Frequency-Temperature Compensated Cuts of Crystalline-Quartz Acoustic Cavity Within the Cryogenic Range [4 K, 15 K].

New temperature coefficients of quartz elastic coefficients particularly relevant at liquid-helium temperature have been reported recently. Based on this result, frequency-temperature compensated cuts are predicted by calculation and then demonstrated by experiment. Such compensated cuts can definitely fix the issue of remaining temperature sensitivity of crystalline-quartz acoustic cavities unbeatable for their extremely low mechanical loss, as low as $10^{-9}$ , when operated at liquid-He temperature.

A new method of probing mechanical losses of coatings at cryogenic temperatures.

A new method of probing mechanical losses and comparing the corresponding deposition processes of metallic and dielectric coatings in 1-100 MHz frequency range and cryogenic temperatures is presented. The method is based on the use of high-quality quartz acoustic cavities whose internal losses are orders of magnitude lower than any available coating nowadays. The approach is demonstrated for chromium, chromium/gold, and multilayer tantala/silica coatings. The Ta2O5/SiO2 coating has been found to exhibit a loss angle lower than 1.6 × 10-5 near 30 MHz at 4 K. The results are compared to the previous measurements.

Quality Factor Measurements of Various Types of Quartz Crystal Resonators Operating Near 4 K.

Quartz crystal resonators can exhibit huge quality factors (in excess of 1 billion) at liquid-helium temperature. However, they must satisfy a set of conditions to meet this high level of performance. With the help of experimentation, the main conditions are identified, such as the material quality, the energy trapping due to the vibrational mode structure, as well as the corresponding influence of the support mechanism and the effects of the electrodes.

Extremely low loss phonon-trapping cryogenic acoustic cavities for future physical experiments.

Low loss Bulk Acoustic Wave devices are considered from the point of view of the solid state approach as phonon-confining cavities. We demonstrate effective design of such acoustic cavities with phonon-trapping techniques exhibiting extremely high quality factors for trapped longitudinally-polarized phonons of various wavelengths. Quality factors of observed modes exceed 1 billion, with a maximum Q-factor of 8 billion and Q × f product of 1.6 · 10(18) at liquid helium temperatures. Such high sensitivities allow analysis of intrinsic material losses in resonant phonon systems. Various mechanisms of phonon losses are discussed and estimated.

Volume dependence in Handel's model of quartz crystal resonator noise.

Although criticized by many, Handel's quantum model for 1/f noise remains the only model giving a quantitative estimation of the level of intrinsic 1/f noise in quartz crystal resonators that is compatible with the best experimental results. In this paper, we reconsider the volume dependence in this model. We first argue that an acoustic volume, representing the volume in which the vibration energy is trapped, should be used instead of the geometrical volume between the electrodes. Then, we show that because there is an implicit dependence of the quality factor of the resonator with its thickness, the net effect of Handel's formula is not an increase of noise proportionally to the thickness of the resonator, as could be naïvely expected, but a net decrease when thickness increases. Finally, we show that a plot of Q(4)Sy versus the acoustic volume, instead of the usual Sy plot, could be useful to compare the quality of acoustic resonators having very different resonance frequencies.

Thermal sensitivity of elastic coefficients of langasite and langatate.

Thermal coefficients of elastic constants of langasite and langatate crystals have been determined from frequency-temperature curves of contoured resonators operating in thickness modes. The effect of the trapping of the vibration has been taken into account to improve the accuracy. In a first step, the thermal sensitivities of stiffness coefficients in Lagrangian description are obtained. Thermal sensitivities of the usual elastic constants are further deduced. Predictions of thermally compensated cuts are given.

Temperature compensated cuts in LGT crystal microresonators using length extensional mode.

In this letter, experimental investigation of frequency-temperature effects in langatate rectangular cross-section beams are presented. It is shown that a first-order temperature compensated cut exists for the first vibrating mode of length extension.

Investigations on LGS and LGT crystals to realize BAW resonators.

The LGS family are promising materials for the design of high quality bulk acoustic wave resonators. We have manufactured many plano-convex 10 MHz 5th overtone Y-cut resonators using langasite (LGS, La(3)Ga(5)SiO(14)) and langatate (LGT, La(3)Ga(5.5)Ta(0.5)O(14)) crystals. We observed that the quality factor strongly depends on the polishing method, the supplier of the material, and on the energy trapping. For quartz crystals, we have found that resulting IR spectra exhibit absorption peaks more or less deep, linked to defects. These predominant criteria are not surprising, but they have to be defined in manner similar to that used for quartz crystal. A satisfying machining and polishing method has been first applied to elaborate high Q resonators, and a comparison between samples of LGS and LGT materials from different suppliers is established. In addition, LGT resonators are characterized by their motional parameters and frequency-temperature curves. Nevertheless, one of the main results is that the measured Q x f product is not the expected one. We present results of Q-factor versus radius of curvature: it appears that an optimization should be performed and that this last one cannot be directly transposed from that of quartz crystal resonator. Currently, the best resonator that we have made has a Q x f product of 1.4 x 10(13) on its 5th overtone (1.7 x 10(13) on its 9th overtone). This result is slightly higher than the similar parameter obtained on a state-of-the-art SC-cut quartz crystal resonator working at the same frequency.

Development of a 10 MHz oscillator working with an LGT crystal resonator: preliminary results.

Presently, to our knowledge, measurement of the noise of langatate (LGT) crystal oscillators has not previously been reported. First results of such a measurement are given in this paper. They have been obtained from 10 MHz resonator prototypes tested with a dedicated electronics. The main steps of the resonator manufacturing are described in this paper. Good quality factors, close to 1.4 10(6), have already been achieved on the 5th overtone of the thickness shear mode of LGT Y cuts, even if the energy trapping should still be optimized. The motional parameters of these resonator prototypes are quite different from those of usual quartz crystal resonators. As a consequence, dedicated sustaining electronics have been designed. The explored options are reported to justify the implemented one. Moreover, the high thermal sensitivity of LGT crystal resonators (parabolic f-T curve) requires that particular attention be paid to the oven thermal stability. This important feature is also pointed out in the paper. The preliminary version of the resulting system exhibits a relative frequency stability of 6 10(-12).

Thermal compensation in GaPO4 beam resonators: experimental evidence for length extensional mode.

Temperature effects in gallium orthophosphate (GaPO4) vibrating beams are reported. In addition to the well-known, thickness-shear AT-cut, temperature-compensated cuts exist in GaPO4 for length extensional modes. Experimental evidence of a temperature-compensated cut in GaPO4 rectangular beam resonator vibrating in length extension is given.

Frequency-temperature behavior of Langasite rectangular beam resonators vibrating in length extension.

This paper shows that first order temperature compensated cut exists in Langasite rectangular cross-section beam vibrating in length extensional mode. Theoretical and experimental investigations of frequency-temperature effects are given.

Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model.

The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoulli's classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)theta cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation and this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared.

Temperature-compensated cuts for length-extensional and flexural vibrating modes in GaPO4 beam resonators.

Flexural modes are the basic vibrating mode of tuning forks used in quartz wrist watches; they also can be used as the basis for sensors. Very little work, if any, has been done for vibrating beam resonators in GaPO4. In this paper, the possibility of temperature-compensated cuts in GaPO4 is investigated for length-extensional and flexural vibrating modes. A theoretical investigation of rectangular cross-section GaPO4 vibrating beam resonators is accomplished by analytical methods. Modeling temperature effect is achieved by the approximate but classical method in which the effective elastic constants, beam dimensions, and crystal mass density are varied as a function of temperature. Temperature-compensated cuts are given in GaPO4 for length-extensional and flexural modes. Some temperature-compensated cuts of GaPO4 exhibit inversion points at high temperatures.

Isochronism defect for various doubly rotated cut quartz resonators.

It has been shown in earlier works that the amplitude-frequency effect [also called isochronism defect (ID) or anisochronism] could be a limitation factor on ultrastable oscillators. Theoretical studies based on the nonlinear theory of piezoelectricity have already been developed to explain the amplitude-frequency effect. So, it is possible to estimate the dependence of the ID versus various parameters of the resonator design (overtone rank, radius of curvature, electrodes diameter, etc.). However, because of the lack of available fourth-order elastic coefficients, it is not possible to predict the ID of any resonant frequency of a given trapped energy resonator. To tentatively find orientations of plates exhibiting a quasi-null ID, we have realized electroded resonators with different orientations and curvatures. We present results that verify, particularly, the R(-1/2) dependence of the amplitude-frequency effect versus radius of curvature. Moreover, we show that the ID can be positive or negative, that it can vary from one orientation to other one of about one order of magnitude, and that there exists a thermal compensated mode for which the amplitude-frequency effect is null.