A substrate energy dissipation mechanism in in-phase and anti-phase micromachined z-axis vibratory gyroscopes

Transcription

1 IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING J. Micromech. Microeng. 18 (2008) (10pp) doi: / /18/9/ A substrate energy dissipation mechanism in in-phase and anti-phase micromachined z-axis vibratory gyroscopes Alexander A Trusov, Adam R Schofield and Andrei M Shkel MicroSystems Laboratory EG2110, Mechanical and Aerospace Engineering Department, University of California, Irvine, CA 92697, USA atrusov@uci.edu, aschofie@uci.edu and ashkel@uci.edu Received 1 May 2008, in final form 18 June 2008 Published 5 August 2008 Online at stacks.iop.org/jmm/18/ Abstract This paper analyzes energy dissipation mechanisms in vacuum-operated in-phase and anti-phase actuated micromachined z-axis vibratory gyroscopes. The type of actuation is experimentally identified as the key factor to energy dissipation. For in-phase devices, dissipation through the die substrate is the dominant energy loss mechanism. This damping mechanism depends strongly on the die attachment method; rigid die attachment minimizes the loss of energy at the cost of reduced vibrational and stress isolation. For anti-phase actuated devices, dissipation through the substrate is suppressed and immunity to external vibrations is provided. However, even in anti-phase actuated devices fabrication imperfections introduce structural non-symmetry, enabling dissipation of energy through the die substrate due to momentum imbalance. Based on the experimental investigation, an analytical model for energy dissipation through the die substrate is proposed and used to study the effects of the actuation type, die attachment and fabrication imperfections. The limiting Q-factor for in-phase devices is generally below while Q-factors much higher than can be achieved with balanced anti-phase actuated gyroscopes. (Some figures in this article are in colour only in the electronic version) 1. Introduction The quality factor, or Q-factor, defines the sensitivity of many vibratory sensors and even the feasibility of their applications [1]. In vibratory MEMS gyroscopes, a higher Q-factor improves the rate precision and bias stability and lowers power consumption [2]. For gyroscopes operated or hermitically sealed at atmospheric pressure, damping caused by the surrounding gas dominates other energy loss mechanisms [3]. Extensive literature on gas damping in different types of MEMS is available, e.g. [4 9]. Dissipation of energy in micromachined vibratory gyroscopes operating in a vacuum is defined by a combination of several mechanisms, such as thermoelastic damping, surface loss, dissipation through support [2] and other mechanisms, such as electronic damping [10]. The properties of thermoelastic damping [11] in various types of vibratory MEMS devices were studied in the literature. Thermoelastic damping for small amplitude flexural vibrations in thin beams was analytically modeled in [12] using equations of linear thermoelasticity. Thermoelastic damping and its sensitivity to silicon etch-stop composition were experimentally characterized in [10]. Thermoelastic dissipation in thin out-of-plane torsional resonators was considered in [13]. The effect of annealing on internal friction in torsional out-of-plane resonators was experimentally studied in [1]. The effects of size and boundary conditions on thermoelastic damping in beam resonators were discussed in [14]. A fully coupled thermo-mechanical approach and a decoupled approximation were used to study thermoelastic /08/ $ IOP Publishing Ltd Printed in the UK

2 Anchors Sense electrode 2D coupling suspension Sense direction, y Drive shuttle Detection mass Proof mass Sense shuttle Drive electrode 0.75 mm Drive direction, x Figure 1. SEM images of testbed MEMS gyroscopes with 1-DOF and 2-DOF drive-modes. Gyroscope [23] with a 1-DOF drive-mode. Gyroscope [24, 25] with a 2-DOF drive-mode. damping in structures of arbitrary geometry in [15], and an application of a thermoelastic Q-factor to temperature sensing was proposed in [16]. A single degree of freedom (DOF) model was successfully used to estimate thermoelastic loss in a vibrating body and it was hypothesized that such simple models may be applicable to other forms of dissipation in [17]. While thermoelastic dissipation often becomes the Q- factor limiting mechanism in micromechanical resonators due to their decreasing size and increasing frequency (MHz range) [18], it does not necessarily dominate damping in micromachined gyroscopes which usually have a bulky proof mass and operate in the khz frequency range, thus separating the vibrational frequency from the thermal relaxation rate and reducing thermoelastic dissipation [19]. The limiting nonviscous Q-factor, Q lim, of most micromachined gyroscopes in a vacuum is in the range of Finite element modeling of an SOI gyroscope shows that the Q-factor due to thermoelastic damping exceeds 10 6 for both in-phase and anti-phase modes, which is several orders of magnitude higher than experimentally measured. The other important non-viscous energy loss mechanism, dissipation of energy through the substrate, is not sufficiently studied, especially in the context of MEMS gyroscopes [2]. Limited literature is available on this dissipation mechanism and mostly focuses on propagation of vibrational energy of micromachined beams into a much larger substrate. For instance, a two-dimensional theoretical model for support loss in thin in-plane beam resonators was derived in [20] based on elastic wave theory. A semi-analytical computational model for dissipation through support in clamped clamped out-ofplane beam resonators on a multi-layer isotropic substrate was proposed in [21, 22]. Different energy loss mechanisms including dissipation through support were studied in [19] using in-plane and out-of-plane beam resonators, and a decoupling frame design was proposed to maximize the Q- factor by isolating vibrations in the beams from the bulk of the substrate. In this paper, we analyze dissipation of energy through the die substrate in in-phase and anti-phase actuated z- axis micromachined vibratory gyroscopes and the effects of die attachment method on the observed Q-factors. In section 2, we describe the experimental setup and procedures for characterization of Q-factors in vibratory gyroscopes. In section 3, we report the measurements of drive-mode Q-factors in gyroscopes with in-phase and anti-phase actuation packaged using different die attachment methods. In section 4, we propose a lumped element model that augments the dynamics of a gyroscope with an additional mass-spring-damper system representing mobility of the die substrate and provides insight into experimentally observed effects of the actuation type, die attachment and fabrication imperfections on the effective Q-factor. The paper is concluded with a summary of the obtained results and a discussion on the high-q sensor design and packaging tradeoffs in section Experimental study: methods In this section, we describe the experimental setup and measurement procedure for characterization of Q-factors at different pressure levels and identification of the limiting nonviscous Q lim values using drive-modes of two structurally different micromachined z-axis gyroscopes. These methods are then used in section 3 to experimentally characterize the effects of the actuation type and die attachment on energy dissipation Testbed gyroscopes Two different SOI micromachined gyroscopes were used for the experimental study of Q-factors: a gyroscope [23] with a single DOF drive-mode shown in figure 1 and a gyroscope [24, 25] with a 2-DOF drive-mode shown in figure 1 respectively. The first gyroscope [23] has only a single degree of freedom drive-mode corresponding to a single resonance of the proof mass. In contrast, the 2

3 Gyro DIP-24 Bond wires Solder preform Figure 2. A packaged gyroscope used for Q-factor measurements. Table 1. Parameters of the testbed devices (in air). Parameters (unit) Value 1-DOF drive-mode of a gyroscope [23] Mass (kg) Stiffness (N m 1 ) 125 Damping (N (m s 1 ) 1 ) DOF drive-mode of a gyroscope [24, 25] Tine mass (kg) In-phase stiffness (N m 1 ) 75 Anti-phase stiffness (N m 1 ) 285 Damping (N (m s 1 ) 1 ) second gyroscope [24, 25] allows for two distinct drive-mode resonances: the lower frequency in-phase mode and the higher frequency anti-phase mode. In the in-phase mode of the gyroscope [24, 25], the two proof masses translate in unison, making it functionally identical to operation of the gyroscope with a 1-DOF drive-mode. Because of this similarity between the in-phase mode of a device with a 2-DOF drive-mode and the single resonant mode of a device with a 1-DOF drivemode, we will refer to both cases as in-phase or in-phase actuated. The anti-phase resonance of a 2-DOF drive-mode will be referred to as anti-phase or anti-phase actuated. The main properties of the two testbed gyroscopes are given in table 1. The test gyroscopes were fabricated using an in-house micromachining process based on p-type SOI wafers with a 50 µm thick device layer, a 5 µm buried oxide layer and a 500 µm thick silicon substrate. After photoresist spin coating, exposure and development, the wafers were subjected to a deep reactive ion etching (DRIE) using a Surface Technology Systems (STS) tool. The fabricated wafers were then cleaned and diced, and individual devices were released in an HF acid bath. The fabricated devices were packaged using ceramic DIP-24 packages and wirebonded for experimental characterization, figure 2. In order to investigate the effect of the die attachment on the measured Q-factor, the gyroscopes were attached to the packages using three different methods: SPI conductive double-sided carbon adhesive tape, Circuit Works two-part conductive epoxy and SPM Au-Sn 80/20 solder preform Experimental setup In order to experimentally measure the dependence of the Q-factor on pressure and identify the limiting non-viscous value Q lim, the packaged gyroscopes were characterized in a vacuum chamber, figure 3. Structural characterization of the gyroscopes was accomplished by using a parasitics-free frequency response acquisition method [26, 27]. The device under test was driven into linear vibrations by a combination of a fixed dc bias voltage V dc and an ac driving voltage V d (t) of variable frequency ω d, figure 3. The motion of device modulated by the carrier ac voltage V c (t) was picked up by the transimpedance amplifier. A lock-in amplifier was used to demodulate the amplified voltage signal. Finally, a dynamic signal analyzer operating in a swept-sine mode was used to measure and record the frequency response of the device under test. The frequency response measurement procedure was repeated for each tested gyroscope at different pressure levels between atmospheric and 10 mtorr Structural characterization The measured frequency response curves, as shown in figure 4, were post-processed in MATLAB to automatically extract the values of the undamped natural frequency, ω n, and the quality factor, Q. We denote a resonator s mass by m, k stiffness by k and damping by c. Then, ω n = m, Q = km, c and the motion of an ideal linear time invariant (LTI) resonator is governed by ẍ(t) + ω n Q ẋ(t) + ω2 nx(t) = A(t), (1) where A(t) is acceleration. The transfer function of this system is iω d TF(iω d ) = ωn 2 ω2 d +iω n Q ω, (2) d where the applied acceleration is the input to the system and velocity is the output. The amplitude of the frequency response is TF(iω d ) = ω d (ω 2 n ) 2 ( ω2 d + ωn Q ω ). (3) 2 d A fitting function in the form of equation (3) was used to automatically extract the undamped natural frequency and the Q-factor of each experimentally measured frequency response, as illustrated in figure Measurement and analysis of Q-factors at different pressures For each tested packaged device, frequency response data were acquired and processed to extract values of Q-factors versus pressure. As figure 4 illustrates, curves for Q- factor as a function of pressure P typically show three different regimes: viscous damping at low Knudsen numbers K n 1, (P 1 )-proportional molecular flow damping at K n 1[7] and a non-viscous asymptote visible at K n 1. This qualitative behavior is similar to experimental observations reported on other micromachined vibratory devices, such as single-crystal silicon resonant beam accelerometers [19], polysilicon resonant microbeams [28], high-frequency bulk mode resonators [29] and vibratory gyroscopes [10]. 3

4 Pick-up voltage Input voltages Drive V +V d (t) dc Sens e - + TZA TZA circuit Packaged gyroscope pump Leak control valve V dc ~ V (t) c d ~ Response 90 Frequency Lock-In Amp. Figure 3. Experimental setup for characterization of Q-factors at different pressures. Vacuum chamber. Electromechanical schematic of structural characterization. LPF x P=18 mtorr Q=9.3e3 data fit 10 4 Q 10.6e3 18 mtorr Q=9.3e3 K n <<1 data fit Gain, db P=atmospheric Q=68 P and Q Q K n >>1 atm. pressure Q= Frequency, Hz Pressure, Torr Figure 4. Identification of the Q-factor as a function of pressure for the in-phase mode of the gyroscope with a 2-DOF drive-mode attached to the package using epoxy. Frequency responses and identification of structural parameters (some curves are omitted for clarity). Measured Q(P ) and identification oon-viscous asymptote Q lim (K n stands for the Knudsen number). Since the focus of this work is analysis of the Q- factor limiting non-viscous energy dissipation mechanisms, an empirical fitting expression was proposed for Q(P ) in the form of Q(P ) = ( 1 Q lim + BP ) 1, (4) 1+CP where Q lim, B and C are the fitting parameters. Here, the viscous Q-factor is approximated by (1/B) P +( ) C B in a wide range of Knudsen numbers; however, more complex, first principlebased models for gas damping are available in the literature, e.g. [4, 5, 8, 6]. As figure 4 shows, equation (4) correctly captures the three different damping regimes and allows us to identify Q lim. Since the described method uses a least squares fit of multiple Q(P ) data points, the resulting measurement of Q lim has an improved precision compared to the often-used single point estimation of Q lim by measuring the Q-factor at a fixed reduced pressure [10, 19]. 3. Experimental study: results In this section we report the experimental analysis of Q- factor measurements in the in-phase actuated gyroscope [23], figure 1, and the gyroscope [24, 25], figure 1, which allows for both in-phase and anti-phase drive-mode actuation. Based on the measurements, the effects of the actuation mode and die attachment on the limiting Q-factor are studied Effects of the actuation type and die attachment In order to study the effect of the actuation type and die attachment, Q-factors of the drive-modes of the gyroscopes 4

5 10 5 Q 200e3 eutectic, Q 18e fits Q 10 3 Q 10.6e3 Q 3.6e3 atm. pressure Q =131 Q =68 Q 10 3 adhesive Q 4.3e3 epoxy, Q 9.6e3 measured Q fit Pressure, Torr Pressure, Torr 10 2 Figure 5. Experimental study of the Q-factor in in-phase and anti-phase actuated gyroscopes packaged using different die attachment methods. Gyroscope [24, 25] with a 2-DOF drive-mode allowing for both in-phase and anti-phase actuation. In-phase actuated gyroscope [23] with a single-dof drive-mode. were characterized, figure 5. Q-factors of the in-phase and anti-phase resonant modes of the gyroscope [24, 25]are shown in figure 5. The in-phase mode Q lim strongly depends on the die attachment method, with Q lim and for adhesive and epoxy die attachment, respectively. At the same time, Q lim of the anti-phase mode is not significantly affected by the die attachment method. In the tested range of vacuum, the damping is dominated by the molecular gas flow damping mechanism and shows Q lim , more than an order of magnitude higher than for the in-phase mode. The effect of die attachment on the limiting Q-factor, Q lim, was further investigated using a batch of identical inphase actuated gyroscopes [23] with a single-dof drivemode, figure 5. These devices were packaged using three different die attachment methods: (1) carbon adhesive, (2) epoxy, (3) eutectic solder preform. The measurements reveal an increasing Q lim of approximately , and , respectively. The experimental results suggest that the limiting energy dissipation mechanism in in-phase actuated devices is dissipation of energy through the die substrate into the package via the die attachment interface with the more rigid and less viscous attachment material resulting in higher Q lim Fabrication imperfections in anti-phase actuated gyroscopes Measurements of the Q-factors in figure 5 suggest that for well-balanced devices, the anti-phase mode Q-factor is not affected by the loss of energy through the die substrate due to effective cancelation of stresses applied to the substrate by the vibrating structure. However, fabrication imperfections are unavoidable in MEMS gyroscopes and can cause momentum imbalance between the two tines vibrating in anti-phase to each other. Figure 6 shows measurements of the anti-phase mode Q lim in two batches of gyroscopes [24, 25], designed in two different die sizes and fabricated using the same process. Figure 6 shows measurement obtained using gyroscopes designed for 7 7mm 2 ; the devices have identical layout except for the suspension elements resulting in three different operational frequencies. Figure 6 shows measurements for a batch of identically designed gyroscopes [24, 25] implemented in a mm 2 die size. Analysis of the Q-factors in figure 6 indicates that imbalance induced by fabrication imperfections can enable substrate dissipation even in anti-phase actuated devices. For the tested anti-phase actuated gyroscopes, the standard deviation of frequency-normalized limiting Q-factors is on the order of 40% if adhesive is used for die attachment. Based on the experimental analysis of the in-phase actuated devices in the previous subsection, the scattering of measured anti-phase mode Q-factors can be improved to approximately 10% using rigid die attachment. 4. Modeling Two modeling aspects of energy dissipation mechanisms are discussed in this section. First, we report a numerical simulation of thermoelastic dissipation to verify the experimental conclusions on the nature of the dominant Q-factor limiting loss mechanism. Second, we propose a lumped element model for dissipation of energy through the substrate by augmenting the dynamics of a MEMS vibratory gyroscope with an extra degree of freedom representing dynamics of the die substrate attached to the fixed package. This modeling approach agrees with experimental results and allows to analyze the effects of die attachment and fabrication imperfections on dissipation through the substrate Finite element modeling of thermoelastic dissipation The limiting non-viscous Q-factor, Q lim, of most micromachined gyroscopes in a vacuum is in the range of 5

6 10 5 =2.7 khz, Q 230e3 =1.5 khz, Q 64e3 measured Q fit 10 5 =7.1 khz, Q 185e3 =6.9 khz, Q 89e3 Q 10 4 Q 10 4 =6.8 khz Q 68e3 measured Q fit =1.8 khz, Q 30e Pressure, Torr Pressure, Torr Figure 6. Experimental study of the effect of fabrication imperfections on the anti-phase mode Q-factor in gyroscopes [24, 25] packaged using adhesive. Gyroscopes designed in a 7 7mm 2 die size. Gyroscopes designed in a mm 2 die size. Figure 7. COMSOL FEM of thermoelastic damping in a gyroscope [24, 25], which allows for both in-phase and anti-phase actuation, predicts Q-factors higher than 10 6 (colors represent x-displacement). In-phase mode, Q TED = , the resonant frequency is 1.46 khz. Anti-phase mode, Q TED = , the resonant frequency is 2.18 khz , which is often attributed to thermoelastic dissipation (TED) [10]. In this study, TED in the gyroscope [24, 25] was modeled using the finite element software package COMSOL Multiphysics. A two-dimensional model was realized starting from the device layout which was then meshed with triangular elements resulting in 671,220 degrees of freedom; smaller quadratic elements were used to form the suspension elements while larger linear elements were used for the mobile masses. A multiphysics problem was set up with two-way coupling between the structural and thermal domains. A damped eigenfrequency analysis with thermal expansion was solved in the structural domain which was coupled to the conductive heat transfer problem by a strain-based heat generation term. Complex eigenvalues λ were obtained using a direct linear system solver and the Q-factors due to thermoelastic damping, Q TED, were calculated using Q TED = Im(λ) 2Re(λ) [12]. Finite element modeling (FEM) of the gyroscope s inphase and anti-phase modes is shown in figures 7 and, respectively. In the in-phase mode, the two proof masses translate in unison and only their individual suspension elements are strained, while the coupling suspension has zero strain. As a result, the operation frequency of the in-phase mode is 1.46 khz and Q TED = In the antiphase mode, however, the proof masses translate in opposite directions causing the coupling spring to deflect twice as much compared to the individual suspension springs, which 6

7 coupling with substrate dynamics reduces Q to 115e3, 43e3, 6e3 and 0.6e3, respectively c 2 k 2 c 1 k 1 mobile mass m 1 force F 2 Gain, db (force to displacement) sub =104 sub =103 sub =102 sub =101 Q=200e Frequency, khz Figure 8. Modeling of energy loss through the substrate by coupling a high-q gyroscope with a mobile die substrate with a damped die attachment interface. Lumped model augmented with the die substrate DOF. Nominal and coupled frequency responses for different die attachment damping values. increases the operational frequency to 2.18 khz and decreases the Q-factor to Q TED = due to the increase of the thermoelastic loss in the coupling springs. For both the in-phase and the anti-phase modes, the Q- factor due to thermoelastic damping, Q TED, exceeds 10 6, which is orders of magnitude higher than the experimentally observed Q lim. These modeling results support the experimental conclusions on importance of energy dissipation through the substrate as the Q-factor limiting mechanism Dissipation through the substrate in in-phase actuated devices It is apparent from the experiments discussed in section 3 that dissipation of energy through the vibrating die substrate via the die attachment interface can be the dominant energy dissipation mechanism. Thus, the dynamics of the die substrate needs to be considered together with the dynamics of a MEMS gyroscope s drive-mode. Figure 8 showsa lumped model of a single-dof vibratory element m 1 on a mobile die substrate m 2. The ratio of the device to the die substrate masses depends mostly on the relative thicknesses of the layers and the relative area of the device; for the in-phase actuated SOI devices discussed in this paper, m 2 20 m 1. The interaction between the vibratory element m 1 and the die substrate m 2 is described by stiffness k 1 and damping c 1 ;the die attachment interface between the die substrate m 2 and the fixed package is represented by k 2, c 2. According to this approach, an ideal single mass MEMS resonator is in fact a coupled dynamic system with 2-DOF, where the observed characteristics of the main resonant mode are affected by the coupling from the die substrate dynamics. Figure 8 illustrates the concept by comparing the nominal 1-DOF and the coupled frequency responses of a single-dof resonator with an intrinsic Q-factor of on a mobile die substrate. For this simulation, the following numerical parameters, similar to the values reported in table 1, were used: m 1 = kg, m 2 = kg, k 1 = 115 N m 1,k 2 = N m 1 and c 1 = N(ms 1 ) 1. The damping value c 2 associated with the energy loss at the die attachment interface was defined by c 2 = k 2 m 2 /Q sub, where the quality factor of the die attachment Q sub was iterated through , , and As a result of the die substrate coupling, the observed Q-factor of the MEMS device can drop drastically. For instance, for Q sub = 10 [19], the simulation shows that an effective Q-factor can drop from to below a thousand. Values of Q sub on the order of several hundreds result in effective Q-factors observed in the experiments. The observed shift of the gyroscope s effective frequency due to coupling with the die substrate dynamics is similar to the effect noted in [22] Dissipation through the substrate in anti-phase actuated devices Similar to the case of a drive-mode with a single degree of freedom discussed in the previous subsection, dissipation of energy through the substrate in devices with 2-DOF, such as the drive-mode of tuning fork gyroscopes, can be modeled by augmenting the lumped model with an extra DOF representing the mobile die substrate. Figure 9 shows a lumped model, where m 1 and m 2 are the two tines of a tuning fork-type gyroscope and m 3 is the mobile die substrate. Each tine is suspended relative to the die substrate with stiffnesses k 1,2 and damping c 1,2 respectively; the two tines are also coupled together with stiffness k 12 and damping c 12. The die attachment interface between the mobile die substrate m 3 and the fixed package is represented by stiffness k 3 and damping c 3. Typically, the two tines of a tuning fork gyroscope are designed to be structurally balanced, i.e. m 1 = m 2,k 1 = k 2, and c 1 = c 2. For the SOI anti-phase actuated gyroscopes described in this paper, m m 1,2. Figure 9 shows simulation of the substrate dissipation using the lumped 3- DOF model. In the simulation, the frequency response of a balanced tuning fork gyroscope with an anti-phase mode Q-factor of is compared to the response of the 3- DOF system that includes the damped die substrate dynamics. The following numerical parameters, based on the properties 7

8 c 3 k 3 k 2 left k 12 tine m 2 c 2 c 12 right k 1 tine m 1 c 1 Gain, db 40 db Q=200e3 not affected by substrate dynamics drops 100 times due to substrate dynamics Frequency, khz Figure 9. Modeling of energy loss through the substrate in 2-DOF devices by augmenting the system dynamics with a damped die substrate DOF. Lumped model including the two tines m 1,2 and the mobile die substrate m 3. Ideal high-q frequency response and effect of die substrate damping. of the gyroscopes with a 2-DOF drive-mode implemented in a 3.5 mm die size, were used for the simulation: m 1,2 = kg, m 3 = kg, c 1,2 = N(ms 1 ) 1,k 1,2 = 4.6 Nm 1,k 12 = 7Nm 1,k 3 = N m 1 and c 3 1 N (m s 1 ) 1. Similar to the experimental results in section 3, the modeled frequency responses illustrate that the effective Q-factor of the in-phase mode can drastically drop due to dynamic coupling with the damped die substrate, unlike the anti-phase mode Q-factor, which is robust to the die substrate damping Fabrication imperfections and dissipation through the substrate in anti-phase actuated devices The simulation results in the previous subsection show that in a balanced tuning fork gyroscope the anti-phase drive-mode Q-factor is not affected by coupling from the die substrate, which has zero displacement at the frequency of the antiphase resonant mode. However, unavoidable fabrication imperfections can cause structural imbalances between the two tines of a tuning fork gyroscope. The developed three-mass model, figure 9, allows us to study the individual effects of mass, stiffness, damping and forcing imbalance on the effective Q-factor on the anti-phase mode. To study the effect of imperfections, we compared simulated frequency responses of a two-mass imbalanced tuning fork with responses of the same tuning fork augmented by the third DOF representing the die substrate attached to a fixed package. Numerical parameters from the previous subsection were used as the nominal values for the simulation of imperfections. The modeling showed that non-symmetry in damping and forcing terms associated with the two tines does not cause any noticeable reduction of the anti-phase drive-mode Q-factor. In contrast, non-symmetry in tines masses and stiffnesses results in dynamic imbalance which introduces dissipation through the substrate into the anti-phase drivemode and lowers the effective Q-factor. Figure 10 shows the relative reduction of the anti-phase mode Q-factor, denoted as Q rel, due to a 10% mismatch between m 1 and m 2 and log 10 (Relative Reduction of Q) k or m 10 % mismatch, Q rel 0.13 Q sub k and m 10 % mismatch, Q rel 0.40 Q sub log 10 (Q sub ) Figure 10. Modeling of dissipation of energy through the substrate in a tuning fork gyroscope with mass and stiffness imbalances. between k 1 and k 2 for different values of the die attachment damping. The relative decrease of the Q-factor is roughly proportional to the die attachment damping, i.e. Q rel Q 1 sub. For a die attachment interface with fixed properties, mass mismatches and stiffness mismatches have equal effect on Q rel ; simultaneous mismatch in both mass and stiffness causes a three times greater reduction of the Q-factor compared to their individual effects. 5. Discussions and conclusions We presented a study of drive-mode Q-factor limitations in micromachined z-axis vibratory gyroscopes. The type of actuation was identified as the key factor to the dominant energy dissipation mechanism and the maximal achievable Q-factor. For in-phase actuated devices, energy dissipation through the die substrate is the dominant damping mechanism; rigid die attachment minimizes the substrate dissipation and allows for the maximal Q-factor on the order of Well- 8

9 balanced anti-phase operation suppresses dissipation through the substrate due to effective cancelation of stresses applied by the mobile structures to the substrate. However, fabrication imperfections can induce structural non-symmetry and thus enable dissipation through the substrate even in anti-phase actuated devices. Achieving drive-mode Q-factors of 10 5 and higher requires balanced anti-phase actuated gyroscopes that are packaged using rigid die attachment. Increasing a gyroscope s drive-mode Q-factor is beneficial because it reduces the necessary driving voltages, consequently decreasing contamination of electrical signals by parasitics and improving the sensor power consumption. However, the rate resolution and bias stability of a modematched gyroscope are defined by the Q-factor of the sensemode. In most micromachined vibratory gyroscopes, the structures of the drive-mode and the sense-mode resonators are very similar, e.g. [30], or exactly identical, e.g. [31, 32]. Therefore, the results of this paper obtained using drive-modes of the test gyroscopes are equally valid for the sense-mode structures and the attained conclusions allow for systematic design of high performance, low power vibratory MEMS gyroscopes. Based on our analysis, several major tradeoffs in design of high-performance MEMS gyroscopes are apparent. Balanced anti-phase-driven gyroscopes (tuning fork architectures) inherently provide mechanical rejection of common mode vibrations [25] and Q-factors as high as hundreds of thousands, limited by thermoelastic dissipation. In addition, the high Q-factor of anti-phase operational modes does not depend on the die attachment properties, allowing for compliant die attachment in order to minimize the mechanical stress coupling between the package and the sensing element [33]. On the down side, tuning fork-type devices usually require intricate structural design, e.g. [24, 34], and occupy a larger die area, leading to higher development and production costs. Gyroscopes based on a single proof mass, however, typically have simpler structural design and can be implemented in a smaller die area, which reduces the cost of fabrication. At the same time, they are prone to dissipation of energy through the substrate, which limits the maximal effective quality factor to as low as several thousands. Rigid die attachment, such as a thin layer of eutectic solder, can minimize the dissipation through the substrate and allows for Q-factors on the order of tens of thousands. In this case, however, maximization of the Q-factor is achieved at the cost of reduced isolation from the package vibrations and stresses [33], which can reduce robustness of such sensors below the practically required. Acknowledgments This work was supported by the National Science Foundation Grant CMS , Custom Sensors & Technologies Systron Donner Automotive (CS&T SDA, formerly BEI Technologies) contract BEI and UC Discovery program ELE The tested gyroscopes were fabricated at the UC Irvine Integrated Nanosystems Research Facility (INRF). Design of the gyroscopes and experimental characterization was performed at the UC Irvine MicroSystems Laboratory. The authors would like to acknowledge David Virissimo of Semiconductor Packaging Materials (SPM) for the help with eutectic attachment process development, and Cenk Acar and Lynn E Costlow of CS&T SDA for stimulating discussions. References [1] Liu X, Vignola J F, Simpson H J, Lemon B R, Houston B H and Photiadis D M 2005 A loss mechanism study of a very high Q silicon micromechanical oscillator J. Appl. Phys [2] Hao Z, Zaman M F, Sharma A and Ayazi F 2006 Energy loss mechanisms in a bulk-micromachined tuning fork gyroscope Proc. IEEE Sensors Conf. pp [3] Geen J A, Sherman S J, Chang J F and Lewis S R 2002 Single-chip surface micromachined integrated gyroscope with 50 deg/h Allan deviation J. Solid-State Circuits [4] Veijola T, Kuisma H, Lahdenpera J and Ryhanen T 1995 Equivalent-circuit model of the squeezed gas film in a silicon accelerometer Sensors Actuators A [5] Veijola T and Turowski M 2001 Compact damping models for laterally moving microstructures with gas-rarefaction effects J. Microelectromech. Syst [6] Hutcherson S and Ye W 2004 On the squeeze-film damping of micro-resonators in the free-molecule regime J. Micromech. Microeng [7] Sumali H 2007 Squeeze-film damping in the free molecular regime: model validation and measurement on a MEMS J. Micromech. Microeng [8] Veijola T 2004 Compact models for squeezed-film dampers with inertial and rarefied gas effects J. Micromech. Microeng [9] Bao M and Yang H 2007 Squeeze film air damping in MEMS Sensors Actuators A [10] Duwel A, Gorman J, Weinstein M, Borenstein J and Ward P 2003 Experimental study of thermoelastic damping in MEMS gyros Sensors Actuators A [11] Zener C 1937 Internal friction in solids: I. Theory of internal frictioninreedsphys. Rev [12] Lifshitz R and Roukes M L 2000 Thermoelastic damping in micro- and nanomechanical systems Phys. Rev. B [13] Houston B H, Photiadis D M, Vignola J F, Marcus M H, Liu X, Czaplewski D, Sekaric L, Butler J, Pehrsson P and Bucaro J A 2004 Loss due to transverse thermoelastic currents in microscale resonators Mater. Sci. Eng. A [14] Sun Y, Fang D and Soh A K 2006 Thermoelastic damping in micro-beam resonators Int. J. Solids Struct [15] Duwel A, Candler R N, Kenny T W and Varghese M 2006 Engineering MEMS resonators with low thermoelastic damping J. Microelectromech. Syst [16] Hopcroft M A et al 2007 Using the temperature dependence of resonator quality factor as a thermometer Appl. Phys. Lett [17] Rajagopalan J and Saif M T A 2007 Single degree of freedom model for thermoelastic damping Trans. ASME E [18] Candler R N, Li H, Lutz M, Park W-T, Partridge A, Yama G and Kenny T W 2003 Investigation of energy loss mechanisms in micromechanical resonators Proc. Int. Solid-State Sensors, Actuators and Microsystems Conf. TRANSDUCERS 2003 vol 1, pp [19] Le Foulgoc B, Bourouina T, Le Traon O, Bosseboeuf A, Marty F, Breluzeau C, Grandchamp J-P and Masson S 2006 Highly decoupled single-crystal silicon resonators: an approach for the intrinsic quality factor J. Micromech. Microeng

10 [20] Hao Z, Erbil A and Ayazi F 2003 An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations Sensors Actuators A [21] Park Y-H and Park K C 2004 High-fidelity modeling of MEMS resonators: I. Anchor loss mechanisms through substrate J. Microelectromech. Syst [22] Park Y-H and Park K C 2004 High-fidelity modeling of MEMS resonators: II. Coupled beam-substrate dynamics and validation J. Microelectromech. Syst [23] Trusov A A, Schofield A R and Shkel A M 2008 New architectural design of a temperature robust MEMS gyroscope with improved gain-bandwidth characteristics Proc. Solid-State Sensors, Actuators and Microsystems Workshop (Hilton Head Workshop 2008) pp 14 7 [24] Schofield A R, Trusov A A, Acar C and Shkel A M 2007 Anti-phase driven rate gyroscope with multi-degree of freedom sense mode Proc. Int. Solid-State Sensors, Actuators and Microsystems Conf. TRANSDUCERS 2007 pp [25] Schofield A R, Trusov A A and Shkel A M 2007 Multi-degree of freedom tuning fork gyroscope demonstrating shock rejection Proc. IEEE Sensors Conf. pp [26] Cao J and Nguyen C T-C 1999 Drive amplitude dependence of micromechanical resonator series motional resistance Proc. 10th Int. Conf. on Solid-State Sensors and Actuators pp [27] Trusov A A and Shkel A M 2007 Capacitive detection in resonant MEMS with arbitrary amplitude of motion J. Micromech. Microeng [28] Zook J D, Burns D W, Guckel H, Sniegowski J J, Engelstad R L and Feng Z 1992 Characteristics of polysilicon resonant microbeams Sensors Actuators A [29] Palaniapan M and Khine L 2007 Micromechanical resonator with ultra-high quality factor Electron. Lett [30] Sharma A, Zaman M F and Ayazi F 2007 A smart angular rate sensor system IEEE Sensors 2007 Conf. (Atlanta, GA, USA, Oct. 2007) [31] Alper S E and Akin T 2005 A single-crystal silicon symmetrical and decoupled MEMS gyroscope on an insulating substrate J. Microelectromech. Syst [32] Alper S E, Azgin K and Akin T 2007 A high-performance silicon-on-insulator MEMS gyroscope operating at atmospheric pressure Sensors Actuators A [33] Gomez U-M et al 2005 New surface micromachined angular rate sensor for vehicle stabilizing systems in automotive applications Proc. Int. Solid-State Sensors, Actuators and Microsystems Conf. TRANSDUCERS 2005 vol 1, pp [34] Neul R et al 2007 Micromachined angular rate sensors for automotive applications IEEE Sensors J