High-Stiffness Driven Micromechanical Resonators with
Enhanced Power Handling
Supporting Materials
Fabrication Process Flow and Results
Micromechanical clamped-clamped beam (CC-beam) resonators were fabricated using a
previously developed small-vertical gap surface-micromachining technology [S1],
summarized by the process cross-sections of Fig. S1. In this process, a 2-m-thick silicon
dioxide and 350-nm-thick silicon nitride were first deposited on the silicon substrate to serve
as a top layer that electrically isolates mechanical devices and interconnects from the
substrate and from themselves. A 350-nm-thick polysilicon layer was then deposited, doped,
and patterned, to serve as drive/sense electrodes and electrical interconnects. Next, a
100-nm-thick sacrificial silicon dioxide layer was deposited to define the gap spacing
between resonators and electrodes. An RIE etch then defined anchors, as shown in Fig. S1(a),
and a 2-m-thick polysilicon layer was deposited, doped, and patterned, to define the beam
structure shown in Fig. S1(b). Finally, hydrofluoric acid was utilized to remove the sacrificial
oxide and release the structural layer, leaving a free-standing structure shown in Fig. S1(c).
Interconnect
High-Stiffness Driving
Electrode
Low-Stiffness Driving
Electrode
Silicon Nitride
Oxide
Silicon Substrate
(a)
Structural Polysilicon
Oxide
Silicon Substrate
(b)
Anchor
Clamped-Clamped
Beam Resonator
Oxide
(c)
Silicon Substrate
Fig. S1: Cross-sections depicting the fabrication process used to achieve beam resonators in
this work.
1
Driving
Electrode
Clamped-Clamped
Beam Resonator
Clamped-Clamped
Beam Resonator
50μm
40μm
2μm
DC Bias VP
Sensing
Electrode
100nm Gap
(a)
(b)
Fig. S2: (a) Global and (b) zoom-in SEM views of a fabricated 50m-wide clamped-clamped
beam resonator.
Fig. S2(a) presents a global SEM (scanning electron microscope) view for a fabricated
50m-wide CC-beam resonator, identifying its driving, sensing, and dc-bias ports; and
Fig. S2(b) further shows the resonator zoom-in view with detail dimensions and its 100-nm
gap spacing.
Modeling and Simulation Results
To model the high-stiffness driven resonator, the governing equations of a
clamped-clamped beam resonator from [S1] are utilized. As shown in Fig. S3 which
identifies key features for the resonator modeling and equation derivation, the mode shape of
the resonator is governed by
𝑋𝑚𝑜𝑑𝑒 𝑦 = 𝜁 𝑐𝑜𝑠𝑘𝑦 − 𝑐𝑜𝑠ℎ𝑘𝑦 + 𝑠𝑖𝑛𝑘𝑦 − 𝑠𝑖𝑛ℎ𝑘𝑦
(S1)
Referring to [S1] by using the relationship between the kinetic energy and velocity, the
equations for equivalent mass, damping factor, and stiffness are presented in (S2), (S3), and
(S4).
𝑚𝑟 𝑦 =
𝜌𝑊𝑟 ℎ
𝑊𝑟
[𝑋𝑚𝑜𝑑𝑒
0
(𝑦′)]2 𝑑(𝑦 ′ )
[𝑋𝑚𝑜𝑑𝑒 (𝑦)]2
=
𝐾𝐸𝑡𝑜𝑡
1
𝑣(𝑦)2
2
𝑘𝑚 𝑦 𝑚𝑟 (𝑦) 𝜔𝑛𝑜𝑚 𝑚𝑟 (𝑦)
k m (y)
=
=
𝑄𝑛𝑜𝑚
𝑄𝑛𝑜𝑚
𝜔𝑛𝑜𝑚 𝑄𝑛𝑜𝑚
𝑐𝑟𝑒 𝑦 =
2
2
𝑘𝑟𝑒 𝑦 = 𝜔 × 𝑚𝑟𝑒 𝑦 = 𝜔 ×
2
𝜌𝑊𝑟 ℎ
𝑊𝑟
[𝑋𝑚𝑜𝑑𝑒
0
(S2)
(S3)
(𝑦′)]2 𝑑(𝑦 ′ )
[𝑋𝑚𝑜𝑑𝑒 (𝑦)]2
(S4)
Mode Shape
Clamped-Clamped Beam
Lr
y
h
Wed
Led1
d(y)
Wed
(b)
Wes
Led2
Les1
Les2
Fig. S3: Resonator cross-sectional schematic for theoretical derivation.
Therefore, the effective stiffness kre in (S4) is location dependent (inversely proportional
2
to [𝑋𝑚𝑜𝑑𝑒 (𝑦)] ) due to the beam-vibrating mode shape where high-stiffness and
low-stiffness locations correspond to low-velocity (close to the anchors) and high-velocity (at
the middle of the beam) positions, respectively. With that, MATLAB simulation based on
(S1)-(S4) was utilized to generate Table I and Fig. 3(b). As shown in Table I, the use of
high-stiffness driving approach demonstrates 16.9X power handling enhancement based on
simulation as compared to its low-stiffness counterpart. From the simulation of Fig. 3(b), the
effective stiffness near the anchors is much larger than that at the center of the beam. The
power handling of the beam resonator is governed by [S2]
Pomax 
o
Q
k r ( yd )a 2 d o2
(S5)
where a (= 0.56 at resonance) is the fraction of the electrode-to-resonator gap beyond which
the onset of strong nonlinearities ensue and where kr (yd) is the effective stiffness of the
resonator at its driving location yd. From (S5), the maximum handling power of the resonator
before entering strong Duffing nonlinearity is proportional to the equivalent stiffness of the
resonator. As a result, the higher the stiffness, the better the power handling capability (or
linearity).
Table I: Resonator Data Summary
Driven at high-stiffness
Driven at low-stiffness
(low-velocity) location
(high-velocity) location
50μm-Wide CC-Beam Parameters
Units
Young’s Modulus, E
150
GPa
Density, ρ
2,300
kg/m3
3
Beam Length Lr = 40
Beam Width Wr = 50
Beam Thickness h = 2
Driving Location yd = 6 &
Dimensions (see Fig. S3)
Driving Location yd = 20
yd = 34
μm
Driving Electrode Width
Driving Electrode Width
Wd = 4
Wd = 20
Sensing Location ys = 6 &
Sensing Location ys = 20
ys = 34
Sensing Electrode Width
Sensing Electrode Width
Ws = 20
Ws = 4
Electrode-to-Resonator Gap, do
100
nm
DC-Bias Voltage, VP
9
V
9.54
MHz
610
―
Measured Resonance Frequency, fo (at
Low Input Power)
Measured Quality Factor, Q (at Low
Input Power)
Effective Mass, mreff (at Driving Port)
Effective Stiffness, kreff (at Driving
Port)
6.15×10-11
3.65×10-12
kg
1.94×105
1.15×104
N/m
-10 (98.93W)
-22.3 (5.87W)
dBm
Calculated Resonator Power Handling,
Pomax
References
[S1] F. D. Bannon III, J. R. Clark, and C. T.-C. Nguyen, “High-Q HF
micro-electromechanical filters,” IEEE Journal of Solid-State Circuits, vol. 35, no. 4, pp.
512-526, April 2000.
[S2] S. Lee and C. T.-C. Nguyen, “Influence of automatic level control on micromechanical
resonator oscillator phase noise,” Proceedings, 2003 IEEE Int. Frequency Control
Symposium, Tampa, Florida, May 5-8, 2003, pp. 341-349.
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