ORIENTATION DEPENDENT CHARACTERISTICS OF HIGH QUALITY FACTOR
SINGLE-CRYSTAL SILICON BULK MODE RESONATORS
C. Tu* and J. E.-Y. Lee
Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong
ABSTRACT
This paper presents an empirical comparison in the
characteristics between identical square-plate resonators
aligned along different crystal orientations for silicon,
namely the <100> and <110>. We show that Q drops by
42% and power handling capability is reduced resulting
from the change in orientation when exciting the Lamé
mode. In the square-extensional (SE) mode, orientation
shows little impact on Q and power handling for the same
orientation change. The measured eigenfrequencies in
both orientations closely agree with those predicted by
finite element (FE) simulations and the analytical form.
INTRODUCTION
Single-crystal silicon (SCS) resonators are typically
transduced along the <100> or <110> direction due to
efficient transmission of compressional and dilational
acoustic waves [1]. For bulk mode resonators, most of the
research has been so far focused on devices aligned in
<110> direction [2-3]. This orientation has had greater
appeal over other orientations within the (100) plane as it
offers higher Young’s modulus and smaller Poisson’s
ratio [4]. These in turn lead to high Q’s in bulk mode
resonators. However, it has been recently reported that
devices aligned to the <100> will be a better choice when
repeatability of Q across various batches of fabrication is
of prime importance [1]. In addition, it has also been
postulated that <100> oriented devices surpass those in
<110> in terms of obtaining high Q above ~750MHz [5].
Thus, more extensive research on bulk mode resonators
aligned in the <100> is of great interest. This paper
investigates the effect of crystal orientation (<100> and
<110>) on the characteristics of square-plate resonators
excited in the Lamé and SE modes.
METHOD AND RESULTS
Two square-plate resonators, each 360µm in length
were fabricated on the same die in a foundry SOI MEMS
process. Each square-plate was suspended by T-shaped
tethers on all 4 corners. The dimensions of the square
plates were designed to be identical, including those for
the tethers, with each one rotated 45o with respect to the
other. Hence one resonator was aligned along <110>
while the other in <100>. Optical micrographs of the
devices are depicted in Fig 1. Using the stiffness
coefficients for SCS, the eigenfrequencies for the Lamé
and SE modes were computed by FE using COMSOL for
both orientations as shown in Figs 2 and 3 respectively.
The devices were characterized in vacuum. Figs 4 and 5
show the measured electrical transmission for the Lamé
mode for <110> and <100> aligned devices respectively.
Figs 6 and 7 show the measured electrical transmission
for the SE mode for the same pair of devices. Table 1
summarizes the measured Q for each mode for each
device. It can be seen that Q for the device in <100> is
almost half that of the <110> device. However, change in
orientation has no obvious effect on Q for the SE mode.
These trends have been observed in measurements from a
2nd die as well. Table 2 compares the eigenfrequencies
derived analytically, by FE, and by measurement. We can
see a general close agreement between each of them. The
difference between the analytical and FE or measured
value for the SE mode stems from the effects of anchors
unaccounted for in the analytical equation. The change in
Q caused by rotation in the Lamé and not in the SE mode
stems from the effect of orientation on stiffness for each
given mode. In the Lamé mode, stiffness is governed by
the shear modulus which drops from 79.4GPa to 50.8GPa
from <110> to <100> (37% drop). The SE mode stiffness
is governed by the bi-axial modulus which is invariant. In
addition, we have also observed from our measurements
(not shown here) that the onset of mechanical nonlinearity
occurs earlier in the <100> (at -5dBm) compared to the
<110> (at 3dBm) for the Lamé mode under the same DC
bias conditions.
ACKNOWLEDGMENT
This work was supported by a grant from the City
University of Hong Kong (Project No. 9667049)
REFERENCES
[1] A. K. Samarao and F. Ayazi, "Quality Factor
Sensitivity to Crystallographic Misalignments in
Silicon Micromechanical Resonators," Hilton Head
2010, pp. 479-482, 2010.
[2] V. Kaajakari et al, “Square extensional mode
single-crystal silicon micromechanical resonator for
low-phase noise oscillator applications,” IEEE Elect.
Dev. Lett., vol. 25, no. 4, pp. 173–175, Apr. 2004.
[3] J. E.-Y. Lee, J. Yan, and A. A. Seshia, "Low loss HF
band SOI wine glass bulk mode capacitive
square-plate resonator," J. Micromech. Microeng.,
vol. 19, 074003, July 2009.
[4] M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What
is the Young’s modulus of silicon?” J. MEMS, vol. 19,
pp. 229-238, 2010.
[5] R. Tabrizian, M. Rais–Zadeh, and F. Ayazi, “Effect
of Phonon Interactions on Limiting the fQ Product of
Micromechanical Resonators,” Proc., Transducers
‘09, June 2009, pp. 2131–2134.
CONTACT
* C. Tu, tel: 852-21942656; [email protected]
(a)
(b)
Figure 1: Optical micrograph of the square-plate resonator
aligned in (a) <110> and (b) <100> direction.
Figure 5: Measured transmission magnitude of the Lamé mode
for the resonator aligned along the <100> direction.
Figure 2: The mode shapes of the Lamé mode resonators
aligned in <110> and <100> direction.
Figure 6: Measured transmission magnitude of the SE mode for
the resonator aligned along the <110> direction.
Figure 3: The mode shapes of the SE mode resonators aligned
in <110> and <100> direction.
Figure 7: Measured transmission magnitude of the SE mode for
the resonator aligned in <100> direction.
Table 1: Comparison of measured Q for resonators oriented in
the <110> and <100> direction on the same die.
<110>
<100>
% change
1149941
667215
-42%
Lamé
195600
198679
+1.5%
SE
Figure 4: Measured transmission magnitude of the Lamé mode
for the resonator aligned along the <110> direction.
Table 2: Comparison of the eigenfrequencies derived
analytically, by FE, and by measurement (Unit: MHz).
Analytical
FE
Measurement
11.466
11.510
11.4994
Lamé <110>
9.170
9.190
9.1742
Lamé <100>
12.147
12.001
11.9805
SE <110>
11.767
11.694
11.6724
SE <100>