Air Damping of Oscillating MEMS Structures: Modeling and

Comparison with Experiment

Sergey Gorelick *1, James R. Dekker1, Mikko Leivo 1, and Uula Kantojärvi1
1
VTT Technical Research Centre of Finland, Tietotie 3, Espoo, P.O.Box 1000, FI-02044 VTT, Finland
*Corresponding author: Tietotie 3, Espoo, P.O.Box 1000, FI-02044 VTT, Finland; sergey.gorelick@vtt.fi

Abstract: Air damping can be detrimental to the individual types of flow can be evaluated using
performance of vibrating MEMS components. simplified analytical models [1] or circuit
Quantitative evaluation of the damping is analysis [2,3]. However, due to the existence of
challenging due to the complex interaction of air intermediate flow regimes (Figure 1) and
with moving structures and typically requires geometrical sensitivity of the fluid-structure
numerical simulations. A full three-dimensional interaction, reliable estimation of the damping
analysis can be computationally very expensive, coefficients requires computational flow
time consuming and not feasible. Here, we simulations. COMSOL MultiphysicsTM was
present a simplified two-dimensional modeling previously used successfully for air damping
of damping per unit length of selected MEMS simulations in angular comb-drive micromirrors
structures. The simulated air damping results [4]. To evaluate the applicability of the Fluid-
were compared with experimental measurements Structure Interaction interface of COMSOL
of corresponding piezoactuated resonators: in- MultiphysicsTM for estimation of the air damping
plane and out-of-plane tuning forks, two types of for a variety of other MEMS devices, the
out-of-plane cantilevers and a torsional experimental and simulated performance of
micromirror. The applicability of the simplified several typical resonator MEMS systems in this
model is verified by a good (2-30%) agreement paper are compared: two types of out-of-plane
between the simulated and measured Q-values. cantilevers, in-plane and out-of-plane tuning
forks, and a torsional mirror (Figure 2). A full
Keywords: MEMS, air damping, FEM, tuning three-dimensional damping analysis is
fork, micromirror computationally very expensive and time-
consuming. Instead, a simpler two-dimensional
1. Introduction analysis of the damping per unit length of the
structures and as a function of initial
Various MEMS devices, resonators and sensors displacement was performed (Figure 2 and 3).
are designed to operate in air and viscous fluids. Such an approximation neglects edge effects and,
Excessive air damping can be detrimental to the in case of a bending cantilever, also the rotary
performance of oscillating MEMS components. motion of the beam cross-section.
Quantitative evaluation of the air damping is,
therefore, required already during the design
stages to obtain accurate predictions of the
device performance. The interaction of fluid flow
with moving structures is complex such that
analytical approximations of the damping
coefficient are available only for simple
structures and trivial boundary conditions. More
complex systems with structures moving with
respect to each other and in proximity to
stationary objects typically require simulations to
reliably evaluate the air damping. Thus, a simple
out-of-plane cantilever oscillating inside a pre-
etched cavity (Figure 1) experiences damping Figure 1 . Air flow simulation time-snapshot of an
due to the interaction with several types of flow out-of-plane cantilever oscillating inside a pre-etched
(squeeze-film, shear flow and drag due to the cavity. The arrows indicate the velocity field of the air
direct air resistance). The damping due to the flow around the cross-section of the cantilever. The
arrow within the cantilever indicates instantaneous 2. Experimental measurements
velocity of the cantilever. Several regions of flow can
be identified (shear flow, squeeze-film flow, drag due The simulated systems aimed at modeling the air
to the direct air resistance) as well as transitional
regions.
damping of selected MEMS devices. The test
devices, such as cantilever and torsional
The validity of the approximation is, however, resonators (Fig 2), in this study were fabricated
justified by the consistency of simulated and using c-SOI technology (cavity silicon on
experimental results. The good agreement of the insulator) technology in 50-µm-thick device
experimental and corresponding simulation layers. The devices were actuated by means of
results validate the applicability of the simplified thin (~1 µm) aluminium nitride (AlN)
flow model for evaluation of the air damping in piezolayers processed on top of the released
other more complex system. structures (cantilevers or actuating flexures). The
air damping in the following devices was
characterized experimentally:

· Wide cantilever C1 (width, height,

length 100×50×900 µm3); Resonance
frequency ~80 kHz (Figure 2a).
· Narrow cantilever C2 (width, height,
length 50×50×900 µm3). Resonance
frequency ~80 kHz. (not shown)
· In-plane tuning fork TF1 (width, height,
length 100×50×700 µm3). Resonance
frequency ~250 kHz (Figure 2b).
· Out-of-plane tuning fork TF2 (width,
height, length 26×50×238 µm3).
Resonance frequency ~1 MHz (Figure
2c).
· Torsional mirror M (width, height,
length 400×50×800 µm3). Resonance
frequency ~50 kHz (Figure 2d,
actuators not shown to facilitate clarity).

The characterization of the devices’ performance

was based on measuring their electrical
admittance in a frequency range around their
resonances both in vacuum and in air. The
mechanical Q-values were then derived from the
fits of the equivalent RmLmCm-C0 circuit [3,5] to
the admittance-frequency curves (Figure 3).
Measurement of the Q-value of the same device
both in air (Qair) in vacuum (Q vacuum) allows
Figure 2 . Simulation of air damping of various isolating the damping effect due to interaction
MEMS systems resonating inside pre-etched cavities
with air flow (Qflow) alone from other damping
using simplified two-dimensional model with
numerical “springs” that replace the deformable effects (anchor loss, thermoelastic damping,
flexures and generate restoring forces. Shown are surface losses, etc).
schematic deformation of each system in 3D with the
resonance frequency and corresponding simplified 2D
model air flow simulation time-snapshots. a) out-of- (1)
plane cantilever, 80 kHz. b) In-plane tuning fork, 250
kHz. c) Out-of-plane tuning fork, 1 MHz. d) Torsional
mirror, 50 kHz.
 where ρ is the fluid density, v and L –
characteristic velocity and dimension of the
system, respectively, and µ is the dynamic
viscosity of the fluid. The maximum of the
Reynolds numbers for the studied systems are
summarized in Table 1. With Re<<2000 it can be
concluded that the air flow is laminar for all the
studied systems and no turbulence effects needed
to be included in the simulations.

device f0 Q in Δx v Re
[kHz] air (AC) (AC)
[µm] [m/s]
C1 80 150 0.75 0.38 0.25
C2 80 180 0.90 0.45 0.30
TF1 250 1200 0.28 0.45 0.30
Figure 3. Measurement of electrical admittance of a TF2 1000 5600 1.26 7.91 5.27
piezodriven cantilever (Fig. 2a) around the resonance M 50 175 0.05 0.02 0.01
frequency with corresponding fits of equivalent circuit Table 1. Measured devices (C1 – wide cantilever; C2
elements (a) in air with a derived Q-value of ~150, (b) – narrow cantilever; TF1 – in-plane tuning fork; TF2 –
in vacuum with a derived Q-value of ~6500. (Phase of out-of-plane tuning fork; M – torsional resonator,
the admittance not shown.) micromirror. See text for further details on the
devices) with simulated maximum amplitude of the tip
3. Fluid flow properties and boundary displacement and velocity using the measured Q-
conditions values in air. The velocity can be used to estimate the
Reynolds number and categorize the type of the fluid
Choosing the appropriate flow properties and flow.
boundary conditions is crucial for reliable
estimation of the air damping. Several The Mach number, M, is defined as the ratio
dimensionless quantities can be used to of the speed of structures moving through a fluid
categorize the type of the flow. The Knudsen to the speed of sound in the fluid. If M<0.3, the
number (Kn) is defined as the the ratio of the gas compression effects can be neglected. From
mean free path of fluid molecules to the Table 1, the maximum velocities of the
characteristic dimension in the system. It can be structures are considerably smaller than 30% of
used to verify the applicability of the viscous the speed of sound in air 0.3×343≈100 m s-1,
fluid dynamics. The typical gap width in the such that the simplified incompressible flow
studied system is 10-20 µm (Fig. 2c). With the model could be used in simulations for all the
mean free path of gas molecules in air at 300 K studied systems.
of ~70 nm, the Knudsen number equals to 3.5-
7×10 -3. Since 10-3<Kn<10 -1, it can be concluded 4. Simplified air flow model
that the slip flow dynamics formalism is
applicable to model the air flow [6]. The Figure 4 depicts the principle of the simplified
stationary surfaces adjacent to the narrow gaps in 2D model that replaces the more complex 3D
the model were, therefore, defined as walls with simulation. Initially deformed beam with a tip
slip boundary condition while no-slip condition displacement A (Fig. 4a) is subdivided into
was applied to further lying walls (Fig. 4c). narrow cross-sections of width dx and mass dm,
The Reynlods number Re can be used to each displaced by an initial amplitude Ai, such
categorize the flow as either laminar or turbulent. that the combined shape of the cross-sections
It is defined as resembles the original beam’s mode shape (Fig.
4b). The cross-sections oscillate vertically and
(2) synchronously at the resonance frequency of the
original beam due to the action of spring forces
dk (Fig 4b). Replacing a three-dimensional air
flow simulation with such a two-dimensional
model is justified when the beam length is much
greater than its other dimensions since the air
flow profile can also be assumed to be two-
dimensional. For a laminar type of flow, the
interaction of different oscillating cross-sections
and the air flows induced by them can be
considered negligible, such that the total air
damping force is a sum or an integral of viscous
damping forces experienced by individual cross-
sections. Modeling a cross-section of an
extended torsional resonator (Fig. 2d) is
appropriate due to the symmetry of the proof
mass about the rotation axis along the longer
dimension (neglecting the edges). In the case of a
bending beam, assuming the straight motion of
the cross-section is an additional approximation
because the neutral axis of the beam moves not
only translationally but also rotationally. For
small amplitudes of vibrations of a long beam
the rotational motion is, however, relatively
small and can be neglected. The validity of the
approximation was demonstrated in a similar
approach where the tines (beams) of the tuning
forks were modeled as spheres or strings of
spheres in straight motion [7-12]. However,
modeling rectangular cross-sections is more
appropriate as they reproduce closer the original
geometry of the structures.
The simulations in time domain were initiated
by displacing the cross-sections of the structures
by initial amplitude A from the equilibrium
position (in case of torsional resonators, tilting
them by angle A). Due to the restoring forces or
torques, the simulated systems began to oscillate.
The oscillation amplitude decayed due to the
interaction with air as the simulations progressed
in time, and the Q-values were estimated from
the logarithmic decrement of amplitude (Fig.
4d). The restoring spring forces/torques due to Figure 4 . Schematic presentation of the simplified air
the springs’ deformation were replaced by damping simulation of (a) vibrating cantilever beam
numerical forces/torques. The restoring with an initial tip displacement A. (b) The beam is
forces/torques were defined as, e.g., boundary divided into cross-sections of width dx and mass dm,
load (Fig. 3c) -dk(A+u) or -dk(A+v), where dk is displaced from the equilibrium position by
the spring constant per unit area (adjusted to corresponding amplitudes Ai. The cross-sections move
result in the required resonance frequency of the synchronously at the resonance frequency of the
system), and u and v are the structural original beam due to the action of spring forces dk. (c)
displacement of the centre of mass horizontally Cross-section of the beam inside a pre-etched cavity
displaced by an initial amplitude Ai. The spring forces
(in-plane) and vertically (out-of-plane),
due to the structural deformations are replaced by
respectively. boundary loads to model the restoring forces. (d)
Time-domain simulation of a cantilever (Fig. 2a)
cross-section displacement initially displaced by 1 µm
from the equilibrium position. Due to the interaction
 with air, the amplitude decays. The Q-value can be Q(Ai) with the resonant mode shape of the
evaluated from the logarithmic decrement of the structure over its length. However, the simulated
amplitude. Q-values’ variations over the typical amplitude
ranges (Table 1) were insignificant.
The numerical springs effectively replaced the The agreement between measured and
action of actual restoring forces thus eliminating simulated values is very good (>90%) for
the need to simulate the structural deformations structures with rectangular cross-sections: wide
and reducing the simulation complexity. cantilever (cross-section 100×50 µm), in-plane
tuning fork (cross-section 100×50 µm), and the
5. Comparison of experimental and out-of-plane tuning fork (cross-section 26×50
simulated results µm). The agreement between the measured and
simulated results is worse (70%) for the narrow
Several devices of each type were cantilever with a square cross-section (50×50
characterized both in vacuum and air. The µm). Even though the agreement is adequate for
averaged Q-values derived from the electrical practical purposes, the reason for the larger
measurements are summarized in Table 2. The mismatch is not clear. Presumably, edge effects
Q-values obtained in simulations are given in the are more pronounced in beams having square-
same table for comparison with the experimental shaped rather than rectangular cross-sections.
values. The simulated Q-values account only for The agreement between the measured and
losses due to the interaction of the structures simulated results is relatively good (82%) for the
with the air flow, Qflow, while the Q-values micromirror torsional resonator with a
measured in air, Qair, include also losses due to rectangular cross-section (400×50 µm), even
other loss mechanisms (e.g., anchor and support though the 2D model is more applicable for this
losses). In order to isolate the Q-value due to the type of device than for a bending beam where the
air damping alone, other contributions to the rotational motion of the cross-sections is
overall damping need to be filtered from the neglected. The discrepancy can be partially
damping measured in air. These additional loss explained by the simplified simulation where the
mechanisms can be estimated by performing the actuator beams were not modeled. The
characterization of the devices in vacuum. Using agreement can potentially be improved if the
Equation 1, the damping due to the interaction effects of the more complex geometry on the air
with air flow alone, Qflow, can be deduced and flow patterns are taken into account.
compared directly with the corresponding The simplified 2D simulation results,
simulation results (Table 2). however, generally fit adequately well the
experimental results over a wide range of
devi f0, Q in Q Q Q Agree frequencies (from tens kHz to MHz) and for a
ce kHz air vacuu flow simula ment
m ted
variety of moving structures (in-plane, out-of-
plane and torsional motion), verifying the
C1 80 150 6500 153 168 91% validity of the model for estimation of the air
C2 80 180 6530 185 265 70% damping in other more complex MEMS
TF1 250 1200 40000 1240 1190 96% geometries and structures.
TF2 1000 5600 16000 8615 8820 97%
M 50 175 50000 176 214 82%
7. Conclusions
Table 2. Comparison of measured and simulated Q-
values for the studied systems (C1 – wide cantilever; We investigated the air damping in several
C2 – narrow cantilever; TF1 – in-plane tuning fork; test MEMS piezoactuated resonating structures
TF2 – out-of-plane tuning fork; M – torsional
resonator, micromirror. See text for further details on
both experimentally and numerically. The
the devices). studied test systems involved various types of
motion within pre-etched cavities (in-plane, out-
The simulated Q-value were evaluated for of-plane and torsional) and in a wide range of
different initial amplitudes of vibration, such that frequencies (104 – 106 Hz). The simplified 2D
the total Q-value for a given structure could be fluid-structural interaction model of the systems
calculated by weighing the amplitude-dependent was used to estimate the Q-values of the studied
systems that were compared with the
experimentally obtained Q-values. The vibrating microcantilevers, Joutnal of Vibration
simulation and experimental results generally and Acoustic, 121, 64-69 (1999)
agree very well or adequately well, with the 12. J.-H. Lee, S.-T. Lee, C.-M. Yao, W. Fang,
discrepancies attributed to approximations in the Comments on the size effect on the
model or simplifications done in the simulations. microcantilever quality factor in free air space, J.
The good agreement of the simulation and Micromech. Microeng., 17, 139-146 (2007)
experimental results validates the model and
proves its feasibility for estimation of the air 9. Acknowledgements
damping in other MEMS systems.
The research has been funded by Tekes
8. References (Finnish Funding Agency for Technology and
Innovation), Bruco Integrated Circuits BV,
1. X. Zhang, W.C. Tang, Viscous air damping in Innoluce BV, SILEX Microsystems AB,
laterally driven microresonators, IEEE Workshop VINNOVA and NL Agency (Department of
on Micro Electro Mechanical Systems, MEMS International Research & Innovation), and VTT
‘94, 199-204 (1994) under the collaborative project of Eurostars
2. T. Veijola, T. Tinttunen, H. Nieminen, V. HighResSensing. The fabrication of the samples
Ermolov, T. Ryhänen, Gas damping model for a used in this study was funded by TEKES,
RF MEM switch and its dynamics Murata Electronics, TDK-EPC and Okmetic
characteristics, IEEE Microwave Symposium under the collaborative project of PiezoMEMS.
Digest, 2, 1213-1216 (2002) We are thankful to Diederik van Lierop and
3. V. Kaajakari, Practical MEMS, 184-197. Marijn van Os from Innoluce BV, Thorbjörn
Small Gear Publishing, Las Vegas (2009) Ebefors from SILEX Microsystems AB and
4. T. Klose, H. Conrad, T. Sandner, H. Schenk, Gerard Voshaar from Bruco Integrated Circuits
Fluidmechanical damping analysis of resonant BV for helpful discussions. We are also grateful
micromirrors with out-of-plane comb drive, to Dennis Alveringh and Sofia Lemström for
Proc. COMSOL Conf., online, help with the measurements.
http://www.comsol.com/papers/5208 (2008)
5. K.S. van Dyke, The piezo-electric resonator
and its equivalent network, Proceedings of the
Institute of Radio Engineers, 16, 742-764 (1928)
6. S. Roy, R. Raju, Modeling gas flow through
microchannels and nanpores, Journal of Applied
Physics, 93, 4870-4879 (2003)
7. K. Kokubun, M. Hirata, Y. Toda, M. Ono, A
bending and stretching mode crystal oscillator as
a friction vacuum gauge, Vacuum, 34, 731-735
(1984)
8. K. Kokubun, M. Hirata, M. Ono, H.
Murakami, Y. Toda, Unified formula describing
the impedance dependence of a quartz oscillator
on gas pressure, J. Vac. Sci. Technol. A, 5, 2450-
2453 (1987)
9. F.R. Blom, S. Bouwstra, M. Elwenspoek,
J.H.J Fluitman, Dependence of the quality factor
of micromachined silicon beam resonators on
pressure and geometry, J. Vac. Sci. Technol. B,
10, 19-26 (1992)
10. H. Hosaka, K. Itao, S. Kuroda, Damping
characteristics of beam-shaped micro-oscillators,
Sensors and Actuators A, 49, 87-95 (1995)
11. H. Hosaka, K. Itao, Theoretical and
experimental study on airflow damping of