Silicon
https://doi.org/10.1007/s12633-022-01853-x
ORIGINAL PAPER
Design and Simulation of Vertical Bi-Directional Fringe Field Tuning
of New Improved MEMS Accelerometer Using SOI Technology
for Stress Compensation
Manoj Kumar Dounkal 1
&
R. K. Bhan 2 & Navin Kumar 3
Received: 9 December 2021 / Accepted: 27 March 2022
# The Author(s), under exclusive licence to Springer Nature B.V. 2022
Abstract
A new conceptual utilization of Silicon on Insulator (SOI) wafer is reported for bi-directional vertical electrostatic fringe field
tuning of the Micro Electro Mechnical Systems (MEMS) micro accelerometer for compensating stress induced curling (up and
down), sensitivity and mechanical dynamic response. The buried oxide (BOX) layer-based SOI wafer provides bi-directional
electrodes for applying bias voltages independently. Residual stress induced curved deflection due to stress gradient is targeted
for tuning and reducing its effects using fringe field electrode configuration in SOI wafer technology. Movable silicon structure is
electrostatically (utilizing fringe field) brought back near to original mean position with softened stiffness (increase in sensitivity)
and reducing drastically the effects of stress gradients. The simulations are carried out using COVENTORWARE and COMSOL
Multiphysics software. The deflection results obtained by both software agree within 7.69% for maximum deviation. There is a
deviation in change in capacitance (del C) of 5.89% when stress gradient of 0.1 MPa/μm and 17.62% when stress gradient of
4 MPa/μm is applied on the structure at 30 g. This deviation can be tuned by above mentioned Bi directional tuning.
Additionally, non-linearity induced by stress gradient in sensitivity can also be tuned by electrostatic fringe field effectively
upto 18.64% when higher stress gradient (4 MPa/μm) was affecting the structure. Maximum disagreement of 4.72% between
analytical and simulated results promises the design of proposed tuning concept. The proposed tuning concept can be utilized for
other MEMS devices suffering from stress gradient issues.
Keywords Silicon on insulator . Accelerometer . Tuning . Electrostatic actuation . Fringe field . Stress gradient . Deflection
1 Introduction
Micro Electro Mechanical Systems (MEMS) device tuning
has been a challenging field and much needed domain since
their inception in industrial applications. Different
* Manoj Kumar Dounkal
mnjdpt@gmail.com
R. K. Bhan
bhan_rk2003@yahoo.com
Navin Kumar
nkumar@iitrpr.ac.in
1
Solid State Physics Laboratory, Lucknow Road,
Delhi, Timarpur 110054, India
2
Institute of Defence Scientists and Technologists, Majumdar Marg,
Delhi, Timarpur 110054, India
3
Indian Institute of Technology, Ropar 140001, India
configuration actuation mechanisms are reported in literature
[1–3]. Electrostatic repulsive force mechanism has been utilized by various groups for active actuation and tuning of
microstructures. Towfighian et al. [4] has utilized repulsion
method with total four electrodes for many applications viz.
accelerometers, gyroscopes, switches, mirror etc. A levitation
force mechanism was used by Quakad et al. [5] for tuning
pressure MEMS sensor. A combined use of electrostatic repulsion and attraction mechanism was used by Yao et al. [6]
for out of plane actuator. Large amplitude deflection was reported by Linzon Y. et al. [7] for SOI based cantilever by
fringing electrostatic field. Pull in free microphone modelling
and characterization utilizing levitation-based electrode configuration was reported by Ozdogan M. et al. [8]. Independent
tuning of linear and non- linear stiffness coefficient for uniaxial micro mechanical device was reported by Adams S.G.et al.
long back in 1996 [9] by combination of electrostatic
actuators. Ultra-thin silicon wafers ranging from several micron thickness to tens of micron thickness reported by Tian Y.
Silicon
B. et al. [10] whose finite element analysis for deflection and
residual stress was carried out by the group.
Among various techniques like thermal [11–13], magnetic
[14, 15] and Peizo [16–18], electrostatic actuation has been
very fruitful due to many advantages over others like compactness, better precision control on movements and minimum reactivation time etc. However, all the above reported
devices used techniques like electrostatic attractive or repulsive techniques have one limitation viz. movable structure can
be deflected only in one direction. The change in direction is
only within the planes of electrodes shifting the whole device
in plane only exhibiting possible tunning accordingly, thus
limiting the scope of tuning in 3 dimensions. Whereas electrostatic attractive actuation always decreases the gap, repulsive mode increases the gap however, there are situations
where bi-directional tuning is required for example deflection
in cantilevers, RF switches or fingers of accelerometers due
+ve or –ve stress gradients (SG) induced by fabrication. Such
types of curled deflections (or curvatures) are possible in
cantilever-based devices as studied by us earlier [19] wherein
axial force compression and tension were used to tune the
structure and a good agreement was reported between experimental, simulated and our proposed analytical model.
Further, recently we reported [20] a new improved
microaccelerometer based on SOI technology. It was shown
that our design has a higher ~200 times overall normalized
figure of merit (FOM) (190 fF g−1 μm−2 Hz−2) compared to
the existing designs in the literature assuming no curling or
stress effects in the device. However, curling induced post
fabrication due to stress gradients is always a reality unless it
is controlled by careful design and fabrication parameters
[21–23]. Here, in the present study we show further advantages in our design by proposing a new vertical bi-directional
fringe field tuning mechanism to reduce the effects of curling
due to stress gradients in these microstructures.
Figure 1 shows the 3-D structure of differential micro accelerometer proposed by us earlier [20] with addition of side
SOI ear electrodes walls (SEEW) with BOX layer. Four number of electrode sets are used, two on each side.
Top view shown in Fig. 2 clearly shows one set of SOI
electrodes with the gap ‘h’ between SOI vertical and horizontal part and gap ‘s’ between fixed SOI horizontal and movable
ear on proof mass (MEPM). The designed gap ‘h’ is 5 μm and
‘s’ is 4 μm. The configuration of electrodes is chosen symmetric to proof mass on both side and positioned in such a way
that proof mass pull in on either side is avoided and effect of
field on fingers pull down limit is remote and sensing fingers
are also not affected due to their presence.
In literature, many researchers have carried out studies on
effect of stresses in thin films and methods for compensating
it. The various ‘technical methods’ which help in reducing
residual stress generated in thin films have been compared
for different MEMS devices in Table 1. The advantages of
the proposed SOI wafer technology using electrostatic actuation, justifies its novelty as electrostatic tuning techniques offers varoius advantages like fast switching, small footprints
and negligible weight etc.Furthermore, in this study, SOI bidirectional electrodes have been designed for along with steps
of fixed combs SOI structure. It will be shown here that one
can use these electrodes for tuning and reducing the curling in
MEMS sturcures induced by processing. The proposed
MEMS accelerometer and its tuning find use in a wide variety
of applications such as navigation, smartphones, automobiles,
bio medical instruments, industrial systems, etc. However,
current design is targeted or navigation application.
2 Analytical Formulations
2.1 Tuning Electrode Formulations
An attempt has been made to develop the analytical equations
from the basics that are used for capturing the physics of the
tuning by using the electrodes shown in Fig. 2. An electrical
relation between charge (qt) and voltage (Vt) on tuning SOI
tuning electrodes is shown in Eq. 1. The proportionality constant between applied voltage and related charge is ‘Ct’ called
capacitance and they are related by
qt ¼ C t V t
ð1Þ
where capacitance is given by
Ct ¼
ε At
h
ð2Þ
where ‘ε’ is the permittivity of dielectric material (air), ‘At’ is
the area of facing electrodes and ‘h’ is gap between them.
Ut ¼
1
C t V 2t
2
ð3Þ
Equation 3 is relation showing between energy, capacitance and voltage. In this equation, Ut is the energy stored in
the capacitor. Further the electrostatic force Fe as a result of
this energy is given by Eq. 4.
Fe ¼
∂U t
1 ∂C t 2 1 V 2t
¼
V ¼ εA 2
2 ∂h t
2
∂h
h
ð4Þ
The electrostatic force equation is given by taking the derivative of energy stored in a capacitor given by Eqs. 3 and 4
shown respectively. It may be mentioned here that Eq. (2) is a
simplified equation wherein the effects of edges and fringe
fields are neglected. However, in our case and in particular
comb type of electrodes, additional contribution to capacitance due to these fringe fields cannot be neglected, we use
improved equation for calculating capacitance. The effect of
Silicon
Fig. 1 Micro accelerometer
structure with fixed side
electrodes (4 Nos.) in SOI wafer
fringe field for asymmetric electrode position is simulated and
discussed in results section. Equation 5 [31] using fringe field
effect in respect of tuning electrodes (Fig. 2) is shown below
where ‘wc’ and ‘tc’ are cross sectional width and half height of
cantilever attached to the proof mass (movable electrode) and
‘h’ is the distance between fixed and movable electrode as
shown in Fig. 2.
0:23 !
w t 0:23
wc
c
c
Ct ¼ ε
þ 0:73
ð5Þ
−1:06 þ 3:31
h
h
tc
Fig. 2 Top view of SOI
electrodes showing set of tuning
electrodes with gaps ‘h’ and ‘s’
In this Eq. 5, the ratio condition within dimension range
0.1 < wc/h and 0.1 < tc/h < 10 is satisfied [Ref. 31, Eq. 4 in
where b = wc, g = h, and h = tc in our case]. Here, since we
are interested in finding out electrostatic force generated due
to the combined effect of fringe field on the proof mass and
accelerometer, a gradient has been calculated with respect to
the distance (gap) between fixed (SOI) and movable electrodes (ears on proof mass). The derived equation is shown
below and has been used in Eq. 4 for calculating the electrostatic force acting for tuning the residual stress.
Method of
compensating
stress
Stress level compensated
Tuning action
Application
Device example
Dounkal et al.,
(Present Study)
Dounkal et al., [19]
Singh J. et al., [30]
Haseeb et al.,2008
[24]
Magnetic actuation Nullify the residual stress for Application of
stress compensated
Combination of
actuator
compressive and tensile
stress material
Axial Force
Stress gradient effect removal Application of
application
Combination of
compressive and tensile
stress material
SOI electrostatic 500 MP in plane and upto
40–200 V
actuation, BOX
4 MPa/μm stress gradient
layer utilization
compensated
Micro accelerometer
Micro accelerometer,
SOI wafer biased and
electrostatic
force used
MEMS
400 μm length
Mechanical resonators 50 μm width
1 μm thick cantilever
Mechanical resonators Magneto strictive layer
Tuned to
−400 MPa
Tuned
to −600 MPa
Tuned to almost flat
deflection
−200 to
−650 (approx..)
−2000
Max 29 μm
deflection
Three-dimensional
solution of a static
shape control by
actuation
−242 in SiO2 and+ Tuned near 100 MPa
with reduced
283 in Fe65Co35
magnetic layer strip
thickness
Stress gradient of the Resonant frequency
order of
tuned upto 30 kHz
0.27 MPa/μm
using 120 V
handled
500 and stress
Tuned to almost flat
gradient upto
cantilever structure
4MPA/μm
Tuned to 1290 MPa
after release
Compressive 2000
Tensile
1000
Nonlinear hysterics
behavior
Does not eliminate all
internal stress
Remarks
307.74, Max.
deflection
330.30 μm
Maximum stress,
MPa
Comparison of tuning methods and techniques for reducing residual thin film stresses in MEMS devices as reported in literature including present study
Thermal annealing Polyamide stress in buckled Ramp up temperature from Micro-accelerometer, 1000 μm x100μm
cycle
compressed state reduced
250 deg to 350 deg by
Micro-gyroscopes accelerometer released
450 deg/h, holding hot
from lock position.
plate temp. at 350 for
Full stress NOT released.
20 min.
Fork et al., [25]
Controlling
Control stress anisotropy
Thermal annealing
Integrated circuits
Balanced spring structure
Saturation stress
(MoCr alloy) with
bath parameters
0.2 μm thickness and
0.57 μm thickness
Combs type micro
Sacrificial layer gap-based
H. Kattelus et al.,
Nitridation in
Lateral stress variation in
Introduction of
mechanical
device with
[26]
Reactive
films/improvement in
N2(Nitrogen)
environment
structures, micro
0. 3 μm thick films
sputtering
micro crystallinity
actuators,
uniformity
Zhao et al., [27]
Control of
Improving films’
Dimensional regulation
RF MEMS
2.2 μm with dynamic
geometry
homogeneity
By static and dynamic
geometry
deposition geometry
2.0 μm with static
geometry
Kouroshk et al., [28] Variable substrates Stress gradient across
RF source to bias growing Optical, thermal and 500 μm long free-standing
voltage
thickness compensated
film, Voltage range
RF system
microstructure
50–110 V
Irschik H. [29]
Piezoelectric
Deflection compensation,
Vary the polarization
Smart structures,
Mechatronics actuators
actuation
spatial distribution
profile of layer
Shape memory
alloys
Authors
Table 1
Silicon
Silicon
1 ∂C t 2
V
2 ∂h t
1
wc
tc
¼ ε* − 2 −0:7613* 0:77
*V 2t *l
2
h
h
Fe ¼
ð6Þ
where ‘Vt’ is applied voltage on the electrode and ‘l’ length
multiplied as Eq. (5) is for per unit length configuration.
Analytical results using above Eq. 6 will be discussed later
(Figs. 12 and 15) in Section 4.
2.2 Main Proof Mass Formulations
The effect of stress and stress gradient in the structure resulting in deflection δtotis given by [32].
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!ffi
u 2 2
u
SG*lm
ð7Þ
δtot ¼ t
þ ðR1 Þ2
2*E
where.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
σ σ 2 σ σ 2
y
y
x
x
R1 ¼
−v
−v
lm þ
wm
E E
E E
SG ¼ 2μ=L2 E
ð8Þ
ð9Þ
where ‘SG’ is stress gradient, ‘μ’ is deflection due to normal
bending and ‘δ tot’ is combined deflection (normal+ stress
gradient). σx is stress in ‘x’ axis and σy is stress in ‘y’ axis, ν
is Poisson’s ratio, E is Young’s modulus, lm is length of proof
mass and wm is width of proof mass.
As with our tuning mechanism we propose an SOI wafer
based electrostatic method, this may shift the design parameters of earlier optimized accelerometer [20]. A cognizance has
been taken for sensitivity change and bandwidth specially and
results shown in last part of Section 4. Mathematical equation
used for sensitivity of micro accelerometer discussed in calculation are as follows:
θ¼
M lm l b
g
4 α G tw3b z
ð10Þ
where ‘θ’ is the angular deflection, ‘M’ is mass of structure,
‘lm’ is length of proof mass, ‘lb’ is length of torsional beam,
‘α’ is geometric factor, ‘G’ is shear modulus, ‘t’ is thickness
of structure, ‘wb’ is width of torsional beam and ‘gz’ is acceleration due to gravity [20].
n ε lf
ð2 lm þ l f Þθ
d
S ¼ ΔC=gz
ΔC ¼
ð11Þ
ð12Þ
where ΔC is change in capacitance, ‘n’ is number of fingers
‘lf’ is length of fingers and ‘S’ is sensitivity. Results discussed
later in Figs. 16, 17 and 18 in Section 4 have been generated
utilizing Eqs. 7–12.
For 3d B bandwidth following equation has been used
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ωn
4ζ4 −4ζ2 þ 2 −2ζ2 þ 1
ð13Þ
f 3dB ¼
2π
where ωn is natural frequency of the accelerometer structure
and ζ is damping ratio. Here as the gap below the proof mass
is of few microns only, so effect of squeeze film damping
dominates. Slide film damping has also been taken care and
reported in earlier studies [20] done by our group. And in
simulation first we find the numerator ‘damping coefficient
(c)’ by defining parameters of squeeze film damping for structure and its vertical air gap beneath. Then using natural frequency ‘wn’ we find denominator, critical damping as Cc =
2*m*wn where ‘m’ is mass of the structure. Then ratio of
damping coefficient (c) and critical damping (Cc) is used as
damping ratio (ζ) in simulation. Dynamic analysis by varying
frequency is carried out for obtaining displacements. Then
using dB scale -3d B bandwidth is achieved.
Next in Section 3, the basic SOI tuning electrode unit concept has been explained and demonstrated by simulations results showing fringe field effects and movements of structure
in vertical direction. Section 4, discusses the results for
optimised meshing comparison between
COVENTORWARE and COMSOL Multiphysics simulations, tuning voltage application for stress mitigation and applications of stress gradient and acceleration on the differential
micro accelerometer parameters (viz sensitivity, bandwidth,
range of operation etc.). Section 5 finally concludes the summary of the proposed new tuning idea of the research work.
3 Conceptual Working of Tuning Electrodes
The proposed electrostatic actuation used by utilizing SOI
wafer as shown in Fig. 3 is new in its nature of application.
The SOI wafers have in built BOX layer providing two separate electric zones for bias applications [20]. In single Tshaped electrode set, there are three units of structures having
total five electrodes. The four electrodes are from two SOI
wafer walls (1 and 2) and fifth electrode is from proof mass
ear.
When the bias is applied on either upper or lower side of
electrode of wall 1, the fringe field cloud acts in such a way
that it pulls up or down the full movable proof mass along with
combs structure. The electrostatic force is generated which is
cause for this behaviour. Earlier concepts worked out in the
form of either attraction or repulsion wherein the movement is
in-plane only. Here we are showing a novel technique using
SOI wafer for tuning out of plane Bi-directional vertical
movement controlled by fringe field.
Silicon
Fig. 3 SOI conceptual unit for
electrostatic tuning of micro
accelerometer proof mass ear. All
5 electrodes shown in COMSOL
(left) and actual part from
accelerometer in
COVENTORWARE
Before we explain the functioning of the proposed concept
of tuning curling due to stress using electrostatic fringe field, a
list of parameters and dimensions used is shown in Table 2
below for clear understanding of the concept. From material
point of view, major three material SOI (silicon on insulator)
wafer, BOX (buried oxide) layer and silicon have been used
for design and simulation purpose.
In Fig. 4 shown are the conceptual movement of movable
electrode (EAR only) using COMSOL software when bias is
applied only on the upper and lower side of SOI wafer in wall
1 only respectively. Wall 1 and 2 are fixed on the substrate as
fixed finger set [20]. An SOI wall 2 is provided for making the
Table 2 List of parameters and
dimensions used for design and
simulation of micro accelerometer
and tuning electrodes
electrostatic field assymetric [1]. This generated electrostatic
fringe field is used for tuning the residual stress generated in
movable structure due to fabrication effects and
constraints.Various other stresses like shear stress, planar
stress and warping stress were discussed in detail on micro
accelerometer structure used [32]. Figure 4a showing movement upside and 4(b) showing movement downside on applied bias on upper and lower side of BOX layer in wall 1
respectively.
We see that fringe field here is mixed due to geometric
island (wall 1 and wall 2) with SOI material having BOX layer
sandwitched between silicon layers. Overall effect here is
S.No.
Parameter
Value (in μm)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
lm, length of proof mass
wm, width of proof mass
lf (length of overlap fingers)
L, full length of device
lb, length of torsional beam
wb, width of torsional beam
t, movable finger, proof mass and torsional beam thickness
w, height of initial overlap above BOX layer
Hm, Height of handle and device layer above and below initial overlap
tf, width of fingers (movable and fixed)
800
500
300
1150
150
4.5
18
8.5
11.5
6
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
d, (gap) between fingers
n, number of fingers
tBOX, BOX(oxide) layer thickness
ts, thickness of substrate structure attached to movable sensor
SOI device layer
SOI handle layer(final)
wc, width of movable ear electrodes
tc, half thickness of cross section of ear
length of movable EAR on side of proof mass
h, gap between fixed SOI and movable ear electrode
Length of fixed SOI electrode 1,2 (Fig. 3)
Width of fixed electrode 1,2 (Fig. 3)
Length of fixed SOI electrode 3,4 (Fig. 3)
Width of fixed electrode 3,4 (Fig. 3)
μ, deflection due to stress gradient (SG) (for example @ SG=0.2 MPa/μm) Fig. 12b
2
32 Nos.
1
400
20
20
16
9
100
5
160
32
90
26
0.755
Silicon
Fig. 4 a Unit electrode set with
Up movement and b Down
movement
(a)
attractive due to electrostatic force of field lines.So we can
utilize this new concept in tuning various microstructures in
terms of stiffness, residual stress,deflections etc. In next section, we show how this technique has been used on vertical
area changing differential micro accelerometer [20]. However,
the idea can be extended to other MEMS devices by utilizing
SOI technology.
4 Results and Discussion
4.1 COVENTORWARE Vs. COMSOL Multiphysics (Mesh
Size Comparison)
A detailed comparison of bi-directional fringe field electrostatic tuning actuation has been carried out utilizing
COVENTORWARE: Coventorware MP 1.2 (2019) and
COMSOL: 5.6 (2020) multi-physics software in addition to
analytical formulations. The system configuration used for
simulation has following details: Processor: Intel (R) Xenon
(R)Silver 4210CPU @2.20 GHz 2.19 GHz (2 processors),
RAM: 64GB (64-bit OS, × 64 based processor), SSD/HDD:
1 TB. The simulation time was from 20 min to 2 h depending
upon the mesh type and size.
COMSOL has been mainly used for study of fringe field
patterns because it has a better physics module for such a
study than COVENTORWARE. A different approach using
encapsulating space (in the form of sphere or cube) needs to be
defined in COMSOL for studying the fringe field pattern in
electrostatic regime. The sphere defined (shown later in
Fig. 8a here houses the whole MEMS accelerometer structure.
However, no separate encapsulating (sphere etc.) needs to be
defined in COVENTORWARE. Before attempting the electrostatic fringe field actuation on the main micro accelerometer, initially only basic unit proposed for tuning has been studied in detail to demonstrate the bidirectional deflection in zdirection. A 3D model was created in both the software to see
(b)
the exact behavior of the SOI electrodes made in the form of
ears that will be attached to main structure later on.
Hexahedral (HEX) and Tetrahedral (TET) meshing have
been used for comparison in both the software as shown in
Fig. 5. We have found that one has to optimize the meshing
types and sizes to get the meaningful results within reasonable
time. We have observed that different deflections are observed
in both cases. However, we observed that TET mesh gives
similar trends in both the software. Figure 6 shows the effect
of mesh size on the deflection in case of TET mesh. HEX
mesh was also compared but the deflection behavior was
showing wider gap with the increase of mesh size but with a
similar trend.
As can be seen from this Fig. 6, at higher mesh sizes both
the SWs tend to give similar deflections, whereas at lower
mesh size the disagreement can be as high as a factor two near
10 μm mesh size. Further, it can be seen that CW SW is more
sensitive to mesh size variation compared to COMSOL when
TET mesh is compared. In COMSOL variation is more or less
flat from 30 to 10 um, however beyond 10 um deflections do
rise as in case of CW. The results will be somewhat inaccurate
at higher mesh sizes but simulation time will be low, whereas
deflections at lower mesh will be more accurate but at the cost
very high simulation time. The typical simulation time for 30
um mesh size for this accelerometer structure is 35 min for
CW and 20 min for COMSOL whereas these times are 3 h for
CW and 2 h for COMSOL at 5 um mesh size. TET mesh
shows dissimilar results at lower mesh size but merge at
higher mesh size (near 25-30 μm). A Manhattan meshing with
20 μm element size and parabolic order has been used to
capture an averaged trend for the accelerometer in
COVENTORWARE and 5 μm element size for COMSOL
Multiphysics. To capture similar trends and deflection magnitude in the main accelerometer, this shift of 15 μm has been
used as there is almost flat deflection between 10 μm to 25 μm
(shift of 15 μm) as shown in Fig. 6 for TET mesh. Apart, this
mesh size variation is attributed to the outer sphere (coarse
Silicon
Fig. 5 Hex mesh (a) and Tet
mesh (b)in COMSOL
Multiphysics software
(a)
HEX MESH
mesh, 300-500 μm) used in COMSOL for defining fringe
field electrostatic actuation.
In addition, it may be seen from the above figure that in
order to get the similar trends of increasing accuracy with
decreasing mesh size, that there is a shift of about 15 μm
between the two software. This shift will be applied while
presenting the results of deflection for the main structure (micro accelerometer) as a function of tuning Voltage. In short,
therefore one has to do a tradeoff among meshing type,
meshing size, accuracy and time required for convergence
for simulation. “Physics controlled mesh” optimisation is used
in COMSOL Multiphysics where element size for the mesh
from extremely course to extremely fine can be chosen. All
active physics interfaces (electrostatic(es) and solid
mechanics(solid) had been kept in check in mode before simulation. COMSOL creates a mesh that is adapted to the current
physics interface setting in the model. This software solves
(b)
TET MESH
PDE (partial differential equation) in an integral weak form.
Unknown are described as sums over a set of basic/shape
functions defined on finite element, rather than by
discretization of derivative on a grid of points [33]. For
COVENTORWARE a predefined density of element in terms
of its size defined (x, y, z) with parabolic order has been used.
It has been observed that meshing type and size plays vital
role for converging the solution for complex problem at hand.
Since high aspect ratio (4:1) torsional beams (width 4.5 μm,
thickness 18 μm) have been used in main structure, along with
32 combs fingers with 3:1 aspect ratio (width 6 μm, thickness
18 μm) attached to proof mass making it a complex MEMS
structure. SOI made Electrostatic ears providing separate
zones for biasing (upper and lower side) generate clouds of
fringe field pushing or pulling the main structure UP and
DOWN providing tuning mechanism with the help of 04 cantilever structure attached to proof mass.
In Fig. 7, deflection is plotted against bias voltage
applied to lower side of SOI wafer for seeing the electrostatic field actuation on the movable proof mass of
the accelerometer. We see that deflections are of nonlinear type and are almost same for both the software.
The argument for this has been explained previously. A
2nd order polynomial fit for deflection vs. voltage is
carried out as shown in Eq. 14.
y ¼ 6:74e−4 −3:61e−6 x þ 7:31e−5 x2
Fig. 6 Normalized deflection of tip of micro accelerometer with variation
in mesh sizes
ð14Þ
where y is deflection in μm and x is tuning voltage in
volts. The Eq. (14) is valid for our structure only. For a
rectangular electrode set as shown in Figs. 3 or 5,
above second degree type of polynomial relation can
predict defelection as a function of actuation voltage,
however, the value of constants depends on geometry
of the electrodes. The idea is to show that the deflection
is non linear as a function of tuning actuation voltage.
Silicon
Fig. 7 Effect of tuning Voltage
(Vt) on deflection of micro
accelerometer for COMSOL and
COVENTORWARE
4.2 Boundary Conditions and Deflection in ‘x’, ‘y’,‘z’
Directions
Many MEMS fabricated devices in the form of cantilever,
crab, diaphgram, plates suffer the residual stresses effects in
the form of curling up or down and hamper the performance of
the devices. To tackle and tune these adverse effects, earlier
axial tuning analysis technique was proposed by us for in field
tuning of cantilevers based resonators [19]. Next, we present
the defelection results for the microaccelerometer shown in
Fig. 8.
In Fig. 8a, complete set of micro accelerometer is shown
along with SOI electrodes walls inside spherical air
domain.This outer domain is required to see the effects of
fringe field. COMSOL multiphysics software is utilized for
fringe field simulation effects. In Fig. 8b,100 V bias is applied
on upper side which pulls the movable strucure upside and in
Fig. 8c pulls it downside providing proof of concept of electrostatic fringe field actuation. COMSOL multi Physics SW is
used for these simulations along with COVENTORWARE
software for comparison and verfication.
A perfect insulating surface is modelled using zero charge
boundary conditions making sure the electrical field lines are
tangential to the boundary of sphere in Fig. 8a. To clearly
highlight the applications of boundaries condition applied, in
Fig. 9, the zero-charge node used in COMSOL simulation
adds the condition that there is zero charge on the spherical
boundaries (Fig. 8a) so that (n. D = 0) where ‘n’ is normal to
surface ‘D’ of sphere defined in simulation. At interior boundaries it means that no displacement electric field can penetrate
the boundary and that the electric potential is discontinuous
across the boundary. Field lines are confined inside the domain of the sphere used for simulation.
In addition, zero potential (ground) was applied to electrode position 1,3,4 as shown in Fig. 9a. Electrode 4 is just
below electrode 3 separated by Box layer (as shown in Fig. 3).
Furthermore, another boundary condition, a mechanical fixed
constraint node was added as so the geometric entities (electrodes 1,2,3,4 and torsional beam ends) fixed (fully
constrained) i.e. displacement is zero in all the directions.
Tuning voltage (Vt) has been applied to electrode position 2
as shown in Fig. 9b.
For further investigation for deflections in other directions ‘x’ and ‘y’ side due to bias applications of 100 V on
lower or upper side, simulations carried and results seen as
shown in Fig. 10a-f. Figure 10a shows deflection in ‘x’
direction on full structure. 10(b) and 10(c) showing equal
deflection (0.0012 μm) on zoomed in fingers section on
either side. This verifies the effect of similar magnitude
of electrostatic field in x direction. Figure 10d shows deflection in ‘y’ direction (0.007 μm) on full structure whereas 10 (e) and 10(f) showing fingers and torsional beam side
respectively. The deflection of (0.007 μm) which can be
seen near combs fingers is exactly same (0.007 μm) backside near torsional beam. The complete deflection in y
direction is verified by this measurement on both the ends.
A symmetric defelction in ‘x’ and ‘y’ ensures the correct
application of bias on electrodes on both side of movable
structure. A detailed tolerance analysis due to fabrication
errors has been given in our earlier reference [20].
Additionally relevant tolerance analysis which is possible
due to fabrication constraint is near the SOI electrodes
Silicon
Fig. 8 a Structure inside spherical
air domain b ‘z’ deflection when
bias applied on upper and c lower
electrodes of SOI wall 1
(a)
(b)
zones. So, Table 3 shows the effect of gap variation (h and
s) between electrodes on deflections at 100 V application.
The normal design position is having ‘h = 5 μm and ‘s =
4 μm.’ as shown in S.no 2 in Table 3. We see that gap ‘h’ has
substantial effect on the deflections of the accelerometer in ‘z’
direction giving a 7.98% reduction when gap is increased
from 5 μm to 6 μm. There is an increment of 12.37% in
deflection when ‘h’ gap is reduced from 5 μm to 4 μm.
There is decrease of 2.59% when ‘s’ increased from 4 μm to
5 μm and increment of 2.19% as gap ‘s’ decreased from 4 μm
to 3 μm respectively. So, in summary, gap ‘h’ location is more
critical for fringe field effects in this study.
All the above results show the possibility of out of plane
movement using fringe field electrostatic movment assuming
no curling in proof mass. In next section, we show the results
assuming curling induced due to stress gradients and how its
undesirable effects can be compensated or tuned.
(c)
4.3 Tuning of Residual Stress Using Electrostatic
Actuation (Analytical Vs Simulation)
Next shown in Fig. 11 are two microaccelerometer proof mass
structures. The one on left side is the normal or ideal position
of the microaccelerometer structure wherein SG is zero and
the one on right is curled up micro accelerometer structure due
to stress gradient. To tune this up deflection or curl up, we
have used the concept of SOI based fringe field electrostatic
actuation as discussed in Section 3.
In Fig. 12a we see that for various stress gradients different
tip deflections of structure are there which can be tuned with
different magnitude of voltage appplication. For example for
for SG of 0.2 MPa/μm, intial tip deflection is about 0.755 μm
@Vt = 0 V as shown in Fig. 12b. This deflection can reduced
or tuned electrostatically by Vt =60 V to less than 0.5 μm.
This will be discussed further. Stress gradient from 0.1 MPa/
Fig. 9 a Electrical ground applied
on (1,3,4 electrode) and in 9 b.
tuning voltage applied on
electrode 2
a
b
Silicon
Table 3 Relevant deflection in
x,y,z direction due to change in
gap near electrodes @100 V
S.No
1.
2.
3.
4.
5.
Voltage
applied
Variation in gap ‘h’
and ‘s’
Deflection in
‘x’in μm
Deflection in
‘y’in μm
Deflection in ‘z’
in μm
100
100
100
100
100
s=4, h=4
s=4, h=5
s=4, h=6
h=5, s=5
h=5, s=3
0.010
0.012
0.008
0.008
0.004
0.017
0.007
0.005
0.007
0.006
0.563
0.501
0.461
0.488
0.512
μm to 0.2 MPa/μm has been drawn with different deflection
magnitude. Here we have shown only when the deflection is
up due to positive SG which are brought back to normal near
to flat deflections. Important dimension which have been used
while using Eq. 6 are wc = 16 μm and tc = 9 μm which are
width and half thickness of movable cantilever cross section.
An approximation of fringe field applied electrostatic voltage
force has been utilised.This has been used for mathematical
force calculation. Further it has been multiplied by 4 as four
set of electrodes work simultaneously and divided by the
spring constant of the structure (1.58 N/m) for deflection estimation. This spring constant tries to retard the effect of electrostatic force being applied for tuning.Simulation results
showing tuning of deflection in proof mass due residual stress
generated curling are shown next in Fig. 13. Curling effects
due to generally observed stress gradient of the order of
0.1 MPa/μm are tuned by electrostatic voltage of 60-75 V
applications. Due to positive or negative stress gradient [32]
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 10 x and y side deflection on application of bias voltage on wall 1
only. Figure(a) showing symmetric deflection in x direction of complete
structure, (b) showing deflection shift in left side of fingers zoomed in (c)
deflection in right side of fingers zoomed in, (d) symmetric deflection of
complete structure in y direction, (e) deflection of fingers coming out in y
direction zoomed in, (f) deflection of backside torsional beam in y
direction
Silicon
Fig. 11 Normal micro
accelerometer and SG induced
vertical curl up deflection in proof
mass of micro accelerometer
structure gets curled up or down in fabrication. Stress is
enevitable issue in MEMS and NEMS fabrication, but can
be tuned or compensated by current proposed new tuning
method. Here, the thickness of the movable micro accelerometer is 18 μm. Initially deflected proof mas including its fingers is deflected by 0.36 μm at the tip (Fig. 13a) and idea is to
align it back to normal horizontal position using electrostatic
force with the small steps of actuation voltage applied to
tuning electrodes from 0 to 60 V. We can see as the electrostatic force is increased by applying 40 V, 50 V and 60 V the
deflections reduces to 0.24 μm,0.17 μm and 0.09 μm respectively and structure aligns towards horizontal axis (Fig. 13b, c
and d).
It may be mentioned here that present tuning method does
not uncurl the stress induced bending, instead whole proof
mass can be moved up or down near the finger area, the effect
is to match the overlap area between the fingers to its original
position and this results in compensating the stress gradient
effects. The structure can still respond to deflections induced
by applying ‘g’ force in presence of this actuation voltage and
tunning electrodes do not interfere with basic functiong of the
accelerometer as will be shown in subsquent sections.
Dependence of electrostatic force on actuation voltgae is of
complex nature due dominating fringe field effects. Figure 14
shows the variation of electrostatic force as a function of
tunning voltage. We also found the analytical Eq. 15 (other
than Eq. 6) of electrostatic force acting on the proof mass of
Fig. 12 a. Deflection due to
various stress gradients and
tuning voltage and 12 b. Initial tip
deflection (0.755 μm) due to
stress gradient SG = 0.2 MPa/μm
@Vt = 0 Volt
our accelerometer structure by fitting the simulated results
shown in Fig. 14. Here we have shown the polynomial fitting
of the data in red curve. The electrostatic induced actuation
force either uplifts or downs the structure completely by coupling the force near the four ear electrodes depending upon the
location of application of bias voltage. The force generated in
our case is given by
F e ¼ A þ BeCV
ð15Þ
where Fe is in μN, A = −0.037 μN, B = 0.031 μN and C =
0.025 V−1 and V is varying applied voltage.
Figure 15 shows the comparison (analytical vs simulation)
tuning of deflection caused by residual stress by electrostatic
force by applying voltage. An initial deflection due to stress
gradient (SG) of approximate 0.36 μm is tuned to <0.1 μm by
increasing voltage upto 60-70 V. There is a marginal maximum disagreement of approximate 4.72% and minimum
2.47% between analytical and simulated results w.r.t
analytical.
Further we have seen that for this stress gradient (0.1 MPa/
μm) there is a bounce back of the structure beyond
70 V,which is due to increase in influence of electrostatic
force near the proof mass of the micro accelerometer. The
proof mass starts bending downward near the center which
starts uplifting the finger end of the structure again leading
to increasing deflection at the end of fingers.
a
b
Silicon
(a)
(b)
(c)
(d)
Fig. 13 Tuning of residual stress by electrostatic force by voltage biasing
(CW software). a showing initial deflection 0.35 μm due to SG (0.1 MPa/
μm), b 40 V tuning voltage reduced deflection to 0.24 μm, c 50 V applied
to reduce deflection to 0.17 μm and d showing application of 60 V to
reduce deflection further to 0.09 μm
4.4 Effects of Tuning on Sensitivity, Bandwidth and
Range of Operation
Fig. 14 Plot of force on proof mass due to increasing tuning voltage
To take care of any adverse or beneficial impact of this SOI
electrostatic tuning, effects of electrostatic tuning of residual
stress on the sensitivity, bandwidth and range of operation of
the micro accelerometer is discussed further.
Figure 16 shows the change in capacitance as a function of
applied ‘g’. The details of this behaviour have been earlier
discussed in detail in [20]. It is clearly shown in Fig. 16 that
sensitivity reported in [20] was without any effect of residual
stress (C1 reference and C2 reference), which got little disturbed due to residual stress (C1 with SG (0.1 MPa/μm) applied. We see that with residual stress gradient applied the
structure either curled UP or DOWN. Mean stress as well as
stress gradient affect the deflection of the structure [32] and
Silicon
Fig. 15 Comparison of analytical and simulated data for tuning by
electrostatic actuation
mainly hamper sensitivity. In practical situations, there are
factors affecting residual stress gradient (SG) like thickness
and in shape dimension along with material stiffness and elongation. Cases where SG (4 MPa/μm) has been reported [34,
35], we considered full range effect of these higher stress
gradient also and studied their adverse effect on tuning on
our micro accelerometer.
Figure 17 shows close view where change in capacitance
(del C, C1-C2) is plotted against acceleration applied. In the
three curves shown, bottom most is the normal condition
where there is no stress gradient applied in the 18 μm proof
mass structure and capacitance change is plotted. The top
curve showing a change in capacitance when a stress gradient
Fig. 16 Change in capacitance with reference, with stress gradient and
tuned voltage
Fig. 17 Effect of tuning voltage (75 V) on sensitivity of lower SG
(0.1 MPa/μm)
of 0.1 MPa/μm is applied and a small slope change near 25 g
can be seen easily. The slope of uppermost curve is 47.58 f
F/g, and the exact calculated slope is 45.73 f F/g. This change
in slope introduces a non-linearity of 4.04% in the sensitivity
curve. Now we see that this is a perturbed condition due to SG
which has although increased the sensitivity but has introduced unwanted non linearity which is fatal as far as accelerometer designs are concerned. So, in the middle curve shown
is the application of our concept discussed here, where upper
curve can be brought down or tuned back nearing normal
position eradicating non linearity as well.
Similarly in Fig. 18, a more relevant tuning can be seen
applying same concept against higher stress gradient (4 MPa/
μm). We see that lower curve is normal one without any stress
gradient, which was disturbed heavily by SG and a non-
Fig. 18 Effect of tuning voltage (200 V) on sensitivity of higher SG
applied (4 MPa/μm)
Silicon
linearity in higher range of operation (beyond 20 g) has been
introduced which is undesirable for the functioning of the
device. Then SOI based lower electrode has been used and a
voltage of 200 has been applied for tuning the non- linearity as
well as shift in the sensitivity. We see that normal (original)
sensitivity when SG was zero was 46.09 f F/g which got
shifted to 40.33 f F/g due to effect of stress gradient (4 MPa/
μm). Now after electrostatic fringe field tuning it has been
brought to 47.85 f F/g. This implies that a tuning range as
high as 18.64% has been possible in sensitivity. So here we
claim that by utilizing this new SOI electrode-based fringe
field bi directional tuning a number of devices like accelerometers, cantilever-based sensors and actuator, micro mirror assemblies can be tuned effectively.
We observed that stress also affected the range of operation
for higher values [32] which can be tuned back by applying
bias voltage. Tuned curve by 75 V and 200 V have been
shown which reduce the effects of curling up or down. So
along with reducing the effect of residual stress due to fabrication, range of operation is also taken care. Figure 17 shows
there is bit deviation in slope of sensitivity beyond 25 g range
in SG (0.1 MPa/μm) applied curve and in Fig. 18 beyond 20 g
range in SG (4 MPa/μm) curve. By SOI electrostatic tuning
these changes in slope is also reduced in this range (Figs. 17
and 18).
Figure 19 shows there is negligible effect of tuning voltage
on the Bandwidth (taken at 30 g for maximum influence) of
the structure (3–4 Hz only) ensuring minimum effect of external tuning voltage which is good for dynamic performance
even in external force environment influence. These results
are at optimised value of meshing discussed in Section 4.1
of previous discussion with COVENTORWARE software.
A comparison in Table 4 between parameters of micro accelerometer (MA) with and without Stress gradient effect and
tuning has been shown for completeness of the study.
Equation 13 has been used for calculating 3 dB bandwidth.
In short, we see that inclusion of four cantilever ear to proof
(two on each side) mass does not influence the otherwise
optimised performance of the MA significantly. Little increment in mass just perturbs the system natural frequency
Table 4 Comparison between
normal optimised micro
accelerometer (MA) performance
parameters and micro
accelerometer (MA) with tuning
electrodes added
Fig. 19 Effect of tuning voltage on resonant frequency of the
accelerometer
without much influence on Bandwidth as well. Range of operation is also utilized fully by tuning the structure for required
sensitivity. Due to the differential concept in capacitance used,
almost negligible cross axis sensitivity will be there as reported earlier in [20].
In short, addition of above fringe field based tuning electrodes to proof mass of main micro accelerometer can be very
useful for reducing the undesirable stress gradient effects.
5 Conclusions
A new concept of vertical bi-directional fringe field based
electrostatic tuning using SOI wafer is reported. Differential
micro accelerometer is tuned for reducing the effects of residual stress gradient induced due to fabrication process. Low
stress gradient of the order of 0.1 MPa/μm is tackled using
60–75 V of bias whereas 200 V is used for higher stress
gradient (4 MPa/μm) applications. An effective electrostatic
fringe field tuning for sensitivity of accelerometer upto
18.64% is reported. A maximum deviation of 7.69% in
S.No
Parameter
Normal MA without
tuning electrodes
MA with tuning electrodes
1.
2.
3.
4.
5.
6.
Proof mass configuration
Mass
Resonant frequency
3 dB, Bandwidth
Sensitivity
Range of operation
No Ear
29.9 μgram
1157.22 Hz
306 Hz (analytical)
48.22 f F/g
30 g
7.
FEM Software used
COVENTORWARE
With Ear (04 Nos. cantilever)
30.16μgrams
1152.20 Hz
305 Hz (analytical)
40.33f F/g
30 g (non-linearity induced due to SG)
and then tuned electrostatically
COMSOL and COVENTORWARE
Silicon
deflection while comparing both the softwares
(COVENTORWARE and COMSOL Multiphysics) is presented. Maximum disagreement of 4.72% between analytical and
simulated results promises the design of tuning concept.
Negligible (3–4 Hz) change in dynamic response (bandwidth)
is observed. New reported SOI wafer used for tuning of MEMS
structure is promising for existing and futuristic applications.
6.
7.
8.
9.
Acknowledgements Authors thank Director SSPL, IDST and IIT Ropar
for allowing to write and publish this work on SOI electrode- based
tuning of micro structures.
Author Contributions Manoj kumar Dounkal has carried out all the back
research simulation work at basic level.
Dr. R.K. Bhan has conceptualised the theme of this research work.
Dr. Navin Kumar has thoroughly revised and updated the changes
required.
Data Availability All data generated or analysed during this study are
included in this published article [and its supplementary information
files].
10.
11.
12.
13.
Declarations
Conflict of Interest Authors declare that there is no conflict of interests.
Ethics Approval All ethical compliances have been ensured for submitting this research quality and quantity work. All the work discussed here
is an original idea with fresh results.
14.
15.
Consent to Participate Here, we wish to participate in this journey of
research work through your esteemed journal.
16.
Consent for Publication We also provide our consent and interested in
getting this research work published.
17.
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