Bernard Dulmet

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. I. Theory

After some necessary recalls on the nonlinear theory of thermoelectroelasticity in piezoelectric crystals, asserting the need of constitutive equations which derive from a rotationally invariant energy function, this paper presents the governing equations for a small vibration superimposed on a bias originated by a slow and homogeneous temperature variation from a well-defined reference state. Thereafter, the authors define the effective coefficients appearing in the linearized incremental dynamic balance equations for linear momentum and electrical charge in Lagrange configuration, not omitting associated boundary conditions. The main features of these coefficients are discussed and explicit relations with more conventionally defined coefficients are given. Determination of numerical values of the proposed effective coefficients and examples of their use in the higher order modeling of static frequency-temperature characteristics of either bulk acoustic wave or surface acoustic wave devices are given in a companion paper.

Temperature-compensated cuts for length-extensional and flexural vibrating modes in GaPO/sub 4/ 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 GaPO/sub 4/. In this paper, the possibility of temperature-compensated cuts in GaPO/sub 4/ is investigated for length-extensional and flexural vibrating modes. A theoretical investigation of rectangular cross-section GaPO/sub 4/ 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 GaPO/sub 4/ for length-extensional and flexural modes. Some temperature-compensated cuts of GaPO/sub 4/ exhibit inversion points at high temperatures.

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. II. Applications and numerical values for quartz

This paper presents a set of numerical values of temperature derivatives of Lagrangian effective elastic coefficients suitable for the modeling of small vibrations in quartz devices submitted to slow and homogeneous temperature variations. After a short description of the proper writing of vibration problems in practical applications with the help of this kind of coefficient, we determine the proposed set of numerical values from the frequency-temperature characteristics of an heterogeneous set of bulk and surface acoustic wave devices.

New electrostatically-excited single crystal silicon resonator vibrating in a Thickness-Extensional mode

This paper presents a new patent-pending structure of BAW resonator built on standard silicon wafer and driven by electrostatic force. Thickness-Extensional (TE) modes are exploited, yielding a fundamental frequency near 10 MHz with a 400µm-thick Si wafer. The device is based on a one-port design featuring a 1µm gap submitted to a superimposition of a DC voltage bias and a lesser-amplitude AC excitation. In contrast with conventional bar and plate MEMS resonators relying on structural resonances, our resonators are designed to use an energy-trapping of the TE waves to optimize the Q factor. The structure essentially consists of a single (100) Silicon plate, p-doped and bonded onto a Corning glass substrate by standard anodic bonding. The thin gap required for the electrostatic excitation is machined in the glass layer by Reactive Ion Etching. Extensive electrical and mechanical characterisations were performed. They match rather well the behavior predicted by the theoretical analysis. A compensation of the large static capacitance is needed to give a better access to the motional parameters. Q factors near 9000 have been observed on the fundamental TE-mode, with a sufficient coupling for frequency source applications. The stabilization of a RF oscillator using the resonators is still under development.

Short- and long-term stability of resonant quartz temperature sensors

A new miniaturized design of the thermosensitive quartz resonator (TSQR) using an NLC cut (yxl/-31deg30psila) with a fundamental frequency of 29.3 MHz was created in the Acoustoelectronics Laboratory of ISSP-BAS for use in a wide temperature range (4.2 K to 450 K) as highly sensitive quartz temperature sensors (QTS). This paper presents the results of the investigations of the short- and long-term frequency stability of QTS. The short-term frequency stability of QTS was measured for averaging times up to 150 s at three constant temperatures: liquid helium (4.2 K), liquid nitrogen (77 K), and melting ice (0degC). The short-term frequency stability is 6.8 times 10-9 at 0degC for t = 15 s, which permits a temperature sensitivity of 2 times 10-4 K. The long-term stability (aging) was investigated at room temperature and at 80deg C for 500 days. The aging characteristics at 25 degC and 80 degC are compared. It was observed that the frequency change does not exceed 5 times 10-7 after the 25th day of accelerated aging at 80 degC. This guarantees a reliable operation of the sensor, without additional calibration, for several years.

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 for vibrating beam resonators of gallium orthophosphate GaPO/sub 4/

A theoretical investigation of rectangular cross-section GaPO/sub 4/ vibrating beam resonators is proposed. Flexural modes are the basic vibrating mode of tuning forks used in quartz wrist watches, and can also be used as sensors. Very little work, if any, has been done for vibrating beam resonators in GaPO/sub 4/. The goal is then to investigate the possibility of temperature-compensated cuts for all three kinds of vibrations in GaPO/sub 4/: extensional, flexural, and torsional modes by analytical methods. Modeling temperature effects is achieved by the approximate but classical method of varying effective elastic constants, beam dimensions and crystal mass density versus temperature. Temperature-compensated cuts are found in GaPO/sub 4/ for length extensional modes and flexural modes. For vibrating beams, some of temperature-compensated cuts of GaPO/sub 4/ exhibit inversion points at high temperatures.

SC-cut resonator with reduction of B-mode electrical response

Excitation, by the use of two pairs of cross-connected half electrodes, of SC-cut resonator operating on first anharmonic modes is considered. In order to obtain resonators having the best electrical response of the C-mode, it is shown that two cases can be encountered, depending on the shapes and orientation of modes, i.e. on overtone number. The computation of electrical parameters is based on Lewis formula and use of analytical expressions for the vibration and the electric field.

New capacitive micro-acoustic resonators machined in single-crystal silicon stacked structures

We present a new type of acoustic resonator technology aimed to undoing the technological locks encountered during the realization of capacitive silicon MEMS resonators exploiting true Bulk Acoustic Wave resonances instead of structural ones. The single-crystal silicon resonators are driven through a combination of a static bias and dynamic voltage applied across a 700 nm-thick electrostatic gap parallel to the surface of the resonator substrate. The electrostatic gap cavity is realized by a local over-thickening of the gold layers used in the gold-gold bonding process of a resonant plate from resistive silicon with an external electrode-support structure, also made from resistive silicon. Thus, the electrostatic actuation is actually applied through existing interconnections between the existing conductive layers. This technology aims to the realization of very thin gaps on surfaces in the range of a few square millimeters while avoiding the shallow machining in the silicon that would require an additional technological step. We detail this technology through the process steps and designs implemented during the so-called ORSEPEE R&T project from CNES. Finally, the first characterization results are provided.

Theoretical analysis of the second stop-band of Rayleigh wave propagation on periodically corrugated anisotropic substrates

This paper presents a modeling of the dispersion curves for SAW propagation on periodically corrugated substrates in the second stop-band, which involves a strong coupling with BAW propagating both ways, towards- and from the depth of substrate. The proposed modeling uses series of Bloch functions to obey periodic boundary conditions which are integrated over one period of surface profile. The case of finite slope for trapezoidal profile is considered and the effect of anisotropy is taken into account. All equations are written in Lagrangian formalism, which will facilitate a further simulation of static thermal behavior of homogeneous crystalline structures with etched gratings. The paper presents results for energy reflection and conversion coefficients of a SAW-BAW transducer with a finite number of gratings on Y+/spl theta/ cuts of quartz.