LGS and GaPO4 piezoelectric crystals: New results

LGS and GaPO4 piezoelectric crystals: New results

Solid State Sciences 12 (2010) 318–324 Contents lists available at ScienceDirect Solid State Sciences journal homepage: www.elsevier.com/locate/sssc...

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Solid State Sciences 12 (2010) 318–324

Contents lists available at ScienceDirect

Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie

LGS and GaPO4 piezoelectric crystals: New results O. Le Traon*, S. Masson, C. Chartier, D. Janiaud ˆ tillon, France ONERA, DMPH, 29 avenue de la Division Leclerc, 92322 Cha

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 January 2009 Received in revised form 26 June 2009 Accepted 29 June 2009 Available online 4 July 2009

Theoretical and experimental results of two interesting piezoelectric crystals for vibrating sensors, the La5Ga3SiO14 (also written LGS) and the GaPO4, are presented. First a comparison of different piezoelectric crystals in terms of quality factor and temperature stability is given and shows the theoretical interest of the GaPO4 and the LGS isomorphs (LGX). Then an experimental validation has been undertaken: a dedicated decoupling structure allowing high insulation of two flexural modes is described, as well as its micromachining by ultrasonic machining (USM). The experimental measurements of the LGS resonator gave disappointing results with very low measured quality factors of about 10,000 (under vacuum) for a beam working at 10 kHz. On the other hand, the GaPO4 resonator behaves very well: a quality factor higher than 700,000 (beam frequency 9 kHz) at room temperature and under vacuum has been obtained, as well as a low temperature frequency dependence of 12 ppm/K, in accordance with the theoretical predictions. The impact of the Au excitation/detection electrodes on the quality factor has also been studied in detail and it clearly explains why the measured quality factors are lower than the thermoelastic limit. Ó 2009 Elsevier Masson SAS. All rights reserved.

Keywords: Gallium orthophosphate Langasite LGX Vibrating inertial sensor Quality factor Damping Thermoelasticity Viscosity

1. Introduction Because of the direct coupling between their mechanical and electrical behaviours, piezoelectric materials find a lot of applications in the field of sensors, actuators or resonators (microbalances, accelerometers, gyros, pressure sensors, piezoelectric motors, filters and oscillators.). Since the first development of a quartz resonator by W.G. Cady’s in the 1920s [1], and thanks to its outstanding properties, quartz crystal remains the most widely used piezoelectric material, in particular for high accurate devices like ultrastable oscillators [2,3] or vibrating inertial sensors [4–6]. Today, piezoelectric crystals such as GaPO4 or LGS (La5Ga3SiO14) and its isomorphs LGT or LGN (called LGX family) are emerging [7–9] and a lot of research activities have been undertaking to develop devices with these crystals. Moreover, high and relatively large quality crystals are now available [10–12], and commercial products now exist, for instance in the field of pressure sensors or SAW filters. Concerning vibrating inertial sensors – i.e. Vibrating Beam Accelerometer (VBA) and Coriolis Vibrating Gyro (CVG) – two main parameters impact their accuracy, the quality factors Q of the useful vibrations and the temperature stability of the material.

* Corresponding author. Tel.: þ33 146734835; fax: þ33 146734824. E-mail address: [email protected] (O. Le Traon). 1293-2558/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2009.06.032

This paper firstly presents a comparison of different piezoelectric crystals in terms of quality factor and temperature stability and shows the theoretical interest of using GaPO4 and LGX crystals. Secondly, an experimental validation is undertaken: the micromachining by USM of dedicated LGS and GaPO4 resonators is described and experimental results are presented. Concerning the quality factor, the theoretical impact of the gold layers deposited on the beams for the piezoelectric actuation is also discussed and compared to the experimental measurements obtained with the GaPO4 resonator. 2. Piezoelectric crystals for vibrating inertial sensors 2.1. Quality factor Vibrating inertial sensors use a resonator as sensing element, and its quality factor Q plays an important part in the final performance: for a vibrating beam accelerometer, the Q factor impacts directly the beam frequency stability – i.e. the bias stability – and a high Q factor of a Coriolis vibrating gyro leads to a better phase control, increasing the parasitic common mode rejection in the synchronous demodulation operation. The vibrating element of such sensors is generally a beam working in a flexural mode. In this case, the thermoelastic damping is most often the main damping factor. This damping results from the temperature gradients induced by the strain field and the heat

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flow during the vibration period. Zener [13] was the first to establish an analytic expression of the thermoelastic quality factor Qthermo of a beam in flexure mode Eq. (1):

Q thermo ¼

rC F02 þ F 2 p l with F0 ¼ 2 rCe2 a2 TE F0 F

(1)

where r is the volumic mass (kg/m3), C the heat capacity (J/kg C), a the thermal expansion (along the beam), T the absolute temperature (K), E the equivalent Young’s modulus (N/m2), F the beam frequency (Hz), l the thermal conductivity (W/mK) and e the beam thickness (m). F0 represents the frequency transition between isothermal (the temperature is in equilibrium during the vibration strain) and adiabatic (no heat flow during the vibration strain) vibrations, and the maximal dissipation is obtained for F ¼ F0 and noticed Qmin Eq. (2) below). When the beam frequency is much lower than the frequency F0, the beam is always in thermal equilibrium during the vibration (isothermal vibration represented by Eq. (3); when the beam frequency is much higher than F0, the temperature has no time to transfer (adiabatic vibration, Eq. (4)):

Q min ¼

2rC a2 TE

pl 1 2a2 TE e2 F

(3)

2r2 C 2 2 e F pla2 TE

(4)

Q isothermal ¼

Q adiabatic ¼

(2)

Three kinds of piezoelectric crystal have been studied: semiconductors (ZnO, GaN, GaAs), dielectrics (Quartz, GaPO4, LGX family) and ferroelectrics (Li2B4O7, LiNbO3, LiTaO3). Their properties [12,14–22] are summarized in Table 1. Because of the anisotropic behaviour of the crystals, an average value of the different properties has been given. For reference, the piezoelectric and dielectric properties are also mentioned, as well as the well-known nonpiezoelectric silicon crystal. The log curves of Fig. 1 exhibit a V shape, corresponding to the isothermal behaviour (Eq. (3)) on the left side and to the adiabatic behaviour (Eq. (4)) on the right side. The minimum Q factor is obtained when the transition frequency F0 is equal to the beam frequency, fixed here at 50 kHz. The high thermal conductivity of semiconductor materials gives a transition frequency F0 higher than the dielectric crystals. The minimum Q factor is thus obtained for a beam thickness from 30 to 50 mm, instead of lower than 10 mm for dielectric materials. That means that semiconductor crystals are particularly well suited when a strong miniaturization is sought [23]. However, dielectric and ferroelectric crystals are more interesting (particularly LBO,

Fig. 1. Comparison of Zener thermoelastic quality factor versus the beam thickness (beam frequency F ¼ 50 kHz).

LGX and GaPO4 materials) since the obtained Q factor for a 50 mm beam is respectively around 106, 5  105 and 8  104, in comparison with 104 for the quartz beam. 2.2. Temperature stability The temperature stability of the crystal is also essential because of its direct link with the resulting temperature sensitivity of the sensors. It means that the resonance frequency of such sensors exhibits very low temperature dependence. In the case of flexural mode, and for an isotropic material, the beam frequency F is given by (Eq. (5)):

e L2

F ¼ k

sffiffiffi E

(5)

r

where e is the beam thickness (m), L the beam length (m), E the young modulus (N/m2), r the volumic mass (kg/m3) and k a constant depending on the boundary conditions (k ¼ 1.028 and 0.16 respectively for a clamped-clamped and a clamped-free beam in fundamental mode).

Table 1 Crystal data taken into account for the Zener thermoelastic quality factor calculation (mean values). Crystal

r, kg/m3

a, 106/K

E, 109 N/m2

l, W/mK

C, J/(kgK)

3, 1012 F/m

e, C/m2

Quartz GaPO4 LGX LNO LBO LTO ZnO GaN GaAs Si

2648 3570 5800 4640 2439 7454 5680 6150 5300 2330

13.7 10 6 15.4 11 16.1 6.51 4 5.73 2.6

78.8 55.8 115 173 114 205 127 325 85 169

6.5 3.9 1.8 5.6 2.5 4.6 100 130 55 148

700 600 415 633 1220 250 522 490 330 700

39 56 165 247 72 379 90 84 86

0.17 0.2 0.4 2.5 0.29 5 0.55 0.33 0.16

320

O. Le Traon et al. / Solid State Sciences 12 (2010) 318–324

Fig. 2. Relative frequency temperature dependence of different crystals (absolute values, around ambient temperature).

The logarithmic derivative of Eq. (5) leads to:

1 vF 1 vE 1 ¼ þ a F vT 2E vT 2

(6)

Eq. (6) is an approached formula of the relative frequency temperature dependence and doesn’t take into account the whole anisotropy of materials. Nevertheless, it allows a good comparison between the different crystals, as shown in Fig. 2. Fig. 2 confirms that the quartz remains the most temperature stable crystal (w8 ppm/ C), but GaPO4 shows a relatively low frequency dependence of 14 ppm/ C and the LGS isomorphs (LGX), around 45 ppm/ C is also interesting. Moreover, compensated cuts have been already related for these two crystals [24, 25], allowing a significant reduction of their frequency temperature dependence. Unfortunately, the ferroelectric LBO crystal exhibits high temperature dependence, as usual for pyroelectric crystals. Thus, because of potential high quality factors and expected low frequency temperature dependence, LGX family and GaPO4 crystals seem particularly attractive materials for vibrating inertial sensors. 3. Experimental investigations of LGS and GaPO4 crystals 3.1. Design and realization of the resonators

the energy losses out of the resonator. The selected resonator (Fig. 3) is based on the VIG CVG structure [6] and is based on of a tuning fork, an insulating system composed of two massive parts and flexible arms, and two mounting areas to fix the structure on a base. Thanks to this structure, two high insulated modes are available, the tuning fork and the out-of-plane flexural modes (Fig. 4), since the energy losses out of the resonator (when fixed on a base) are lower than 108 for the two modes. Fig. 5 shows the LGS and GaPO4 resonators obtained by ultrasonic abrasion micromachining. A specific ultrasonic technique, called ‘‘by generation’’, has been implemented, allowing the relatively easy machining of complex shapes with a simple tool [26]. On the other hand, a minimum groove of about 0.7 mm is needed (the tool diameter), limiting the miniaturization of the devices. Thus, the diameter of the devices (Fig. 5) is approximately of 20 mm with a thickness of 1 mm (along Z axis). The beam tuning fork length is 8 mm (along Y axis) and its section 1 mm  0.9 mm. The excitation/detection electrodes for the piezoelectric actuation of the two modes (tuning fork and the out-of-plane flexural modes) are obtained by photolithography techniques from the chromium–gold layers (20 nm Cr and 200 nm Au) previously deposited on the LGS and GaPO4 wafers by sputtering. After the ultrasonic micromachining, a soft chemical etching is realized in order to remove the amorphous zone generated on the etched sides (w20 mm must be removed on each side).

3.2. Experimental results on the LGS resonator The first results of the LGS resonator have been a little bit disappointing: the measurements via the dedicated electrodes were not possible for the two modes, which leaves a doubt about the real orientation of the crystal. The measurements have been thus undertaken using optical vibrometry techniques [27] allowing in-plane and out-of-plane measurements under vacuum (lower than 105 mbar). The obtained quality factors are very low since the theoretical Zener thermoelastic quality factors are higher than 15 million for these beam dimensions and frequencies (Table 2). An explanation of these poor quality factors could be found in the bad quality of the micromachining, particularly in the sensitive area of the beam built-in ends (Fig. 6). At this time, it is thus necessary to make other measurements on other resonators to evaluate the real potential of the LGS crystal.

The resonator design must be compatible with potential high quality factor and needs a high insulating system in order to reduce

Mounting areas

2nd massive part

Flexible arms Flexible arms

Mounting areas

1st massive part Tuning fork Fig. 3. Monolithic resonator structure composed of a tuning fork and an insulating system between the tuning fork and the mounting areas.

Fig. 4. The two high insulated modes of the VIG structure: the tuning fork (on the left) and the flexural out-of-plane (on the right) modes. The colour represents the strain energy. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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321

Fig. 5. GaPO4 (on the left) and LGS (on the right) devices obtained by ultrasonic micromachining.

3.3. Experimental results on GaPO4 resonator Figs. 7 and 8 respectively give the measured frequency temperature dependence of the tuning fork and the out-of-plane flexural modes. Theoretical results obtained from Eq. (6) and GaPO4 data [12] are also given, as well as those predicted by finite element modelling (FEM). For the tuning fork mode (frequency 8800 Hz at 25  C), the measurements are in accordance with the FEM results and give a frequency dependence of about – 12 ppm/ C. The theoretical results give a slightly weaker sensitivity of 11 ppm/ C. The out-of-plane mode measurements (frequency 8590 Hz at 25  C) give a frequency dependence of 16.5 ppm/ C, also in perfect adequation with the FEM results. Nevertheless, the theoretical results from Eq. (6) do not fit as well than the previous ones, since the obtained dependence is 21 ppm/ C. That shows the limitation of this model which does not take into account the whole anisotropy of the material. As expected, GaPO4 crystal shows low frequency temperature dependence for the two modes, and the good fit between measurements and finite element modelling validates the data. Compensated cuts could be thus precisely determined by finite element modelling. The quality factors of the two modes and their temperature dependence have been measured thanks to the Agilent Impedance Analyzer 4294 (Figs. 9 and 10). High quality factors have been obtained, respectively 710,000 and 470,000 (at 30  C) for the tuning fork and out-of-plane modes. These results prove the efficiency of the resonator design, the quality of its micromachining, as well as the quality of the GaPO4 crystal, but don’t meet the theoretical expectations. Indeed, as shown in Figs. 9 and 10, the expected quality factors resulting from the Zener theory (Eq. (1); squared-shaped points) are respectively 1.55 million and 1.15 million at room temperature for the tuning fork and out-of-plane flexural modes. The Lifshitz and Roukes theory [28], in which the beam transverse temperature profile is more accurately modelled, gives lower quality factors, respectively 1.28 million and 970,000 for the tuning fork and out-of-plane modes. Finite element modelling [29], taking into account the whole thermo-piezo-elastic coupling, the full anisotropy of the crystal and the real 3D shape of

the resonator, leads to a quality factor of 1.1 million for the tuning fork mode and 975,000 for the out-of-plane mode. These values are thus higher than the measured ones, which underline that another dissipation mechanism takes place here. 4. Effect of the gold layers on the beam quality factor 4.1. Theoretical considerations The effect of the viscosity of the gold layers deposited on the beams is studied here. Taken the general Kelvin–Voigt model of a viscous solid:

s ¼ E3 þ h3_

(7)

where s is the stress (N m2), E the Young modulus (N m2), 3 the strain, h the viscosity (N s m2) and 3_ the strain velocity (s1), the elementary energy dEenergy is written as:

dEenergy ¼ sd3 ¼ E3d3 þ h3_ d3 whence the instantaneous power Pinst:

Pinst ¼

dEenergy d3 d3 ¼ E3 þ h3_ dt dt dt

(9)

which gives the dissipated energy per period and per volume unit:

Edissipated=period ¼

ZT

Pinst dt ¼

0

ZT

E33_ dt þ

0

ZT

 2

h 3_ dt

(10)

0

In the case of a harmonic solicitation, Eq. (10) leads to:

Edissipated=period ¼

ZT

 2

h 3_ dt ¼ h320 u2 T ¼ 2ph320 u

(11)

0

where 30 is the amplitude of the strain, and u its pulsation (rd/s). The volume integration of Eq. (11) gives the total dissipated energy per period:

ETOT dissipated=period ¼

Table 2 Measurements obtained with the LGS resonator.

Frequency (Hz) Quality factor

(8)

Z

2ph320 udV

(12)

V

Tuning fork mode (flexure along X axis)

Out-of-plane mode (flexure along Z axis)

12683.5 12,000

12886.5 10,000

while the total stored energy is given by:

ETOT

stored

¼

Z V

E320 dV

(13)

322

O. Le Traon et al. / Solid State Sciences 12 (2010) 318–324

0.002

DF/F

0.001 0 -0.001 -0.002 -0.003 -50

0

50

100

Temperature °C

Measurements

Finite Element

150

Theoretical

Fig. 8. Frequency temperature dependence of the out-of-plane flexure mode: diamond-shaped points correspond to the measurements, the solid line to finite element modelling and the squared-shaped points to the theoretical results given by Eq. (6).

Fig. 6. Beam built-in ends obtained after ultrasonic micromachining of the LGS resonator. Flexure X EStored

e3 ¼ E h 12

ZL

3Flexure 0

X

¼ x

v ux ðyÞ vy2

(14)

!2 dy

(17)

0

With reference to Fig. 11, Eqs. (14) and (15) give respectively the strain expression 30 for the in-plane and out-of-plane flexure modes (Bernouilli hypothesis): 2

v2 ux ðyÞ vy2

Flexure Z ETOT dissipated=period ¼ 2pue

gh2 h3 hcryst: þ h 12 2 Au

! ZL

!2 v2 ux ðyÞ dy vy2

0

(18) 2

v uz ðyÞ vy2

Z 3Flexure ¼ z 0

(15)

where ux and uz are respectively the displacements along the X and Z axis. From Eqs. (14) and (15), the total dissipated energy per period (Eq. (12)) and the stored energy (Eq. (13)) can be evaluated for the two modes:

Flexure X ETOT dissipated=period ¼ 2pu

 e3  hhcryst: þ 2ghAu 12

ZL

v2 ux ðyÞ vy2

!2 dy

0

(16)

Flexure Z EStored

-0.002 -50

0

50

100

150

Temperature °C Measurements

Finite Element

Theoretical

Fig. 7. Frequency temperature dependence of the tuning fork mode: diamond-shaped points correspond to the measurements, the solid line to finite element modelling and the squared-shaped points to the theoretical results from Eq. (6).

v2 uz ðyÞ vy2

!2 dy

(19)

where hcryst and hAu are respectively the intrinsic viscosity of the GaPO4 crystal and the gold layer (N s m2), e and h the beam cross-section (m), g the gold layer thickness (m), L, the beam length (m) and E the Young modulus of the crystal. Considering the weak thickness g of the gold layer in comparison of the beam cross-section dimensions (ratio of about 5000 for e/g and h/g), the stored energy in the gold layer is not taken into account in Eqs. (17) and (18).

Quality Factor

DF/F -0.001

ZL 0

0.001

0

h3 ¼ E e 12

2.20E+06 2.00E+06 1.80E+06 1.60E+06 1.40E+06 1.20E+06 1.00E+06 8.00E+05 6.00E+05 4.00E+05 2.00E+05 -50

-30

-10

10

30

50

70

90

110

130

150

Temperature °C QZener

QLifchitz

F.E.M

Measurements

Fig. 9. Quality factor versus temperature of the tuning fork mode: the squared-shaped points correspond to the Zener theory, the diamond-shaped points to the Lifchitz and Roukes theory, the cross-shaped points to the finite element modelling and the starshaped points to the measurements.

O. Le Traon et al. / Solid State Sciences 12 (2010) 318–324

1.80E+06

Table 3 Resulting QAu obtained for the two modes and at different temperatures.

1.60E+06

Quality Factor

323

1.40E+06

Tuning fork mode

1.20E+06 1.00E+06

Out-of-plane mode

8.00E+05 6.00E+05

Temp

Qmeasure

Qth

30  C 90  C 130  C 30  C 90  C 130  C

7.1  105  5% 4.8  105  5% 4.1  105  5% 4.7  105  5% 2.8  105  5% 2.3  105  5%

1.1  106 1.3  106 1.46  106 9.75  105 1.19  106 1.32  106

Fem

Qviscous

QAu

2  106  16% 7.6  105  8% 5.7  105  7% 9.1  105  10% 3.79  105  5% 2.8  105  5%

670  16% 255  8% 190  7% 720  10% 290  5% 220  5%

4.00E+05 2.00E+05 -50

-30

-10

10

30

50

70

QLifchitz

QZener

90

Temperature °C

110

130

150

Measurements

F.E.M

Fig. 10. Quality factor versus temperature of the out-of-plane mode. The square-shaped points correspond to the Zener theory, the diamond-shaped points to the Lifchitz and Roukes theory, the cross-shaped points to the finite element modelling and the star-shaped points to the measurements.

It is thus possible to write the expression of the resulting quality factor Qviscous ¼ 2pEStored/Edissipated/period for the two modes:

1 Flexure QViscous

X

¼

hcryst u E

þ

hAu u 2g hAu u 2g E

h

y

E

1 EAu 2g ¼ h QAu ECryst h (20)

1 Flexure QViscous

Z

¼

hcryst u E

þ

hAu u 6g hAu u 6g E

h

y

E

h

¼

1 EAu 6g QAu ECryst h (21)

where QAu ¼ EAu/(hAuu) is the quality factor of the gold material and EAu its Young modulus. In Eqs. (20) and (21), the term (hcrystu)/E is negligible since hcryst is generally lower than 103 N s/m2. Eqs. (20) and (21) show that the viscosity of the beams’ gold layers impacts the quality factor of the out-of-plane flexural mode 3 times more than the in-plane beam mode. In the case of a tuning fork, the stored energy is not only confined in the 2 beams but also in the stem, and its distribution differs for the in-plane and out-of-plane modes. Taking also into account that the GaPO4 resonator electrodes cover only 90% of the top and bottom sides of the beam (see Fig. 5.), it is possible to estimate more precisely (FEM) the ratio between the stored energy in the gold covered beams and the total stored energy of the two modes: this ratio is 47% for the out-of-plane flexural mode and 60% for the tuning fork mode. That leads to adjust the Qviscous of Eqs. (20) and (21) in the same proportions: Tuning fork mode

Qviscous

¼ 0:83  QAu

EGaPO4 h EAu g

(22)

EGaPO4 h Out-of -plane mode Qviscous ¼ 0:35  QAu EAu g

(23)

4.2. Application to the GaPO4 quality factors measurements From the quality factor measurements of the 2 modes (Section 3.3) and their theoretical thermoelastic quality factors (FEM), the Qviscous can be easily calculated by:

1 1 1 ¼  Qviscous Qmeasured Qthermoelastic

5. Conclusion Because of the expected great improvement of the thermoelastic quality factor of vibrating beam in flexure, as well as their good temperature stability, GaPO4 and LGX family crystals are two interesting alternative piezoelectric materials of quartz in the field of vibrating inertial sensors. Due to a bad quality of the micromachining, the first measurements on the LGS resonator were unsatisfactory, and need to be completed by other experiments. However, the experimental results on a dedicated GaPO4 resonator are particularly encouraging: the frequency temperature

X

γ h/2

Z X

Z

h/2

GaPO44 GaPO Y

(24)

The gold layer thickness g of the studied GaPO4 resonator is w200 nm and the gold Young modulus is estimated to 7.8  1010 N/ m2. The beam thickness h is 1 mm and the GaPO4 Young modulus is w5.6  1010 N/m2. From Eqs. (22) and (23), it is thus easy to obtained the QAu which must be equal for the 2 modes and at each temperature. Table 3 summarizes the obtained results for the GaPO4 resonator. The resulting QAu are very similar for the two modes, which clearly validate the previous developments and the impact of the gold layers on the measured quality factors. At the same time, these results also consolidate the predicted thermoelastic quality factors and their temperature dependence. The obtained QAu and its temperature dependence (due to the well-known metal viscosity increase with temperature) is relatively closer to other results [30], even if the QAu depends strongly on the deposition techniques, here by sputtering.

Z

Au

L

For the GaPO4 resonator, the impact of the viscosity of the gold layers is thus 2.35 times higher for the out-of-plane mode than for the tuning fork mode.

e/2 e/2

γ

Y X

Fig. 11. Beam cross-section with the gold layers. On the left, strain in the beam for the in-plane flexure (along X) and, on the right, for the out-of-plane flexure mode (along Z).

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O. Le Traon et al. / Solid State Sciences 12 (2010) 318–324

dependence measurements are in total accordance with the predicted ones, high quality factors have been obtained, lower than the theoretical thermoelastic limit, but clearly explained by the deposited gold layers on the beams. These experimental results totally confirm the high potential of the GaPO4 crystal. Future work will confirm these be focused on new experimentations on LGS and LGT resonators and on characterization at the microscale of GaPO4 resonators. Other work will be also undertaken to develop new micromachining techniques (chemical etching, DRIE,.) suited to LGX and GaPO4, allowing the realization of inertial vibrating sensors. Acknowledgments The authors wish to thank the De´le´gation Ge´ne´rale de l’Armement (DGA) for its financial support. References [1] W.G. Cady, Piezoelectricity, McGraw-Hill Book Company, Inc., New York, 1946. [2] J. Chauvin, P. Weber, J.-P. Aubry, F. Lefebvre, S. Galliou, E. Rubiola, X. Vacheret, A new generation of very high stability BVA oscillators, IEEE International Frequency Control Symposium and 21th European Frequency and Time Forum, Geneva, Suisse, 2007. [3] G. Weaver, M. Reinhart, M. Miranian, Developments in ultra-stable quartz oscillators for deep space reliability, 36th Annual Precise Time and Time Interval (PTTI) Meeting, Washington, USA, 2004. [4] R. Jaffe, Quartz MEMS GPS/INS technology Developments, Symposium Gyro Technology, Stuttgart, Germany, 2003. [5] O. Le Traon, The VIA vibrating beam accelerometer: concept and performances, Proceedings of the Position, Location and Navigation Symposium, Palm Spring, April 1998. [6] D. Janiaud, The VIG vibrating gyrometer: a new quartz micromachined sensor, Symposium Gyro Technology, Stuttgart, Germany, 2003. [7] P.M. Worsch, GaPO4 crystals for sensor applications, sensors 2002, Proceedings of IEEE 1 (June 2002) 12–14. [8] J. Imbaud, A. Assoud, R. Bourquin, J.J. Boy, S. Galliou, J.P. Romand, Investigations on 10 MHz LGS and LGT crystal resonators, IEEE Frequency Control Symposium, 2007 Joint with the 21st European Frequency and Time Forum, May 29 2007–June 1 2007, pp. 711–714. [9] O. Le Traon, R. Levy, F. Deyzac, D. Janiaud, B. Lecorre, S. Muller, M. Pernice, Preliminary results about GaPO4 vibrating inertial sensors, J. Phys. IV France 126 (2005) 123–126 EDP Sciences. [10] J. Luo, D. Shah, C. Klemenz, M. Dudley, H. Chen, The Czochralski growth of large-diameter La3Ga5.5Ta0.5O14 crystals along different orientations, J. Cryst. Growth 287 (2) (25 January 2006) 300–304. [11] http://www.newpiezo.com. [12] www.piezocryst.com.

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