JMEMS Letters - IEEE Xplore

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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 22, NO. 2, APRIL 2013

JMEMS Letters Crystallographic Effects on Energy Dissipation in High-Q Silicon Bulk-Mode Resonators Cheng Tu and Joshua E.-Y. Lee Abstract—This letter reports measured results of quality factor (Q) for single-crystal silicon square-plate resonators in relation to their crystal orientation within the (100) plane. Results from both the Lamé and extensional modes oriented in the 110 and 100 axes are presented. Q appears to be dependent on orientation for the Lamé mode. In the extensional mode, there is no significant difference in Q between the two axes due to the dominance of anchor loss. These trends agree well with the brief theoretical analysis that we have described here to provide a quantitative basis for interpreting the trends between Q and orientation observed in our measurements. [2012-0342] Index Terms—Akhiezer loss, anchor loss, elastic anisotropy, microelectromechanical systems (MEMS) resonator, quality factor.

I. I NTRODUCTION The potential of microelectromechanical systems (MEMS) resonators is of interest as basic vibratory units for microsensors [1] and timing references [2]. MEMS resonators can be designed to exhibit a high quality factor (Q), of which values of over 106 have been demonstrated in silicon-based bulk-mode resonators [3]. Most implementations of bulk-mode resonators align the device to the 110 direction of the silicon crystal lattice for a (100) silicon wafer [4], [5]. This orientation has had greater appeal over others within the (100) plane as it offers the highest Young modulus and smallest Poisson ratio [6]. These, in turn, generally lead to high Q’s in bulk-mode resonators. On this note, it has been reported that resonators aligned to the 100 direction will be a better choice when repeatability of Q across various fabrication batch runs is a primary concern [7]. More recently, it has been reported that reducing temperature coefficient of frequency by degenerate doping is much easier for silicon resonators aligned in the 100 direction [8]. It has also been postulated that 100oriented resonators surpass those oriented along 110 in terms of Q above ∼750 MHz [9]. While theoretical predictions on crystal orientation effects on Q can be found in the literature, there have been little empirical results for further validation. In this letter, we compare the measured Q’s of square-plate resonators fabricated along the 110 and 100 directions. Both the Lamé and square-extensional (SE) modes, shown in Fig. 1, are considered here. On this note, it has been hypothesized in [10] that a Lamé-mode resonator aligned in the 100 direction would suffer a much lower Q than that aligned in the 110 direction due to higher acoustic losses. No empirical validation has been provided until now. In this letter, we report results of Q’s measured along different orientations for the aforementioned two modes. Consistency and repeatability in the trends are observed as we extend our sample pool to include larger resonators. Q for the Manuscript received November 16, 2012; revised January 2, 2013; accepted January 4, 2013. Date of publication January 31, 2013; date of current version March 29, 2013. This work was supported by a grant from the City University of Hong Kong (Project 9667062). Subject Editor J. A. Yeh. The authors are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JMEMS.2013.2239259

Fig. 1. Lamé- and SE-mode shapes simulated for resonators aligned against the 110 and 100 directions. “Min disp” and “Max disp” denote minimum and maximum displacements, respectively.

Lamé mode is found to halve when the resonator is aligned along 100 compared to when the resonator is aligned along 110[11] but appears as invariant between the two orientations for the SE mode. These observations are explained according to the theoretical analysis described in Section II. The measured results are presented in Section III. II. A NALYSIS ON O RIENTATION -D EPENDENT L OSS Under vacuum, for a bulk-mode resonator with a feature size of several hundreds of micrometers, its net Q (Qnet ) is given by [12] 1 1 1 1 1 1 = + = + + . Qnet Qint Qanc QAKE QTED Qanc

(1)

QAKE , Qanc , and QTED denote the respective Q’s set by Akhiezer effect (AKE) loss, thermoelastic dissipation (TED), and anchor loss. The first two loss mechanisms are both intrinsic to bulk-mode resonators and can be combined into a single parameter denoted by Qint . In this section, we derive theoretical estimates of Qint and Qanc so as to compute Qnet theoretically. This will provide a theoretical reference to compare against the measured Q’s of our fabricated devices described in Section III. As can be seen from Fig. 1, each square plate is suspended by T-shaped tethers on all four corners. For the Lamé mode, it can be seen from Fig. 1(a) and (b) that the nodes of this vibration mode are located at the corners of the square plate where the structure can be conveniently clamped; this characteristic applies to both orientations. This ultimately translates to significantly reduced anchor loss. As such, it is expected that Qnet for the Lamé mode will be governed largely by Qint . In contrast, the displacement is at maximum at the corners for the SE mode in either orientation, as shown in Fig. 1(c) and (d). This leads to considerable anchor loss, and thus, Qanc will have to be considered when computing Qnet for the SE mode. For acoustic waves propagating in single-crystal silicon, Qint can be determined using the relation reported in [13]

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Qint = nQa = n

Ce . ωηe

(2)


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TABLE I C OMPARISON B ETWEEN C HARACTERIZED S QUARE -P LATE R ESONATORS O RIENTED A LONG THE 110 AND 100 D IRECTIONS W ITHIN THE (100) P LANE.

Qa denotes the acoustic quality factor per wavelength, while n is a correction factor. In our case, the values of n are one for the Lamé mode and two for the SE mode. This is because the acoustic wave propagates over one wavelength in the Lamé mode (for a shear wave across the diagonal of the square plate). In the SE mode, the acoustic wave propagates over half wavelength (defined by the length of the square plate). Ce and ηe represent the effective stiffness constant and the effective viscous damping parameter in the wave-propagating direction, respectively. For longitudinal waves, Ce equals C11 from the stiffness matrix [6]. For shear waves, Ce equals C44 . Lastly, ω is the angular frequency, and ηe is given by

ηe =

αCe 4.343ω 2

Ce ρ

(3)

where α denotes the acoustic attenuation in decibels per meter and ρ is the material density. Given the values of Ce , α, and ω in the wavepropagating direction, one can then obtain the corresponding value of Qint from (2) and (3).

A. Analysis on Q for Lamé Mode The computed values of Qint for the Lamé mode of resonators in either orientation are shown in Table I. The values of α in Table I are quoted from measured results reported previously in [14] and [15] and normalized to the resonant frequencies of our measured devices. We computed Qnet for 360 × 360 μm2 square plates. We can see from Table I that Qint drops by 79% when the resonator is aligned to the 100 direction compared to when the resonator is aligned to the 110 direction. For reference, we also computed Q for 800 × 800 μm2 square plates. For comparison, the values of Qanc for the Lamé mode have also been derived using the method of perfectly matched layers (PMLs) that we have reported previously in [16]. It can be seen from Table I that the values of Qanc are orders of magnitude larger than those of Qint for both orientations and either size. This further indicates that anchor loss is comparatively negligible for the Lamé mode. Moreover, it has been postulated that, since the Lamé mode is uniformly isochoric, the driving mechanism for TED is absent [17]. Hence, QAKE can be approximated to Qint and, thus, Qnet .

B. Analysis on Q for SE Mode As previously mentioned, anchor loss for the SE mode will be substantial, and thus, Qanc needs to be considered when deriving Qnet for the SE mode. Details of implementing the PML have already been described elsewhere [16] and thus omitted here for concision. Table I summarizes Qanc computed for resonators in the two orientations. For 360 × 360 μm2 resonators, Qanc reduces by about 10% for devices in the 100 direction in comparison to those in 110. By using an alternative approach to PML for analyzing anchor loss, which considers the relative distribution of strain energies between the tethers and square plate [18], we obtain a similar difference in Q between devices in the two orientations. For reference, we also computed Q for 800 × 800 μm2 devices in the 100 direction. Qint for the SE mode is likewise obtained from (2) and (3), and summarized in Table I. It can be seen that Qint is about two times larger than Qanc for all the resonators. This suggests that anchor loss is dominant over AKE loss and TED in the case of the SE mode. The values of Qnet for the SE mode are also given in Table I. These values all show a significantly smaller reduction in Q (up to 13%) due to changes in orientation when compared to those for the Lamé mode (about 81%).

III. M EASUREMENT AND D ISCUSSION The devices were fabricated in a foundry silicon-on-insulator (SOI) MEMS process. The measurement results from a total of ten die samples are reported here. Six of the ten die samples had square-plate resonators that were 360 × 360 μm2 ; each die sample contained a pair of square-plate resonators, one oriented to 110 and the other along 100. All ten die samples were designed with the same electrostatic transduction gap (2 μm) and thickness (25 μm) with the only exception of one 360 × 360 μm2 device which has a thickness of 10 μm. Optical micrographs of the 360 × 360 μm2 SOI devices are shown in Fig. 2. All devices were electrically characterized in vacuum with a dc bias voltage of 50 V. Figs. 3 and 4 show the extracted (feedthrough contribution removed using the method in [19]) electrical transmission for the Lamé mode and the SE mode for one of the die samples with 360 × 360 μm2 devices oriented along the 110 and 100 directions. The measured frequencies agree well with those obtained by finiteelement analysis using COMSOL according to Table I. Furthermore, good agreement is generally achieved between the measured Q and

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limited by acoustic attenuation (e.g., Lamé mode), Q is dependent on the crystal orientation due to silicon’s anisotropy. IV. C ONCLUSION

Fig. 2. Optical micrographs of the square-plate resonators (with side length of 360 μm) aligned in the 110 and 100 directions within the (100) plane.

This letter has reported measurements of Q for Lamé- and SEmode resonators in relation to their crystal orientation. Q for the Lamé mode appears to halve between the 110 and 100 axes, while Q for the SE mode shows an invariance with orientation. These empirical observations agree well with our theoretical analyses, which points to attenuation anisotropy with orientation. R EFERENCES

Fig. 3. Extracted transmission magnitude of the Lamé mode for a 360-μmlong square-plate resonator aligned along the 110 and 100 directions. (Dotted line) Lorentzian fit to the measured electrical transmission.

Fig. 4. Extracted transmission magnitude of the SE mode for a 360-μm-long square-plate resonator aligned along the 110 and 100 directions. (Dotted line) Lorentzian fit to the measured electrical transmission.

theoretical Qnet for all devices except for the Lamé mode in the 100 axis. The measured f · Q product drops by ∼52% going from 110 to 100, while theory predicts an 84% drop. Interestingly, the theoretical f · Q product for 100 is notably lower than the measured value, while the measured f · Q product for 110 is close to the theoretical value. The remaining four die samples contain 800 × 800 μm2 devices in 100. On this note, the measured f · Q product in the 100 is consistent between all the die samples, indicating that the limit on Q is imposed by intrinsic losses. In short, for vibration modes that are

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