount to allow a volume expansion during the heat treatment to densify the material and fill the gaps pores of the anode. The operating conditions are summarized in Table III, line2. In these condition, the deposition rate is of the order 1200 to 1500 nm.h -1 which is about twice as large as in the first series and in agreement with our previous work it is approximately 7 times higher in the absence of the use of the PEM system ( 13). This higher sputtering rate is directly related to the modification of the set point value which makes the film more oxygen deficient compared to the first series. In magnetron sputtering the deposition rate of a metal is much greater than that of a ceramic compound [START_REF] Billard | Technique de l'ingénieur[END_REF]. The operating conditions allowing the synthesis of a denser film being now defined, it will be sufficient to modify the deposition time to obtain the thickness of the desired film on the anode made according to the same protocol as the series 1. All the films are crystallized under the cubic face-centered structure of the zirconia and the heat treatment makes it possible to increase the crystallization as well as to relax the stresses. Brittle cross section observation by SEM of the cells after the electrochemical test reveals clear interfaces, as well as a dense electrolyte (figure 5). Moreover, it appears that the electrolyte doesn't have a columnar appearance as in the first measurement series (figure 3) and that the modification of the synthesis parameters made it possible to densify the YSZ layer. This observation is confirmed by the determination of complete cell performance (figure 6). Indeed, the OCV obtained for the thinnest film (1.8 μm) is almost equivalent to that of the first series which has a thickness approximately 5 times greater (Table IV). The performance of the cell increases with the reduction of the electrolyte which agrees with the literature (6). The maximum power is obtained from 1060 mW.cm -2 at 650°C for the cell having an electrolyte of 1.8 µm. The modification of the experimental parameters allowed to increase the performances of the cell because of the increased density of YSZ films.

Conclusion

In this work, the influence of bias power applied to the substrate holder and the thickness of YSZ coatings deposited by reactive magnetron sputtering has been studied. The application of PEM system could allow to obtain a relatively high sputtering rate, up to seven times higher than without PEM system. It should be noted that the density of the YSZ increase with the bias power applied on the substrate and annealing treatment relax stresses appears during the elaboration and increase the oxygen content into the coating. All coatings present f.c.c structure of zirconia before and after thermal treatment. SEM observation of the brittle cross section of the cells after the test shows the good quality of all the layers as well as the interfaces of the unit cell. The performance of the cell increases with the decrease of the thickness of the electrolyte (1060 mW.cm -2 at 650°C then the thickness is 1.8 µm)

Figure 1 .

 1 Figure 1. XRD measurements of (a) as-deposited YSZ coatings and (b) after annealing during 2 h under static air (orange).

Figure 2 .

 2 Figure 2. SEM view of brittle fracture cross section of (a) as-deposited and (b) annealed samples.

Figure 3 .

 3 Figure 3. (a) Potential and Power versus Current of complete cell and (b) Nyquist diagram for different temperature. (c) SEM view of surface of cathode side and (d) brittle fracture cross section of complete cell after electrical measurement.

Figure 4 .

 4 Figure 4. SEM view of brittle fracture cross section of YSZ coatings annealed 2 h at 500°C under static air on anode substrate with different bias powers.

Figure 5 .

 5 Figure 5. (a, b, c) SEM view of brittle fracture cross section of complete cell with different YSZ thicknesses after electrochemical measurements and (d, e, f) SEM view of the surface of YSZ coating after annealing.

Figure 6 .

 6 Figure 6. (a, c, e) Nyquist diagram and (b, d, f) Potential and Power versus Current of complete cell for different temperature and different YSZ electrolyte thicknesses.

Table I .

 I Sputtering parameter.

	Target	Intensity	Rf bias power	Ar flow rate	O2 Flow rate	Set Point	Total pressure
	(at.%)	(A/kHz)	(W)	(sccm)	(sccm)	PEM (%)	(Pa)
	Zr-16Y	2.5/50	20	20	adjust	35	0.2

Table II .

 II electrical measurements data vs thermodynamics calculus.

	T (°C)	OCV (V)	Ideal OCV (V)	Ohm. ASR at OCV (Ω-cm 2 )	ASR of ideal 10 µm YSZ (Ω-cm 2 )	Pol. ASR at OCV (Ω-cm 2 )	Power Density at 0.7V (mW/cm 2 )	Peak Power Density (mW/cm 2 )
	66600℃ 1.051	1.13	0.481	0.226	0.706	220.1	293.6
	550	1.061	1.14	0.900	0.502	1.749	108.2	149.5
	500	1.067	1.148	1.960	1.238	5.395	44.7	64.0
	450	1.065	1.157	5.330	3.858	18.283	17.3	22.5
	650	1.038	1.122	0.296	0.111	0.372	369.8	474.4

Table III .

 III , line 1. Sputtering parameter for the second study.

	Target	Intensity	Rf bias power	Ar flow rate	O2 Flow rate	Set Point	Total pressure
	(at.%)	(A/kHz)	(W)	(sccm)	(sccm)	PEM (%)	(Pa)
	Zr-16Y	2.5/50	0 to 80 80	10 15	adjust	30 45	0.1 0.15

Table IV .

 IV Electrical measurements data of complete cell with different electrolyte thicknesses for two temperature.

	650 C
	600 C
	550 C
	500 C
	450 C
								650 C
								600 C
								550 C
								500 C
								450 C
	650 C
	600 C
	550 C 450 C 500 C							550 C 600 C 650 C
								500 C
								450 C
	650 C
	600 C
	550 C 450 C 500 C							550 C 600 C 650 C
								500 C
	Temperature	Thickness (µm/study)	OCV (V)	Ohm. ASR at OCV (Ω-cm 2 )	Pol. ASR at OCV (Ω-cm 2 )	(mW/cm 2 at 0.7V Density Power	)	(mW/cm 2 ) Power Density Peak 450 C
		1.8 (2)	1.066	0.637	3.159	100.1		157.1
	66500℃	6 (2) 9 (1)	1.147 1.067	3.339 1.960	4.750 5.395	47.1 44.7		54.8 64.0
		13.8 (2)	1.127	4.030	4.610	40.0		46.4
		1.8 (2)	1.034	0.076	0.292	801.5		1060.5
	650	6 (2) 9 (1)	1.123 1.038	0.284 0.234	0.636 0,372	508.6 369.8		648.6 474.5
		13.8 (2)	1.114	0.359	0.609	434.6		549.3

Acknowledgments

This study was granted by the Pays de Montbéliard Agglomeration.


Rodolphe Boudot
email: rodolphe.boudot@femto-st.fr

Vincent Maurice

Eric Kroemer

Moustafa Jaroslaw Rutkowski

Abdel Hafiz

Sylwester Bargiel

Nicolas Passilly

Christophe Gorecki

M Abdel Hafiz

Ramsey-CPT spectroscopy in buffer-gas filled Cs vapor micro-fabricated cells

Keywords: miniature atomic clock, coherent population trapping, Ramsey-CPT spectroscopy, microfabricated cell

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.

INTRODUCTION

Over the last decade, significant efforts have been pursued in numerous laboratories and industries towards the development of miniature atomic clocks based on coherent population trapping (CPT) [START_REF] Knappe | A microfabricated atomic clock[END_REF][2][START_REF] Zhang | Rubidium chip-scale atomic clock with improved long-term stability through lihgt intensity optimization and compensation for laser frequency detuning[END_REF]. These sources are attractive candidates through their ability to combine a reduced volume, power consumption and fractional frequency stability at the level of 10 -11 at 1 day integration time. These clocks are based on a micro-fabricated buffer gas-filled alkali vapor cell. Atoms in the cell interact with a bi-chromatic optical field generated by a vertical-cavity surface-emitting laser modulated at half the atom ground-state hyperfine splitting frequency. Using the CPT phenomenon, a narrow so-called dark resonance can be detected at the output of the cell using a photodiode and be exploited to stabilize the frequency of a local oscillator driving the VCSEL. In miniature atomic clocks, continuous atom-light interaction is generally performed. In this regime, the clock short-term fractional frequency stability is optimized for low laser intensities, resulting from a trade-off between the CPT signal and linewidth and detection noise.

In this study, we investigate the use of a pulsed CPT interaction scheme [START_REF] Zanon | High-contrast Ramsey fringes with coherent population trapping pulses in a double-lambda atomic system[END_REF][START_REF] Liu | Ramsey spectroscopy of high-contrast CPT resonances with push-pull optical pumping in Cs vapor[END_REF] to detect narrow Ramsey-CPT fringes in Cs-Ne micro-fabricated vapor cells. We study the impact of different experimental parameters on the characteristics (linewidth, amplitude, contrast) of the central Ramsey-CPT fringe. At quite high laser intensity, narrow Ramsey-CPT fringes with a reasonable signal-to-noise ratio are detected.

II. EXPERIMENTAL SET-UP

Figure 1 shows the experimental setup used for the detection of Ramsey-CPT fringes in the microfabricated cell. The setup combines a distributed feedback diode laser tuned on the Cs D1 line (894.6 nm), a Mach-Zehnder electro-optic modulator (MZ EOM) driven by a microwave frequency synthesizer at 4.596 GHz for optical sidebands generation, an acousto-optic modulator to compensate for the buffer-gas induced optical frequency shift in the CPT cell and to produce the pulsed interaction by switching on and off the microwave frequency, a Michelson-based polarization orthogonalizer and delay-line system to produce the PPOP scheme [START_REF] Jau | Push-pull optical pumping of pure superposition states[END_REF][START_REF] Liu | Coherent population trapping resonances in buffer-gas filled Cs vapor cells with push-pull optical pumping[END_REF]. The laser is frequency-stabilized on an evacuated Cs cell using the technique described in [START_REF] Hafiz | Doppler-free spectroscopy on Cs D1 line with a dual-frequency laser[END_REF]. The output laser beam is sent into a microfabricated Cs-Ne cell [START_REF] Hasegawa | Microfabrication of Cs vapor cells with buffer gas for MEMS atomic clocks[END_REF], shown in Fig. 2, with a diameter of 2 mm and an optical path length of 1.4 mm. The buffer gas pressure in the cell is about 70 Torr. The cell is temperature stabilized at 75°C where the CPT contrast is maximized. In the pulsed regime, atoms interact with a sequence of optical CPT pulses. The typical pumping time is The transmitted light through the cell is detected 10 µs after the beginning of the pulse in a 10 µs-length detection window. A data acquisition card and personnal computer allows to pilot the whole experiment.

III. RESULTS

Figure 3 shows Ramsey-CPT fringes detected in the Cs-Ne microcell. The free-evolution time is 300 µs and the laser power incident in the cell is 114 µW. The central fringe linewidth is 1358 Hz. Its contrast is about 3.8 %. Further tests will be performed up to the conference time. We expect also to compare results obtained in the continuous and pulsed regimes. Preliminary frequency stability results will be performed in both regimes. Latest results will be shown at the conference.

Fig. 1 .

 1 Fig. 1. Schematic of the Cs CPT clock. DFS: Dual-frequency Doppler-free spectroscopy.

Fig. 2 .

 2 Fig. 2. Photograph of a Cs-Ne microfabricated cell.

Fig. 3 .

 3 Fig. 3. Ramsey-CPT fringes detected in a Cs-Ne microfabricated cell.

Figure 4

 4 Figure 4 reports the contrast of the central fringe versus the laser power. The latter is found ton increase for laser powers up to 130 µW.

Fig. 4 .

 4 Fig. 4. Fringe contrast versus the incident laser power. Experimental parameters are TR = 300 µs, Tb = 400 µs.

Figure 5

 5 Figure 5 reports the evolution of the central fringe amplitude versus the free-evolution time TR. Experimental data are wellfitted by an exponential decay function with a time constant of about 396 µs. The latter can be considered as an estimation of the CPT microwave coherence relaxation time T2 in the microfabricated cell.

Fig. 5 .

 5 Fig. 5. Fringe amplitude versus the free-evolution time TR.

ACKNOWLEDGMENT

The authors thank colleagues in FEMTO-ST (P. Abbé, C. Rocher), ANR-DGA ISIMAC project and LabeX FIRST-TF.


Kevin Rosenziveig
email: kevin.rosenziveig@femto-st.fr

Jérémy Bon

Philippe Abbé

Valérie Soumann

Nicolas Passilly

Serge Galliou

Pierre-François Cohadon

Quartz Crystal Resonator Used as an Optical Fabry-Perot Cavity

Keywords: Optics, Quartz Resonator, Fabry-Perot, Optical Cavity, Acoustic Cavity, Optomechanics

come

I. Introduction, Background

Quartz crystals are used in time standards routinely. Moreover, the quality of synthetic materials have been improved up to the point where the product Q × f gets to 10 13 for the best ones at room temperature.

Furthermore, such quartz resonators (QR) in the cryogenic 4K region can have quality factors (Q) as high as a few billion [START_REF] Galliou | Properties related to Qfactors and noise of quartz resonator-based systems at 4k[END_REF].

Earlier studies [START_REF] Gagnepain | Relation between 1/f noise and Q-factor in quartz resonators at room and low temperatures, first theoretical interpretation[END_REF] have shown that the flicker frequency level of the power spectral density of the fractional frequency fluctuations behaves as S y (1 Hz) ∝ 1 Q a with a ≈ 4. Very good performances can therefore be achieved, with possible fractional frequency deviation floor, or Allan deviation floor, as low as 10 -16 .

All these considerations qualify cryo-cooled quartz as a candidate for ultra-stable clock. We need to implement some vibration control and a way to impede performancelimiting frequency instabilities due to temperature fluctuations. The latter problem has been solved [START_REF] Bon | Frequency-Temperature Compensated Cuts of Crystalline-Quartz Acoustic Cavity Within the Cryogenic Range [4 K, 15 K[END_REF] by the finding of a specific compensated cut for the quartz crystal in this temperature region.

To be able to actuate the quartz crystal, piezoelectricity is usually called upon. In our setup however, this would bring further noise to account for, because of variable temperature fluctuations at different stages of the solid link between the oscillation sustaining system (at room temperature) and the QR which sits on the cold finger of the cryorefrigirator. A way to circumvent this is to actuate the quartz optically instead. Indeed, if we are able to use the radiation pressure, we can avoid thermal dilation fluctations in the excitation link. Because the radiation pressure is very weak, we use the fact that the two electrodes used for actuating the quartz piezoelectrically also are two mirrors "glued" together with the QR. By sending a laser on these, we can use the radiation pressure of the hence amplified light intensity inside the cavity to create vibrations of our QR. For this, we should modulate the laser intensity at the acoustic frequency [START_REF] Bon | Cryogenic optomechanic cavity in low mechanical loss material[END_REF]. We will first demonstrate the feasability of using such QR as Fabry-Perot cavities.

II. Quartz Crystal as a Fabry-Perot Cavity

Typical QR manufactured for frequency references here at FEMTO-ST are plano-convex crystal cavities, with a thickness of few mm with a thin layer (around 5 -15 nm) of Chromium on the quartz on top of which a much larger layer of Gold (around 200 nm) is deposited. The reason for the Chromium layer is that it sticks well to the quartz, whereas the Gold does not. Gold, in turn, sticks well onto the Chromium layer.

The particular QRs we use are plano-convex 1 mm thick lenses, with a radius of curvature of 250 mm. Figure 1 shows such a metallized QR which does not seem of surface quality incompatible with our uses. In these conditions, the g stability factor for a Fabry-Perot cavity is g = 0.996, very close to 1 when instability for optical cavities starts [START_REF] Kogelnik | Laser Beams and Resonators[END_REF]. Gold, in the infrared region, is reflective to about 98% [START_REF] Yakubovsky | Optical constants and structural properties of thin gold films[END_REF] for ≈ 100 nm layers. It is generally considered [START_REF] Yakubovsky | Optical constants and structural properties of thin gold films[END_REF] that films more than ≈ 200 nm are certain to behave just like their bulk counterpart whereas quantum effects need to be taken into accout for films thinner than 20 nm.

All these considerations lead us to believe that a common metallized QR could be used in a straightforward manner as an optical Fabry-Perot cavity. From these, we expect a finesse F = π √ R 1-R with R the intensity reflection coefficient of the whole Fabry-Perot cavity. In our case, the Finesse should amount to about F = 155. Furthermore, the typical Fabry-Perot cavity gives a transmission maximum and a reflection minimum for the resonance condition λ = 2L m with m ∈ N * .

III. Experiment, Results

In our experimental setup, the laser, optical isolator and EOM are all fibered as seen in figure 2. We have put a λ 2 plate before the beam splitter to be able to compensate for the birefringence of the QR, and align the polarization of our beam to one of the QR's fast or slow axes.

Because of the quartz having an index of refraction n quartz ≈ 1.54, the wavelength inside is modified with respect to that of vacuum. This phenomenon accounted for, the mode-matching demands that we have a waist of 71 µm at the (entrance) plane side, so that the beam's radius of curvature matches that of the end mirror.

Our first results with Chromium-Gold coated QR were discouraging, and it was found that Chromium was a poor candidate for making a mirror [START_REF] Rakić | Optical properties of metallic films for verticalcavity optoelectronic devices[END_REF]. It is indeed the Chromium that the light inside the quartz first "sees". Eliminating altogether the Chromium layer was investigated, but did not turn out well, as a "blistering" [START_REF] Dahmane | Cloquage de films minces ductiles sur substrat rigide[END_REF] effect on the gold was observed.

This lead us to use Silver instead, whose intensity reflection coefficient is comparable to that of Gold at our wavelengths, and whose main defect with respect to gold is that it gets oxydated whereas gold does not. On the other hand, Silver sticks better to the quart and the "blistering" effect does not seem to happen as much.

The results obtained for a 60-60 nm Silver metallized QR were surprising, as there was a frequency shift between the extrema of the transmitted and the reflected peaks. As it turns out, this is in fact a known phenomenon [START_REF] Monzón | On the concept of absorption for a Fabry-Perot interferometer[END_REF] and it is due to the absorption of the mirrors. The measured difference between the peaks is 58 MHz and the theoretical value is 37.5 MHz Contrary to the typical case, the theoretical extrema of transmission and reflection are limited. In our case, the transmission and reflection maxima are theoretically limited to T = 4 and R min R max = 12%. We measure, however, a reflection visibility of ≈ 3 and a transmission of ≈ 3.10 -4 .

IV. Conclusions

The common QR can be used as a Fabry-Perot cavity, by adapting the metallization. Investigation is under progress to better understand the discrepencies between the theoretical and experimental values. A systematic investigation of the bandwidths for the transmission and reflection peaks will be lead.

MENESR

  France for K.R. research scholarship Région Franche-Comté and Labex First-TF for their funding dedicated to this program Oscillator IMP platform (Equipex) for reference frequencies

Fig. 1 .Fig. 2 .

 12 Fig. 1. Surface of one of our metallized Quartz Resonator

Fig. 3 .

 3 Fig. 3. Transmission and reflection for a 60-60 nm Ag metallized QR (noise filtered data).

Acknowledgment K.R. would like to thank R. Giust, J. Millo and C. Lacroûte for the fruitful discussions.


Rémy Vicarini

Jaroslaw Rutkowski

Vincent Maurice

Eric Kroemer

Serge Galliou

Christophe Gorecki

Rodolphe Boudot
email: rodolphe.boudot@femto-st.fr

Characterization of 894.6 nm VCSELs and application to a microcell-based atomic clock

Keywords: VCSEL, miniature atomic clock, coherent population trapping, micro-fabricated cell

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Over the last decade, the combination of coherent-population trapping (CPT) physics, micro-electro-mechanical systems (MEMS), and semiconductor lasers technologies has led to the development of miniature atomic clocks (MACs) demonstrating a fractional frequency stability lower than 10 -11 at one-day integration time, a hundred times better than those of widely used quartz crystal oscillators for a similar volume (~15 cm 3 ) and low power consumption (~150 mW) [START_REF] Knappe | A microfabricated atomic clock[END_REF][2][START_REF] Zhang | Rubidium chip-scale atomic clock with improved long-term stability through lihgt intensity optimization and compensation for laser frequency detuning[END_REF]. These MACs are of great interest for a number of civil and industrial applications such as mobile telecommunications, power networks, satellite-based navigation systems, or sensing applications. To date, a CPT-based MAC proposed by Microsemi is even commercially-available [2]. However, many laboratories or industries pursue their efforts towards the development of alternative MAC products. Laser sources used in MACs are vertical-cavity-surface emitting lasers (VCSELs). The short-term fractional frequency stability of CPT-based MACs is known to be significantly improved by using a laser tuned on the alkali D1 line [START_REF] Stähler | Coherent population trapping resonances in thermal 85Rb vapor: D1 versus D2 line excitation[END_REF]. For Cs MACs, different 895 nm VCSELs were developed [START_REF] Serkland | Vertical-cavity surface-emitting lasers[END_REF][START_REF] Gruet | Metrological characterization of custom-designed 894.6 nm VCSELs for miniature atomic clocks[END_REF]. Nevertheless, to our knowledge, finding commerciallyavailable VCSELs at this wavelength for Cs MACs has long been a significant issue. In this study, we report the characterization of novel 894.6 nm VCSELs dedicated to be used in a Cs MAC. These measurements include output optical power versus dc current and temperature, spectral linewidth, relative intensity noise (RIN) or frequency noise [START_REF] Kroemer | Characterization of commercially-available vertical cavity surface emitting lasers tuned on Cs D1 line at 894.6 nm for miniature atomic clocks[END_REF]. For application, a VCSEL is used in a Cs microcell-based CPT atomic clock. A preliminary fractional frequency stability of 3.8 10 -11 τ -1/2 up to about 1000 s is reported, achieving a value lower than 3 10 -11 at 60 000 s.

II. VCSEL CHARACTERIZATION

VCSELs used here are resonant with Cs atom for a temperaturecurrent couple is about 17°C-1.4 mA. Figure 1 reports the measured RIN of the VCSEL for a laser power of 50 µW. Figure 2 shows a measurement of the beat-note spectrum between the VCSEL and a DFB laser. The spectrum is fitted by a Lorentzian function. The VCSEL spectral linewidth is measured to be 31.6 MHz.

III. CLOCK SETUP AND RESULTS

Figure 3 shows the microcell-based CPT clock experimental setup. The cell is fabricated using the dispenser-based technology described in [START_REF] Hasegawa | Microfabrication of Cs vapor cells with buffer gas for MEMS atomic clocks[END_REF]. It is filled with Ne buffer gas, temperature-stabilized at about 79°C. A static magnetic field is applied in order to split the different Zeeman transitions. The ensemble is surrounded by a single-layer mu-metal magnetic shield. Atoms in the cell interact with a bi-chromatic optical field generated by direct microwave modulation at 4.6 GHz of the VCSEL injection current through a bias-tee. Atoms are pumped in a dark state when the frequency difference between both first-order optical lines equals the Cs ground-state hyperfine splitting (~9.192 631 GHz). The light power transmitted through the cell is detected by a photodiode. The photodiode output signal is used both for laser frequency stabilization and local oscillator frequency stabilization. A noise budget will be reported at the conference to evaluate the main contributions to the clock fractional frequency stability. Some advanced light-shift cancellation or reduction techniques inspired from [START_REF] Zhang | Rubidium chip-scale atomic clock with improved long-term stability through lihgt intensity optimization and compensation for laser frequency detuning[END_REF] or [START_REF] Shah | Continuous light-shift correction in modulated coherent population trapping clocks[END_REF] will be tested to improve the clock mid-term frequency stability. Moreover, we think that the clock frequency stability for integration times higher than 1000 s could be improved thanks to the use of a further miniaturized MAC physics package with optimized thermal design. Latest results will be shown at the conference.

Fig. 1 .

 1 Fig. 1. RIN of the VCSEL. The VCSEL power is here 50 µW.

Fig. 2 .

 2 Fig. 2. Spectral linewidth of the VCSEL.

Fig. 3 .

 3 Fig. 3. Schematic of the Cs CPT clock. The laser frequency-stabilization is not shown here.

Figure 4

 4 Figure 4 shows the Allan deviation of the Cs-Ne microcellbased CPT clock. It is measured to be 3.8 10 -11 τ -1/2 up to 1000 s and lower than 3 10 -11 at 60 000 s averaging time. These performances are encouraging and demonstrate the potential of the tested VCSELs, associated with microcells, for the development of high-performance MACs.

Fig. 4 .

 4 Fig. 4. Allan deviation of the CPT clock.

ACKNOWLEDGMENTS

The authors thank colleagues in FEMTO-ST (P. Abbé, C. Rocher, S. Bargiel, N. Passilly, M. Abdel Hafiz), Délégation Générale de l'Armement (DGA, Rapid HABAC project), Région de Franche-Comté and LabeX FIRST-TF.


J Bon

Ph K Rosenziveig

C Abbé

S Rocher

Galliou

L Neuhaus

S Deléglise

T Briant

P.-F Cohadon
email: cohadon@lkb.upmc.fr

Optomechanical coupling in a quartz crystal resonator for cryogenic clocks

Keywords: Optomechanics, Low Mechanical Loss, Quartz Resonator, Optical Cavity, Acoustic Cavity

Plano-convex quartz crystal resonators can exhibit quality-factors close to a billion at 𝟒𝑲. They are also suitable to operate as optical cavities. As a consequence they are good candidates for optomechanic experiments including quartz-based cryogenic clocks. The paper is focused on the latter application and demonstrates its feasability.

INTRODUCTION

Devices under consideration in this paper are mainly bulk acoustic wave (BAW) resonators made in quartz crystal SC-cuts, and designed to operate typically at room temperature (RT), at 5 or 10 MHz on their low-speed (quasi-pure) thickness shear mode, the C-mode. Resonators under test are put inside a pulsetube cryorefrigerator.

II.

QUARTZ RESONATOR BEHAVIOR AT 4 K At helium temperature very low mechanical losses 𝜙 can be achieved [START_REF] Galliou | Losses in high quality quartz crystal resonators at cryogenic temperatures[END_REF], i.e. very high quality-factors 𝑄 1 𝜙 ⁄ . Ultimate intrinsic losses in the range [3 K-15 K] and at frequencies greater than a few MHz in our case are those originating from the interaction of the acoustic wave and phonons according to the Landau-Rumer theory [2]. The latter predict a Q-factor that does not depend on frequency anymore. Q-factors are usually greater than a few millions close to 4 K and can go beyond a billion on the extensional mode, the A-mode, for high-quality quartz resonators [START_REF] Galliou | Extremely Low Loss Phonon-Trapping Cryogenic Acoustic Cavities for Future Physical Experiments[END_REF]. Moreover, it has been observed from a set of experimental results that the trend of quartz noise, described by its power spectral density of fractional frequency fluctuations 𝑆 𝑓 1𝐻𝑧 , at the offset frequency 1 Hz, against the unloaded qualityfactor 𝑄 can be written as 𝑆 𝑓 1𝐻𝑧

1 𝑄 ⁄ with 𝑛 close to 4 [START_REF] Gagnepain | Relationship Between 1 𝑓 ⁄ Noise and Q-factor in Quartz Resonators at Room and Low Temperatures, First Theoritical Interpretation[END_REF], [START_REF] Walls | A New Model of 1/f Noise in BAW Quartz Resonators[END_REF]. This suggests that flicker noise decreases as Qfactor increases, as expected.

Consequently, high-Q quartz resonators seem like good potential candidates as the heart of ultra-stable cryogenic clocks. Nevertheless a couple of issues regarding the sensitivity of BAW resonators to their environment should be fixed before implementing them inside a cryogenic clock. Indeed, it has been shown that at least temperature first and then vibrations should be minimized [START_REF] Goryachev | Quartz Resonator Instabilities Under Cryogenic Conditions[END_REF].

Vibrations of a cryogenerator can be processed to reduce displacements with the machine pulsing period to less than 1𝜇𝑚 at the second stage level, where the resonator sits.

Regarding temperature, even if the temperature coefficients are lower at low temperature than at RT, the fractional frequency sensitivity remains typically close to 10 𝐾 ⁄ because of the non-compensated behavior of the tested modes and cuts, mainly the SC-cut, at cryogenic temperature. The use of a compensated cut is mandatory for implementing an ultra-stable frequency source, i.e. a cut whose frequency-temperature curve has a turning point inside the range [3 𝐾-15 𝐾]. To identify such a cut a campaign has been carried out to measure the temperature sensitivity of the elastic coefficients of quartz within the operating temperature range [START_REF] Bon | Temperature coefficients of crystallinequartz elastic constants over the cryogenic range [4 K, 15 K[END_REF]. In a second step, those results have been used to calculate the temperature coefficients of a selected resonance frequency with respect to cut angles. This calculation has successfully lead to a compensated cut whose frequency-temperature curve is more flat around its turning point than the C-mode of the SC-cut at RT [START_REF] Bon | Compensated cut of crystalline-quartz over the cryogenic range [4 K, 15 K[END_REF].

Thus, all the ingredients would be gathered to consider the design of a cryogenic quartz-based source.

III. QUARTZ RESONATOR OPERATING AS AN OPTICAL CAVITY

Quartz is a low mechanical loss material but not the only one: sapphire, diamond, silicon, calcium fluorite... might also be considered [START_REF] Nawrodt | Mirror thermal noise calculation for ET[END_REF]. Some of them are non-piezoelectric and an optical excitation could be particularly relevant in that case, which stands true for every material. A light beam incident to a quartz resonator surface can induce an acoustic response by means of various basic processes, or a mix of them, such as Brillouin diffusion, thermal diffusion, radiation pressure, etc. The possible mechanical displacement from the radiation pressure of some incident laser beam results in an extremely weak effect with usual laser power. Nevertheless, one way to magnify efficiently the radiation pressure consists in using the acoustic cavity also as an optical cavity. Indeed, while the acoustic cavity becomes an optical Fabry-Perot, i.e. when proper conditions between the optical length and the cavity thickness are met, multiple round-trips can be described as light trapping between both reflecting surfaces. Accordingly, the displacement resulting from a modulated incident laser beam is much bigger than before, as demonstrated in the following section.

Firstly, the tested devices are typically plano-convex resonators whose thickness is 𝑡 0.5𝑚𝑚 or 1𝑚𝑚 and radius of curvature in the order of 𝑅 300𝑚𝑚. Consequently, the condition of stability [10] of the optical cavity, 1 𝑔 0 with 𝑔 1 𝑡 𝑅 ⁄ , is close to the boundary of the instability domain but validated anyway.

Secondly, the ability of the optical intracavity field to actuate the mechanical displacement should be checked. The mean intracavity photon flux is 𝐼 𝐹𝐼 where 𝐼 is the mean incident laser flux and 𝐹 the cavity finesse, i.e. the amplifying factor. The latter depends on the optical properties of both reflecting surfaces: the usual gold electrodes can be used simply as basic reflectors with a reflectivity of about 96% at 1550𝑛𝑚. Thus, the finesse 𝐹reaches 80. Reflectivities could be improved by means of Bragg mirrors for example, and consequently the finesse could be increased, but their impact on mechanical losses should be considered. A trade-off would likely be the solution.

The effect of those data on the mechanical displacement is assessed below.

IV. OPTOMECHANICAL COUPLING

When relating the laser power 𝑃 to the light intensity 𝐼, and assuming that both mirrors have the same transmission factor and that optical losses are negligible, it can be shown that the radiation-pressure force from the intracavity beam is:

𝐹 2𝑃 𝑐 2𝑃 𝐹 𝜋𝑐
Moreover, the radiation-pressure force induces a mechanical displacement along the axis 𝑦 normal to the resonator plane (see Fig. 1) given by 𝛿𝑦 𝜔 𝜒 𝜔 𝛿𝐹 𝜔 , where 𝑛 stands for the overtone 𝑛00 of the extensional A-mode, and 𝜒 is the mechanical susceptibility close to the resonance of overtone 𝑛at 𝜔 exhibiting a Q-factor 𝑄 . The resulting displacement is then:

|𝛿𝑦 𝜔 | 4𝐹 𝜋𝑐 𝑄 𝑚 𝜔 |𝛿𝑃 |
where 𝑚 is the effective mass. It turns out that the resulting displacement sensitivity would be 1.310 𝑚 𝑚𝑊 ⁄ for a golded quartz resonator with 𝐹 80. This is compatible with the noise floor which is assessed to 210 𝑚 √𝐻𝑧 ⁄ for a 1𝑚𝑊 laser power at 1550𝑛𝑚 [11].

V. CONCLUSION

A typical plano-convex quartz crystal resonator can be used both as an optical and an acoustic cavity , theoretically. The practical proof of concept is in progress.

Fig. 1 .

 1 Fig. 1. Optomechanic sketch and typical device.

ACKNOWLEDGMENT

The authors would like to thank the MENESR France for JB research scholarship, Région Franche-Comté and LabEX FIRST-TF for their funding dedicated to this program, Oscillator IMP platform (EquipEX) for providing reference frequencies.


Rodolphe Boudot
email: rodolphe.boudot@femto-st.fr

Gilles Martin

Jean Friedt

Enrico Rubiola

J-M Friedt

Characterization of high-overtone bulk acoustic resonators: applications to ultra-low noise microwave oscillators and miniature atomic clocks

INTRODUCTION

The generation of microwave signals with MEMS bulk acoustic wave (BAW) resonators is an exciting challenge. Such resonators allow the development of high-performance microwave sources combining low power consumption, small size, autonomy and ultra-low phase noise performances making them well-suited for applications such as radars, embedded electronics systems, telecommunications or embedded sensors. In the family of MEMS acoustic resonators, high-overtone bulk acoustic (HBAR) resonators [1,2] are valuable candidates by demonstrating a possible direct operation frequency of a few GHz, state-of-the-art Qf products up to 10 14 [3,4] and high-potential for wafer-level fabrication. In this paper, we propose the description and characterization of AlN-sapphire HBAR resonators. Additional details are reported in [5]. Some detailed investigations are reported on a HBAR mode of interest at 2.3 GHz. We report the measurement of S-parameters, temperature coefficient of frequency, typical loaded Q-factor, sensitivity of the HBAR frequency to the input microwave power highlighting the existence of non-linear effects and residual phase noise measurements of the HBAR. The residual phase noise of the HBAR is found to increase with the input microwave power. A residual phase noise of -130 dBrad 2 /Hz is measured at 1 Hz offset frequency at 2.3 GHz for low input microwave power (~6Bm). The HBAR resonator is inserted in a 2.3 GHz oscillator loop, allowing the demonstration of excellent phase noise performances at the level of -145 dBrad 2 /Hz at 10 kHz offset frequency, in excellent agreement with the Leeson effect. In the last section of the manuscript, the HBAR-oscillator prototype (not integrated) is used as a local oscillator in a microcell-based coherent population trapping (CPT) Cs atomic clock laboratory-prototype. For this purpose, a low noise frequency doubling stage is implemented to up-convert the HBAR oscillator 2.3 GHz signal to 4.6 GHz (half of the Cs atom frequency). Fine tuning of the oscillator frequency to the atomic transition frequency is performed by adjusting of the HBAR temperature and implementation of a voltage-controlled phase shifter (VCPS) in the oscillator loop, preventing the use of a power-consuming direct digital synthesis (DDS). The HBAR-oscillator is successfully stabilized to the atomic transition, allowing to demonstrate when locked to the atoms, a preliminary clock short-term fractional frequency stability at the level of 7 10 -11 τ -1/2 up to about 10 s, limited by the detected atomic resonance signal to noise ratio.

DESCRIPTION AND CHARACTERIZATION OF THE HBAR RESONATOR

Figure 1 shows the architecture of the HBAR resonator, designed by CEA-LETI, Grenoble, France. It consists of a substrate made of sapphire with a thickness of 550 µm, a width of 1700 µm and a length of 875 µm. On this substrate is found a piezoelectric film made of AlN material with a thickness of 1 µm. This resonator is a dual-port resonator and allows to operate in the transmission mode [START_REF] Reinhardt | Proc. IEEE International Ultrasonics symposium (IUS)[END_REF]. The idea is to ensure the coupling of acoustic waves between two adjacent resonators, achieved by implementing two resonators close to one another, allowing for evanescent waves between the resonator electrodes to overlap. The HBAR is connected through SMA connectors and the ensemble is implemented in a small duralumin packaging box (see figure 2). A thermistance and heating resistance are implemented in order to ensure the thermal control of the resonator at the mK level. Figure 4 reports the HBAR mode frequency and loaded Q-factor QL versus the HBAR temperature. The HBAR temperature coefficient of frequency (TCF) is measured to be -24.7 ppm/K. While such a TCF appears large and might prevent high stability oscillator operation, we will actually benefit from this high sensitivity by tuning the oscillator frequency to an atomic transition by selecting the HBAR temperature: the HBAR mode spacing of 10 MHz, determined by the low acoustic loss sapphire substrate thickness, is swept around 2.3 GHz by tuning the temperature in a 180 K range. The loaded Q-factor is in the range of 28 000 and found to increase slightly with temperature between 40 and 90°C. Figure 5 shows the evolution of the HBAR frequency versus the microwave input power. Experimental data are correctly fitted by a linear function with a slope of -1350 Hz/mW (-5.9 10 -10 /µW in fractional value). These values are close to typical amplitude-frequency coefficient values, of few 10 -10 to 10 -9 / µW for quartz resonators [START_REF] Rubiola | [END_REF]. As detailed in [5], we think that the HBAR frequency variation is here caused by non-linear effects and not by power-induced thermal heating of the HBAR resonator. Indeed, with increased input microwave power incident on the HBAR input port, we observed a clear and significant distortion of the resonance curve away from the Lorentzian line shape. In this experiment, the resonance shape was found to be slightly analogous to lineshapes induced by spring-softening Duffing non-linearity phenomena [START_REF] Nayfeh | Nonlinear oscillations[END_REF] We measured the residual phase noise of the HBAR resonator. The setup description is the following. A 2.3 GHz source (Keysight E8257D) is power-split into two arms. The first arm involves a microwave amplifier and the HBAR resonator. The second arm only contains a microwave phase shifter to adjust the quadrature condition between arms of the phase noise measurement system. Both arms are directed to a microwave mixer operating as a phase detector. The output of the mixer is low pass-filtered, dc amplified and sent to a FFT analyzer. In this single-resonator bridge configuration, an issue is that the frequency fluctuations of the source can be converted into phase fluctuations through the resonator-based frequency-phase converter. The higher the Q factor, the higher the contribution of the source noise to the overall output noise. Figure 6 reports the residual phase noise of the HBAR resonator for several input microwave power values. The contribution of the source is shown in grey and is shown to be the main limitation of the measurement for f > 2 kHz. The noise of the mixer-based setup is shown in black and is at the level of -140 dBrad 2 /Hz at 10 Hz offset frequency. The HBAR residual phase noise is clearly visible for 10 Hz < f < 1 kHz and is found to increase with increased input power. It is found at the level of -120 dBrad 2 /Hz at 1 Hz offset frequency for an input power of 9.5 dBm. For an input power of 6.5 dBm, we measured a phase noise of -130 dBrad 2 /Hz at f = 1Hz. In the presence of flicker, the Allan deviation σyq (τ), stability of the resonator frequency, i.e., the time-domain stability of an oscillator in which the resonator is the only source of frequency instability, is given from: with b-1 the flicker phase noise value at f = 1Hz. From our experimental phase noise data, with b-1 = -130 dBrad 2 /Hz, we estimate that the Allan deviation σyq (τ = 1 s) is about 6.7 10 -12 .

DEVELOPMENT OF A HBAR-BASED LOW PHASE NOISE OSCILLATOR

We have developed a low phase noise 2.3 GHz oscillator based on the HBAR resonator. Figure 7 shows the architecture of the oscillator. The HBAR resonator is associated in a feedback loop with two sustaining amplifiers of total gain ~ 25 dB, a microwave isolator, a band pass filter and a phase shifter to satisfy Barkhausen conditions. The HBAR resonator can be finely temperature controlled. Moreover, a voltage-controlled phase shifter is implemented in the feedback loop.

On the one hand, it is used for fine tuning of the oscillator output frequency (DC bias). On the other hand, it allows to create modulation sidebands needed for locking the oscillator on the atomic transition frequency. 3 GHz [11]. They are also significantly better than those of a conventional local oscillator [START_REF] Lutwak | The chip-scale atomic clock -prototype evaluation[END_REF] used in miniature atomic clocks (MACs) (see figure 8). GHz. The latter is compared with the absolute phase noise of a state-of-the-art 100 MHz OCXO ideally frequency-multiplied to 2.3 GHz. Also, the phase noise at 2.3 GHz of a conventional MAC LO [START_REF] Lutwak | The chip-scale atomic clock -prototype evaluation[END_REF] is reported for comparison.

APPLICATIONS TO A MICROCELL-BASED ATOMIC CLOCK

The physical phenomenon of coherent population trapping (CPT) [START_REF] Alzetta | [END_REF], combined with the progress of MEMS technologies and semi-conductor lasers, has allowed over the last decade the development of miniature atomic clocks (MACs) [START_REF] Knappe | Comprehensive Microsystems[END_REF] with a typical volume of 15 cm 3 , a power consumption of 150 mW and a fractional frequency stability of 10 -11 at 1 day integration time.

Local oscillators for Cs MAC applications must satisfy many stringent requirements. They should have an output frequency of 4.596 GHz (half-frequency of the Cs atom at 9.192 631 770 GHz) to be resonant with the alkali atom, a tunable output power of -10 to 0 dBm to drive the VCSEL laser, modulation capabilities (frequency modulation and modulation depth up to a few kHz), a frequency resolution at 1-10 mHz level not to limit the clock frequency stability, a phase noise lower than -80 dBc/Hz at f = 1 kHz not to degrade the clock short-term frequency stability at the level of 10 -11 at 1 s through the so-called intermodulation effect [START_REF] Audoin | [END_REF] and a free-running Allan deviation lower than 10 -7 at 1 s.

To date, the local oscillator in a Cs MAC consists in general of a 4.596 GHz voltage-controlled oscillator (VCO) phaselocked to a 10 MHz quartz oscillator through a fractional-phase locked loop. Different designs were proposed in the literature [START_REF] Lutwak | The chip-scale atomic clock -prototype evaluation[END_REF][START_REF] Knappe | Comprehensive Microsystems[END_REF]. Nevertheless, it has to be noted that today, the LO and associated frequency synthesis chain electronics represent more than 50% of the total power budget of a miniature atomic clock. Phase noise performances of these architectures are modest but remain to date sufficient. For information, Figure 8 compares the phase noise of a 2.3 GHz signal ideally-generated from a classical 10 MHz MAC LO to our 2.3 GHz HBAR-oscillator. In that sense, we proposed in an investigation experiment the use of the above-described HBAR-oscillator as a local oscillator of a microcell-based Cs vapour cell atomic clock based on coherent population trapping (CPT).

Figure 9: Principle of the CPT atomic clock. The HBAR LO 2.3 GHz signal is frequency doubled to 4.596 GHz. This signal drives a VCSEL laser which generates a bi-chromatic optical field at 895 nm. The laser beam is shaped, circularly polarized and sent into a Cs vapor micro-fabricated cell filled with Ne buffer gas. When the frequency difference between both optical lines is exactly equal to 9.192 GHz, atoms are pumped in a so-called dark state which increases the light transmission through the cell. The CPT resonance is detected by a photodiode. The output signal of the photodiode is sent into two lockin amplifiers (LA). LA1 is used for laser frequency stabilization by synchronously modulating-demodulating the laser dc current (laser frequency). LA2 is used for the HBAR LO frequency by synchronously modulating-demodulating the VCPS in the HBAR oscillator. The output 4.6 GHz at the output of the doubler is extracted with a coupler (not shown on the figure) and compared to the 4.6 GHz signal from a highperformance commercial microwave synthesizer driven by a hydrogen maser for frequency stability measurement. The inset on the fight of the figure shows the energy diagram involved in the CPT phenomenon.

Figure 9 presents the Cs CPT clock experimental setup. The laser source is a 25 MHz-linewidth custom-designed VCSEL tuned at 894.6 nm on the Cs D1 line, designed by Ulm University [17]. The laser injection current is directly modulated at 4.596 GHz by the HBAR-based local oscillator (frequency-multiplied by 2) to generate two phasecoherent first-order optical sidebands frequency-split by 9.192 GHz for CPT interaction. The output laser beam, circularly polarized, is sent into a micro-fabricated Cs vapor cell [18] filled with a pressure of Ne buffer gas (113 Torr at 80°C). The input power is 30 µW. The microcell temperature is stabilized to within 1mK around 84°C. The laser power transmitted through the cell is detected by a photodiode. The output signal is used both for laser frequency stabilization and local oscillator frequency stabilization. The 2.3 GHz output signal of the HBAR-oscillator is frequency-doubled to 4.596 GHz with a frequency doubler (Minicircuits ZX90-2-36S). The 4.596 GHz signal is bandpass-filtered with a 50-MHz bandwidth filter and amplified to a power up to 10 dBm with an amplifier (Minicircuits ZX60-8000ES+). A variable attenuator is used at the output to adjust if required the microwave power that drives the VCSEL laser. The tuning of the LO output frequency to the atomic line is performed in two steps. A first coarse tuning of the LO frequency is done by adjusting the HBAR temperature. Once around the Cs resonance, a fine tuning of the LO frequency is performed by tuning finely the bias voltage of the VCPS. The tuning voltage-frequency sensitivity was measured to be about 8 kHz/V around the chosen set-point. We succeeded to stabilize the HBAR-oscillator frequency to the atomic transition frequency. This was achieved without the use of any additional direct digital synthesis (DDS) and only with fine tuning of the VCPS. For measurement of the 4.596 GHz LO frequency stability, a comparison is performed with the 4.6 GHz signal from a commercial synthesizer driven by a hydrogen maser. Figure 10 reports the Allan deviation of the HBAR-oscillator in free-running (HBAR temperature stabilized) and locked regimes. The stabilization to the atom frequency allows to improve the short-term fractional stability of the LO from 2 10 -9 to 7 10 -11 at 1 s integration time [19]. At 1000 s integration time, the stability of the HBAR LO is improved by about three orders of magnitude compared to the free-running regime.

DISCUSSIONS

HBARs exhibit several drawbacks. First, an inherent characteristic of HBARs is their multi-mode spectrum (due to materials superposition) that complicates their use in an oscillator and forces one to filter the response to select a single resonance. A second issue is the initial difficulty to tune finely the output frequency of the HBAR resonator due to the difficulty to control accurately thicknesses of materials during the fabrication process. Moreover, the HBAR we used here presents a quite high temperature sensitivity of about -24 ppm/K. Techniques exist to reduce the HBARs TCF but often at the expense of a degradation of the Q-factor and additional technological steps during the resonator fabrication. In this study, after detailed characterization of the HBAR, we tried to convert these drawbacks into advantages to use the HBAR as a local oscillator in a microcell atomic clock application. Our strategy was to use both the HBAR temperature sensitivity and the exploitation of a VCPS in the oscillator loop to tune finely and stabilize successfully the oscillator output frequency to the atomic transition frequency. The VCPS allowed a frequency response of 8 kHz/V. A resolution of 1 µV would then allow a frequency resolution of 8 mHz, which is satisfying for MAC applications. No special integration and packaging efforts were done in this work. Nevertheless, related activities in the literature demonstrate that such HBAR-LO architectures are compatible with integration at the chip level and low power consumption requirements for MAC applications. Ideally, for a MAC application, the temperature setpoint of the HBAR should be fixed at about 90°C to support MACs typical operation temperature ranges. Other potential obstacles exist. In standard MACs, the output useful signal frequency is 10 MHz. In a potential HBARbased oscillator Cs vapor cell MAC, the LO is frequency-stabilized at 4.596 GHz. This microwave frequency is not well-adapted for standard widespread applications. This issue would impose certainly to use a frequency divider to down-convert the 4.596 GHz signal to about 100 or 10 MHz, adding certainly 10-20 mW on the total power consumption of the system. Moreover, in the architecture presented here, modulation of the VCPS at about 1 kHz for tuning and stabilization of the oscillator to the atomic frequency could appear as an undesired spurious signal on the useful signal to be filtered or rejected for correct exploitation. At the end, we note that the combination of a HBAR-LO with stabilization to the atoms could be an interesting solution to propose an extremely small and low power consumption source demonstrating excellent spectral purity (intrinsic phase noise of the LO) and excellent long-term stability (thanks to the atoms).

CONCLUSIONS

We reported the detailed characterization of a 2.3 GHz AlN-Sapphire high-overtone bulk acoustic resonator (HBAR), with loaded Q-factor of about 28000 and insertion losses of 15-20 dB. The temperature coefficient of frequency of the HBAR is about -25 ppm/K. The power-induced fractional frequency variation of the HBAR resonance is measured to be about -6 10 -10 /µW. The residual phase noise of a HBAR is measured in the range of -110 to -130 dBrad 2 /Hz at 1 Hz Fourier frequency. This yields an ultimate HBAR-limited oscillator Allan deviation about 6.7 10 -12 at 1 s integration time. The 1/f noise of the HBAR resonator is found to increase with the input microwave power. The HBAR resonator was used for the development of a low phase noise 2.3 GHz oscillator with a phase noise of -145 dBrad 2 /Hz at 10 kHz offset frequency, in excellent agreement with the Leeson effect. The HBAR LO, with the help of a frequency doubling stage to reach 4.596 GHz, was used successfully as a local oscillator in a CPT-based Cs microcell atomic clock, demonstrating a fractional frequency stability of 7 10 -11 at 1 s integration time.

Figure 1 :

 1 Figure 1: Architecture of the AlN-Sapphire HBAR resonator.

Figure 2 :

 2 Figure 2: Photograph of the HBAR resonator.

Figure 3

 3 Figure 3 reports the S21 parameter (magnitude and phase) of the HBAR resonator mode at about 2.3 GHz. The HBAR exhibits two well-visible eigenmodes, explained by the dual-port structure of the HBAR resonator. Losses of the resonator are about 16 dB at resonance.

Figure 3 :

 3 Figure 3: S21 parameter (magnitude and phase) of the HBAR resonator mode at about 2.3 GHz.

Figure 4 :

 4 Figure 4: Frequency and QL factor of the HBAR versus the temperature.

Figure 5 :

 5 Figure 5: Frequency of the HBAR resonance versus the input port microwave power.

Figure 6 :

 6 Figure 6: Residual phase noise of the HBAR resonator for various values of the incident microwave power.

Figure 7 :

 7 Figure 7: Architecture of the HBAR-based 2.3 GHz oscillatorFigure8shows the absolute phase noise of the HBAR-oscillator at 2.3 GHz The phase noise of the sustaining amplifier is reported for information. The phase noise floor of the oscillator is in good agreement with the phase noise floor of the amplifier[START_REF] Boudot | [END_REF], in excellent agreement with the amplifier thermal noise FkT/Pin where F is the amplifier noise figure, kT the thermal energy and Pin the amplifier input power (-13 dBm). The output oscillator signal phase noise is in excellent agreement with the Leeson effect[10] with a f -3 slope for 5 Hz < f < 1 kHz, a f -2 slope for 1kHz < f < 40 kHz (with υ0/(2QL) = 40 kHz the Leeson frequency, the resonator bandwidth). The HBAR-based oscillator exhibits excellent phase noise performances at the level of -145 dBrad 2 /Hz at 10 kHz. Phase noise performances of the HBAR oscillator are better than those of a state-of-the-art 100 MHz OCXO ideally frequency-multiplied to 2.3 GHz[11]. They are also significantly better than those of a conventional local oscillator[START_REF] Lutwak | The chip-scale atomic clock -prototype evaluation[END_REF] used in miniature atomic clocks (MACs) (see figure8).

Figure 8 :

 8 Figure 8: Phase noise of the sustaining amplifier and HBAR-based oscillator at 2.3 GHz. The latter is compared with the absolute phase noise of a state-of-the-art 100 MHz OCXO ideally frequency-multiplied to 2.3 GHz. Also, the phase noise at 2.3 GHz of a conventional MAC LO[START_REF] Lutwak | The chip-scale atomic clock -prototype evaluation[END_REF] is reported for comparison.

Figure 10 :

 10 Figure 10: Allan deviation of the HBAR oscillator in free-running and locked regime (to the atoms).

  frequently observed in MEMS or NEMS resonators.

		2.29930
		2.29928
		2.29926
	Frequency (GHz)	2.29916 2.29918 2.29920 2.29922 2.29924
		2.29914
		2.29912
		2.29910
		0	10	20	30	40	50	60	70
					P in HBAR (mW)

ACKNOWLEDGMENTS

The authors thank C. Rocher and P. Abbé (FEMTO-ST) for help with electronics stuff, the MOEMS Group of FEMTO-ST (in particular C. Gorecki, V. Maurice, N. Passilly, J. Rutkowski and S. Bargiel) for the Cs microcell fabrication used in this study. The authors would like to thank Vincent Giordano (FEMTO-ST) for fruitful discussions on resonator nonlinearities and phase noise measurements. The authors acknowledge David Rabus and Sébastien Alzuaga (FEMTO-ST) for HBAR TCF calculations. The authors thank A. Reinhardt and P. P. Lassagne (CEA-LETI) for supplying us the HBAR resonators. This work has been partly supported by the Agence Nationale de la Recherche (ANR) Project Equipex Oscillator IMP. The Oscillator IMP Project funded in the frame of the French national Projets d'Investivement d'Avenir (PIA) targets at being a facility dedicated to the measurement of noise and short-term stability of oscillators and devices in the whole radio spectrum (from MHz to THz), including microwave photonics. This work has been also partly supported by LabeX FIRST-TF, Direction Générale de l'Armement (DGA) and Région de Franche-Comté.


Jérémy Bon
email: jeremy.bon@femto-st.fr

Roger Bourquin

Philippe Abbé

Xavier Vacheret

Serge Galliou
email: serge.galliou@femto-st.fr

Temperature Coefficients of Quartz at Cryogenic Temperature

Keywords: quartz, elastic coefficients, temperature coefficients, liquid-helium temperature, low mechanical loss, high quality factor. I

The temperature coefficients of the elastic coefficients of quartz within [4 K, 15 K], the operating range of conventional cryocoolers, are predicted from experimental data. The raw data consist of a set of frequency-temperature curves recorded from bulk acoustic wave resonators made in different plates cut according to various angles relative to the crystallographic axes. Raw data are then processed as described in the paper to provide the 1 st , 2 nd , and 3 rd order temperature coefficients of the seven elastic coefficients of quartz crystal.

INTRODUCTION

Usually temperature sensitivity cannot be neglected when designing a quartz resonator-based device. Even at temperatures lower than 6 K, where the material properties are less sensitive to temperature than at room temperature, the remaining sensitivity has to be considered. The resonant frequency of a quartz crystal resonator is not an exception to this rule: the remaining fractional frequency change versus temperature is typically a few 10 -9 K -1 close to 4 K [START_REF] Galliou | Outstanding quality factors of bulk acoustic wave resonators at cryogenic temperature[END_REF][START_REF] Goryachev | Recent Investigations on BAW Resonators at Cryogenic Temperatures[END_REF]. This is still too much to realize a stable frequency source even with a temperature control exhibiting a typical gain of 1000: definitely, such an application would need a temperature compensated cut. It should be reminded that quartz resonators look very attractive around 4 K because of their very high quality factor that can be greater than one billion [START_REF] Galliou | Extremely Low Loss Phonon-Trapping Cryogenic Acoustic Cavities for Future Physical Experiments[END_REF]. Conversely, quartz resonators could also be used as temperature sensors provided they are made in an appropriate cut, i.e. a very temperature-sensitive one. In both cases, to design a frequency reference or a temperature sensor the knowledge of the temperature coefficients of the elastic constants of the material is mandatory, the resonance frequency depending on these elastic constants. A set of coefficient values already exists for the quartz crystal since 1975 [START_REF] Smagin | Frequency-temperature characteristics of quartz crystal units of different cuts operating over a wide temperature range including helium temperatures[END_REF] but it covers the very wide range [4K, 400 K], and consequently becomes irrelevant inside the shorter operating range of conventional cryorefrigerators. These are the main reasons why we carried out a campaign of measurements to determine the 1 st , 2 nd and 3 rd order temperature coefficients of the elastic constants of quartz within [4 K, 15 K].

II. METHOD OF CALCULATION

The thermal coefficients can be determine from various experimental principles. It can be from resonant ultrasound spectroscopy, phase velocity measurements by pulse echo overlap methods, laser ultrasonic techniques or combined resonance techniques. Nevertheless, their implementation at cryogenic temperatures arises technical difficulties. The selected alternative consists in probing the temperature sensitivity of resonance frequencies of dedicated quartz devices.

A. Theoretical principle

Basically, a flat plate exhibits bulk acoustic waves (BAW) resonance frequencies derived from where n is the overtone order, 2h the plate thickness,  its density, and Ce an elastic coefficient that looks like a combination of the material elastic constants Cijkl depending on the cut orientation with respect to the crystallographic axis in case of quartz because of its anisotropy. All these parameters are temperature dependent yielding to a temperature-dependent frequency. Usually, the frequency-temperature characteristic can be fitted effectively by a 3 rd order polynomial approximation, and this operation can be applied to each parameter. Once experimental frequencytemperature records processed this way, the three corresponding temperature coefficients of the elastic coefficient(s) can be extracted accordingly. The calculation is just somewhat complicated by the fact that there are seven coefficients Cijkl for the quartz crystal. A set of combinations should therefore be considered involving a set of resonator cuts to be tested.

B. Relationships and assumptions

Actually, the tested devices are not flat plates but planoconvex disks, a shape commonly used to trap efficiently the acoustic energy. This is one way to preserve the resonator against disturbances from its environment (i.e. parasitic stresses and then frequency changes), and consequently a kind of guarantee to qualify the material regardless of the device structure. However, the modelling is complicated and the resulting frequency is as follows, according to the Stevens-Tiersten model [START_REF] Stevens | An analysis of doubly rotated quartz resonators utilizing essential thickness modes with transverse variation[END_REF], one of the most accurate theories:

                      ) 1 2 ( ) 1 2 ( 2 1 1 0 2 2 p C P m C M R h n f f e n e n n nmp

  

Where fn are the eigenfrequencies of the corresponding infinite flat plate, R0 the curvature radius of the convex face, m, p the number of modal lines in the resonator plane, Mn and Pn the dispersion constants.

Nevertheless, several assumptions can be made by considering some orders of magnitude [START_REF] Bon | Temperature coefficients of crystallinequartz elastic constants over the cryogenic range [4 K, 15 K][END_REF], and calculation proceeds according to the described process.

III. EXPERIMENT

A. Experimental set-up

Resonators have been tested in a conventional pulse tube cryorefrigerator SRP082B. The operating temperature is controlled at ±3 mK within [4 K, 15 K] by a Lakeshore 335 probing a cernox sensor and actuating a heater embedded inside the copper block where the resonator under test is set.

B. Devices under test

As mentioned above, a set of dedicated BAW energytrapped resonators with gold electrodes, as shown in Fig. 1, has been made and tested.

Table 1 shows, from left to right, from which cut and modes each experimental frequency-temperature characteristic is taken from, the corresponding frequency coefficients of the polynomial fit, and the targeted elastic coefficient(s).

Cut

From experimental data Target Mode T (1) f In 10 -7 T (2) f In 10 -8 T (3) IV. RESULTS

Table II shows the resulting 1 st , 2 nd and 3 rd order temperature coefficients of the Cij's, for quartz, with Tref = 8 K. V. CONCLUSION Resulting data will be used to identify temperature compensated cuts in the very useful range [4 K, 15 K], where quartz resonators exhibit Q-factors of about one billion.

Fig. 1 .

 1 Fig. 1. Set of plano-convex resonators tested for determining the temperature coefficients of the elastic constants of quartz at 4 K.

TABLE I .

 I TARGETED ELASTIC COEFFICIENTS

f In 10 -9

	AT	C300, C500	0.769	0.748	0.105	C14
	SC	A300, A500	-1.178	-2.189	-2.747	C13, C33
	SC	B300, B302 B320, B500	-0.879	-1.672	-1.832	C13, C33
	X	A100, A1300, A1500	-0.143	-0.451	-0.581	C11, (C12 a )
	Y	C300, C500	1.339	1.869	1.289	C66
	Z	C300, C500	-1.465	-2.662	-3.571	C44
					a. Calculated from C11 and C66

TABLE II .

 II RESULTING COEFFICIENTS

	Cij	T (1

) Cijkl in 10 -7 T (2) Cijkl in 10 -8 T (3) Cijkl in 10 -9

	C11	-0.235	1.28	-3.25
	C12 b	-6.39	-10.56	-4.22
	C13	-7.18	-7.42	-0.877
	C14	-1.35	-1.31	-7.93
	C33	-4.62	-5.65	-7.62
	C44	-3.17	-4.37	-11.6
	C66	3.08	5.92	0.486
				b. Calculated from C11 and C66

This work is supported by Conseil Régional de Franche-Comté. J.B. is thankful to Ministère de l'Enseignement Supérieur et de la Recherche, for his grant.