Quality Factor Se Sitivity to Crystallographic Axis Misalig Me T I Silico Micromecha Ical Reso Ators

We study the sensitivity of quality factor in single crystal silicon (SCS) micromechanical resonators to crystallographic axis misalignments that are present due to fabrication non-idealities. Our experimental results, being reported for the first time here, unveil that very small angular misalignment from [110] axis of transduction adversely affects the high Q of a bulk acoustic wave SCS resonator by more than 50%, unlike the misalignment errors about the [100] axis of transduction. Interestingly, when the axis of transduction is intentionally offset by a large angle from either [100] or [110], multiple peaks with comparable relative strength are observed from a single resonator. I TRODUCTIO Single crystal silicon (SCS) micromechanical resonators have shown high frequencies (>100 MHz) and high quality factors (Q>50,000), surpassing those of commercially available quartz resonators [1]. Insertion of such high frequency silicon resonators into low phase-noise frequency references requires stable and reproducible Q. However, the observed Q variation across various batches of simultaneously processed devices under very similar processing conditions suggests that there might be a possible influence from a non-systematic non-ideality in the fabrication process that might be adversely affecting the Q. While many studies have been conducted on determining the various energy loss mechanisms limiting the Q of a SCS resonator [2], the sensitivity of Q to crystallographic misalignment has not yet been systematically studied. Fig. 1: Schematic showing the arrangement of atoms in the silicon crystal lattice as seen along the (a) [100] and (b) [110] directions. Every atom in silicon has a linear chain of atoms aligned along the [100] or [110] direction which facilitates efficient transmission of the compressional and dilational forces of an acoustic wave (Figure 1). Hence, SCS microresonators are typically transduced along either of these axes. However, any angular misalignment from these two axes of transduction disrupts the atomic linearity which might lead to acoustic losses at the atomic level, the ensemble of which might reflect on the Q of the resonator. As will be revealed from this work, the amount of angular misalignment needed to adversely affect the Q of the resonator is imperceptible to the human eye and are beyond the tolerance limits of automated lithography systems as well. A rectangular silicon bulk acoustic resonator (or SiBAR) (Figure 2(a)) that involves a capacitive air-gap based transduction along a very specific crystallographic axis in SCS is the best candidate to study variations in Q with very small angular offsets from the intended axes of transduction. The SiBAR is placed between the drive and sense electrodes separated by a very high aspect-ratio air-gap realized using the HARPSS process [3]. A DC polarization voltage (Vp) applied to the resonator generates an electrostatic field in these capacitive gaps. When an AC voltage is applied to the drive electrode, the resulting time-varying electrostatic force applied to the corresponding face of the resonator induces an acoustic wave that propagates through the bar, resulting in a width-extensional resonance mode (Figure 2(b)) whose frequency is primarily defined by the width of the SiBAR. Small changes in the air gap on the other side of the device induce a voltage on the sense electrode whose amplitude peaks at the mechanical resonance frequency. Fig. 2: (a) SEM and (b) Simulated width-extensional mode (WEM) shape of the SiBAR. Experimental platform to study Q sensitivity to angular offset A batch of 100 MHz SiBARs (width = 40 μm, thickness = 20 μm and length = 10 × width) were fabricated about both the [110] and [100] axes of transduction with intentionally created negative and positive angular offsets in steps of 0.l degrees as illustrated in Figure 3. 0-9640024-8-5/HH2010/$25©2010TRF 479 Solid-State Sensors, Actuators, and Microsystems Workshop Hilton Head Island, South Carolina, June 6-10, 2010 Fig. 3: (a) Schematic and (b) SEM of the intentionally angular offset 100 MHz SiBARs for studying the Q variation with angular misalignment. The variation in Q with the doping level of the SCS substrate and the size of support tethers needs to be taken into account to attribute any variation in Q solely to the angular misalignment. Hence, these SiBARs were fabricated across various batches of wafers that spanned different levels of boron doping (0.01~0.02 Ω-cm [moderatelydoped], 0.001~0.002 Ω-cm [highlydoped] & <0.001 Ω-cm [ultra-highly-doped]) as purchased from the vendor for different widths of the support tethers (3 μm & 1.5 μm). Any pattern in Q variation observed across all these batches of devices can be safely concluded to stem from the angular misalignment in the axis of transduction. Q VARIATIO ABOUT [110] AXIS OF TRA SDUCTIO Among the SiBARs fabricated about the [110] axis, the SiBAR that offers the highest Q is assumed to be transduced exactly along [110]. This assumption is reasonable as the atomic arrangement favoring minimum loss for acoustic transduction would exist only for the [110] axis of transduction. The SiBARs that are angularly offset clockwise from the [110] device are assumed to have a positive offset and vice versa (Figure 3). A very interesting symmetric pattern in the Q variation with angular offsets has been observed in these devices about [110] as shown in Figure 4. These results correspond to moderately doped substrate with a supporting tether width of 3 μm. The Q values were measured in vacuum at an input power of -10 dBm at a Vp of 10 V, and a capacitive air-gap of ~100 nm. Except for a minor variation in each Q value by ±3k, this exact pattern repeats in highly doped and ultra-highly-doped substrates as well, thus confirming the pattern to be a result of angular misalignment. Fig. 4: Measured pattern of Q variation in SiBARs with angular offsets from [110] axis of transduction. The most important conclusion from this study is that the Q drops by ~50% when offset from the [110] axis of transduction by just 0.1 degree. This calls for additional efforts to alleviate misalignment sensitivity when Q is of utmost importance. A 180 ° phase difference has been measured across all the resonance peaks to confirm that the lower-Q is not due to a coupling of adjacent resonant modes of the SiBAR. When these devices were fabricated with a 1.5 μm wide supporting tether to further reduce the support loss, the unloaded Q (i.e., Q at the turn-ON Vp of 1 V) along the [110] direction almost matches the maximum possible fQ in SCS (Figure 5(a)) [4] . Figure 5(b) illustrates that other than the expected ~50% drop with a 0.1 degree angular offset, spurious modes start appearing within a span of 1 MHz from the main resonance peak. One or more of such spurs have been observed in all the devices about the [110] axis except for the one aligned exactly long [110]. Thus, the observation of such spurs can serve to be a good indicator of the existence of an angular misalignment from the fabrication process. A vacuum measurement setup and careful SOLT calibration have been found to be necessary in the case of some devices to observe the spurs. Fig. 5: Measured responses from the SiBARs transduced (a) along [110] direction, and along (b) ± 0.1 ̊and (b) ± 0.2 ̊ angular offset.