A comparison of vibrating beam resonators in Quartz, GaPO(4), LGS and LGT

The present paper reviews and compares the properties of vibrating beam resonators in length-exten- sional, ﬂexural and cylindrical-torsional modes; resonators are designed and built with new piezoelectric materials: Gallium orthophosphate (GaPO 4 ), Langasite (LGS) and Langatate (LGT). Their characteristics are compared with the results obtained with the same structures built in classical Quartz crystal. For all 3 new crystals, temperature-compensated cuts have been found for length-extensional and ﬂexural modes by a theoretical analysis. Experiments have been made on test beams fabricated by conventional precision lapping/machining and photolithography to deposit electrodes. The existence of temperature-compensated cuts for length-extensional resonators is experimentally conﬁrmed. The possibilities of using temperature-compensated cylindrical resonators in the torsional modes are analysed and their temperature sensitivity is computed.


Introduction
Since the last 20 years, new piezoelectric materials have been developed, among them, Gallium orthophosphate (GaPO 4 ) [1] and the Langasite family (Lanthanum Gallium Silicate La 2 Ga 5 SiO 14 LGS [2], Lanthanum gallium tantalate, La 3 Ga 5.5 Ta 0.5 O 14 LGT [3])h a v e been studied in detail for high-performance piezoelectric resonator applications. The reasons of this interest are the material is now available commercially in crystal sizes compatible with production machines used in the industry; GaPO 4 , LGS and LGT can be used at high temperatures without loosing their piezoelectric properties; GaPO 4 , LGS and LGT exhibit temperature-compensated cuts for various modes of vibration: bulk waves, surface waves, vibrating beams and cylinders.
What is the interest of piezoelectric beam resonators? Historically, beam resonators were developed in the years 1930-1940 as the standard piezoelectric element to stabilize the frequency of high-performance oscillators. This kind of resonators was superseded by thickness-shear resonators in the field of precision time and frequency control, however vibrating Quartz beams in flexural modes are the key elements of very successful miniature wrist-watch oscillators [4], a device built by hundreds of millions each year.
This kind of resonators is also used in atomic force microscope as a shear-force sensor [5]. Several other sensor applications built from Quartz crystals, like viscosimeters based on cylindrical resonators in torsional modes could also be foreseen with the new crystals in vibrating modes at high temperatures [6].
In this paper, the properties of vibrating-beam resonators made in GaPO 4 , LGS and LGT are investigated. The temperature-induced frequency shifts are computed and temperature-compensated orientations of the crystal are found for which the first-order temperature coefficient of the resonant frequency (TCF) is zero.
Taking into account that no resonator or sensor applications can be developed for Quartz at high temperatures (due to the a-b phase transition at 573 C), the possibility to use new piezoelectric materials at high temperatures is investigated.
Another rationale is to investigate temperature effects in resonators built with the new piezoelectric materials. For Quartz, material constants and their temperature derivates are well known; however for new materials, only a limited number of vibrating modes have been investigated and tested for frequency-temperature effects, mostly thickness-shear. There is a need for more experimental data to improve the knowledge of temperature derivatives of elastic constants: experiments with beam resonators will allow to expand the database of modes for a precise determination of temperature effects.
Neglecting the influence of piezoelectricity on the resonant frequency, simple analytical models [7,8], yield the lowest-order resonant frequency f 1 of the structures: for length-extensional modes of a free-free beam: for flexural modes of a clamped-free beam:  for torsional modes of a free-free cylinder: Where: f 1 is the resonant frequency of the lowest-order mode; L is the beam length, w is the beam width and t the beam thickness; I is the inertia of the beam, I ¼ wt 3 12 r is the mass density of the crystal, S ij are the compliance elastic constants and C ij stiffness constants.

Modelling temperature-compensated cuts
One of the most important properties of piezoelectric resonators built in Quartz, GaPO 4 , LGT and LGS is their ability to allow the design of temperature-insensitive vibrating devices, taking advantage of anisotropy of the crystals and the specific behaviour of temperature dependence of material constants to cancel the firstorder temperature derivatives of the resonant frequency. In order to investigate for temperature-compensated modes, the following approach has been implemented [7,9]: change length L and lateral dimension t according to anisotropic thermal expansion (f 1 is actually independent from width w); change specific mass r according to thermal expansion; change elastic stiffness constants C ij and compliance S ij according to their 1st and 2nd order temperature coefficients; re-compute frequency for each temperature and fit the dependence to a 2nd order polynomial; find 1st order and 2nd order temperature coefficients of frequency (TCF); look for cut angles for which the 1st order TCF is zero.
The data for mass density, coefficients of thermal expansion, temperature derivatives of elastic constants are taken from Refs. [10][11][12]. Cut angles for quartz are taken orientated according to Ref. [13]; for new crystals, the new IEEEE standard is used [14].
It has been found that for all crystals: Quartz, GaPO 4 , LGT and LGS, temperature-compensated orientations exist for beam resonators. Theoretical and experimental results are now presented for all crystals.

Temperature-compensated cuts: length-extensional
For length-extensional resonators, it is a classical result that Quartz does not exhibit a perfectly temperature compensated orientation for singly-rotated cuts (IEEE angle 4 ¼ 0 [14]). Fig. 2 shows that for new crystals, however, several temperaturecompensated orientations exist for which the turnover point is at room temperature in the case of length-extensional resonators.
For LGT, the model predicts a temperature-compensated singlyrotated cut at 250 C for q ¼À15 (Fig. 3). By varying the cut angle, different operating temperatures with a zero TCF can be selected for each application (Fig. 4).
A comparison between theoretical predictions at room temperature and experiments for LGT is presented in Fig. 5. The model and experiments agree well but not perfectly. The reasons for this are very probably: the fact that temperature coefficients of elastic constants have been extracted from thickness-shear resonator measurements only; Fig. 3. LGT, length-extensional: predictions of frequency vs. temperature for cut angle theta ¼À7.3, À15 and þ35 . Fig. 4. LGT, predictions for the lowest resonant frequency of a 20 mm long lengthextensional resonator. the present model does not take into account piezoelectric effects and could be improved by using a better definition of temperature derivatives of fundamental elastic constants [15].
The Table 1 presents the different values of the frequency constant (in Hz mm) and the second order coefficients for temperature-compensated length-extensional modes. For quartz, no temperature-compensated mode exists among singly-rotated cuts. The values for the BT-cut of quartz and AT-cut of GaPO 4 are given only for comparison of the magnitude.
The (Table 2) presents the experimental motional parameters actually measured on different length-extensional resonators made in LGS and LGT in order to experimentally determine the optimum temperature-compensated cut. All resonators are 20 mm long, cross-section of the beam: 2 mm Â 1 mm. All parameters were measured at room temperature under atmospheric pressure.

Temperature-compensated cuts: flexural
A similar approach is applied to flexural resonators. Note the absolute similarities with the length-extensional of the curves giving the 1st TCF vs. the cut angle q. This behavior is very similar to what was known for quartz (Fig. 6).
For GaPO 4 , a temperature-compensated cut is predicted at 400 C for a cut angle q ¼À18.2 (Fig. 7).
Two temperature-compensated cuts are predicted for LGS at room temperature (Fig. 8).
The following table (Table 3) presents the different values of second order coefficients for temperature-compensated flexural modes. For quartz, no temperature-compensated flexural mode exists.
Experimental verifications are now under test for LGS with devices fabricated by photolithography, gold thin-film deposition and precision diamond saw machining (Fig. 9).

Temperature-compensated cuts: torsional (cylinder)
Temperature effects for the 1st torsional mode of a cylinder are computed according to equation (3). All crystals are expected to have temperature-compensated cuts for singly-rotated cuts and the results are summarised in Table 4. LGT, length-extensional: frequency vs. temperature experiments for cut angle theta ¼À15, 0 and þ15 .  Table 2 Motional parameters of rectangular beam resonators vibrating in length-extensional mode.

Extension
LGS q ¼À25 LGS q ¼ 0 LGT, this is no surprise taking into account the higher mass density. A possible advantage is that for the same frequency, devices can be made smaller (Fig. 10).
All 3 kinds of modes: extensional, flexural and torsional/ cylinder exhibit temperature-compensated orientations at room temperature for singly-rotated cuts. Temperaturecompensated cuts are predicted at high temperatures, but a good reliable set of high-temperature constants is required for LGS and LGT in order to predict the actual orientations precisely. Torsional modes of cylinders are already applied to viscosimeters in Quartz, potential applications exist at high temperatures with other crystals.  It appears that the lowest overall temperature sensitivity for all crystals listed in this paper in GaPO 4 for which the second-order temperature coefficients are smaller than quartz and for some orientations, one order of magnitude is lower for LGS and LGT.
However, the resonant frequencies are lower in GaPO 4 and LGS/ LGT than in quartz, as summarised in Table 5:

Strong points and weak points of new piezoelectric crystals for beam resonator applications
The success of quartz for tuning-fork resonators can be explained as follows: 1. Quartz has been extensively studied for one century, reliable industrial sources of quartz crystal exist. 2. Quartz cuts close to the Z-cut can be batch processed by chemical etching. 3. Temperature-compensated cuts exist for flexural and torsional quartz beam resonators.   9. Fabrication of test devices by photolithography, gold thin-film deposition and precision diamond saw machining. LGT resonators is significantly greater than quartz used in similar vibrating modes; this has been measured for thickness-shear resonators [16].
The strong points of new piezoelectric crystals, to date, are: LGT [19] show that the minimum Qfactor for beam resonators in the frequency range (10-100 kHz) is 4 times higher for LGS and LGT than quartz.
This last point is very important for the development of precision vibrating beam accelerometers and force sensors where LGS and LGT could bring a significant improvement over quartz.

Conclusion
In conclusion, temperature-compensated cuts are found theoretically and experimentally for length-extensional modes of singly-rotated cuts of GaPO 4 , LGS, LGT. The model also predicts the existence of temperature-compensated cuts for flexural modes at high temperatures and temperature-compensated cuts for cylinders in torsional modes.
Possible applications involve miniature sensors at high temperatures, e.g. for the automotive industry. Actual development of those applications is conditioned by the possibility to micromachine miniature beam resonators efficiently. Meanwhile, scaled models can be made by photolithography and precision diamond saw cutting.

Table 5
Frequency range for resonant frequencies of singly-rotated cuts, 20 mm long beam resonator vibrating in length-extensional mode.

Extension
Quartz GaPO 4 LGS and LGT