Stability of Symmetrical Comb-Drive Actuator

Abstract

This paper reports the study, design, and simulation of a symmetrical comb-drive actuator. The approach for definition of the potential energy of the system is proposed. The electrical parameters of the comb-drive actuator are defined in COMSOL Multiphysics® software. Depending on an actuation voltage and an initial design it can form system with one, two, and three stable states. We show that the equilibrium at x = 0 is more stable for the comb-drive actuator with positive overlap than for device with the gap of the same value. The proposed approach will be used for design of the symmetrical actuator, which forms the output of the recently proposed contactless four-terminal MEMS element for capacitive adiabatic logic based on silicon MEMS technology.

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Stability of Symmetrical Comb-Drive Actuator
To cite this article: A Galisultanov et al 2019 J. Phys.: Conf. Ser. 1407 012087
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PowerMEMS 2018
Journal of Physics: Conference Series 1407 (2019) 012087
IOP Publishing
doi:10.1088/1742-6596/1407/1/012087
1
Stability of Symmetrical Comb-Drive Actuator
A Galisultanov1, G Pillonnet1, Y Perrin1, L Hutin1, P Basset2, and H Fanet1
1 Univ. Grenoble Alpes, CEA, LETI, F-38000 Grenoble, France
2 Université Paris-Est, ESYCOM, ESIEE Paris, France
gael.pillonnet@cea.fr
Abstract. This paper reports the study, design, and simulation of a symmetrical comb-drive
actuator. The approach for definition of the potential energy of the system is proposed. The
electrical parameters of the comb-drive actuator are defined in COMSOL Multiphysics®
software. Depending on an actuation voltage and an initial design it can form system with one,
two, and three stable states. We show that the equilibrium at x = 0 is more stable for the comb-
drive actuator with positive overlap than for device with the gap of the same value. The
proposed approach will be used for design of the symmetrical actuator, which forms the output
of the recently proposed contactless four-terminal MEMS element for capacitive adiabatic
logic based on silicon MEMS technology.
1. Introduction
Smooth (adiabatic) switching between logic states is used for recover the signal energy to power
supplies [1]. Unfortunately, the available hardware is a key problem to use the full benefits of
adiabatic logic. For example, energy efficiency in CMOS-based adiabatic logic is limited by an
inherent trade-off between the dynamic and leakage losses [2] and remains a few decades higher than
the theoretical Landauer limit for irreversible logic (3 zJ at 300 K) [3]. The electromechanical relays is
another candidate for adiabatic logic realization [4]. Thanks to metal-metal contact instead of a
semiconductor junction, the leakage becomes almost negligible except in the case of nm-scale
electrostatic gap. However, the main bottleneck of the relay-based adiabatic logic is the mechanical
reliability and performance limit of the scaled switches, due to adhesion force of contact interface [5].
To overcome this limitation, we recently proposed a new logic family called Capacitive Adiabatic
Logic (CAL) [6-7]. Thanks to smooth switching process in adiabatic logic, the resistive elements
(transistors, relays) in a voltage divider circuit can be replaced by capacitive ones. The capacitor value
can be modulated by the variation of relative permittivity, plate surface and gap thickness. In present
paper we discuss an electrostatic actuator as a possibility for the integration of the MEMS relays in
VSLI circuits has already been demonstrated [8].
Electrostatic actuation is widely used in MEMS technology due to simplicity and low power
consumption [9]. There are two main types of electrostatic actuators: gap-closing and comb-drive (c.f.
Figure 1) [10]. The gap-closing MEMS variable capacitor could be a good candidate for CAL purpose
as it offers large capacitance variation if the actuation voltage is higher than pull-in voltage [11].
However, a mechanical contact is required in order to have a high capacitance variation.
Consequently, this solution suffers from a collapse when the actuation voltage exceeds pull-in voltage
during capacitance increasing and from damping losses during capacitance decreasing (release of the
top electrode) [12]. The both cases lead to a loss of control over the moving mass and cause the non-

PowerMEMS 2018
Journal of Physics: Conference Series 1407 (2019) 012087
IOP Publishing
doi:10.1088/1742-6596/1407/1/012087
2
adiabatic losses, which is independent of the operating frequency and cannot be suppressed by the
ramping time increasing. On the contrary, the comb-drive MEMS variable capacitor avoids electrical
and mechanical contacts [13]. Despite the gap symmetry, the comb-drive actuator designed for high
displacement can collapse in lateral direction, perpendicular to the intended travel direction, when
voltage–deflection conditions are favorable [14]. As presented in Figure 1, the same stability issue
appears in gap-closing capacitor with a central moving electrode and two fixed electrodes that enclose
it. For the comb-drive transducer, the problem is usually solved by increasing the mechanical stiffness
in lateral direction.
Figure 1. Stability problem for different types of gap-closing and comb-drive actuators.
The proposed in work [15] MEMS buffer element for CAL is presented in Figure 2a. The device
consists of the moving mass m with two insulated electrodes (GND, D) and two fixed electrodes (G,
S). The moving mass is suspended by four identical springs with total spring constant k. The two pairs
of fixed and moving electrodes form an input and an output comb-drive capacitive transducers. The
input (left) transducer has an initial overlap Lin between the fixed and the moving electrodes. The
output transducer (right) is symmetric and has initial gap at the output (Lout≤0).
Figure 2. Design of the MEMS buffer element for CAL. b) Data and energy transfer in four-phase
quasi
-adiabatic pipeline PC. The maximal PC voltage is denoted as VDD. c)
Electrical schematics of
CAL buffer chain
. d) Evolution of VG0 (first graph), VG1 and VDS1 (second graph), and
moving mass
displacement x1 (third graph).
Adiabatic logic circuits must be able to receive the logic state from the previous gate, process it and
transmit the result to the next gate, i.e. to be cascadable. They basically operate with four-phase power
supplies, called power clocks (PC’s) [14]. These PC's can provide and recover energy, i.e. realize
charge recovery. As shown in Figure 2b, PC's have a π/2 phase shift and synchronize the transfer rate
and information processing. The cascadability of the proposed 4-terminal device is depicted using an
array of buffer elements. Figure 2c depicts the buffer chain circuit. The binary input logic sequence
"01" is transferred through the buffer, as illustrated in Figure 2d. The logic state in further gates is
coded by the moving mass displacement x1, induced by the input voltage VG0. It is important to avoid
of self-actuation by the output voltage VDS1 when VG0 is less than the threshold voltage VTH. When VG0

PowerMEMS 2018
Journal of Physics: Conference Series 1407 (2019) 012087
IOP Publishing
doi:10.1088/1742-6596/1407/1/012087
3
is higher than VTH, x1 is higher than xTH. The high logic state should be maintained during the input VG0
decreasing thanks to the output electrostatic force. Consequently, the gate transmits high state to the
next element of the buffer chain (VG1>VTH).
In this work we would like to investigate the stability in indented direction of symmetrical comb-
drive actuator, which forms the output of the MEMS buffer element to avoid false logic states during
the information transfer. The FEM simulation results, which take into account the fringing field effect,
are compared with the parallel plate approximation results. The lateral stability problem is out of scope
of this paper.
2. Symmetrical comb-drive actuator
We isolate a single symmetrical comb-drive actuator. In order to differentiate the analyzed device
from the MEMS buffer we change the notations here. The transducer capacitance is Cout and the
voltage across actuator is Vout. Figure 3a represents a top view of the segment of comb-drive actuator.
The parameter of the combs are taken from [9]. The length of the comb L is 25 um, the width w is 2
um, the thickness t is 40 um, the air gap g is 2 um, and the number of output fingers Nout is 220. Here
we discuss the case of the thick device, i.e. Lout<<t and g<<t.
Figure 3. a) Top view of the segment of symmetrical comb-drive actuator (L
b
= 2 um). b)
Comparison of analytically calculated
(dashed) capacitance Cout and normalized
electrostatic force
2
Fe/(Vout)2=dCout/dx with FEM simulation results (solid) for Lout = 2 um. c) 3D COMSOL
model
of the output comb
-drive segment without initial overlap. d) Cout as function of x for the
different
overlap Lout.
2.1. Analytical model
The electromechanical transducer has one electrical and one mechanical port. The electrical part
consists of two electrical terminals. Based on a 1st order parallel-plate approximation, the output
capacitance Cout can be defined from:
0
2( )/,if
,if
p out out out
out
p out
CNtxL gxL
CCxL
H
 t
°
®
°
¯
, (1)
where ε0 = 8.854·10−12 F/m is the vacuum permittivity and Cp is the parasitic capacitance, which is
zero if there is a gap between the combs (Lout<0) and equals 4Noutε0tLout/g if there is a initial overlap
between the combs (Lout>0). The mechanical part of the system and can be described by the following
equation of motion:
mẍ = – bẋ – kx + Fe, (2)
where m is the mass of the central moving part, k is the spring constant, and b is the damping
coefficient. The electrostatic force Fe is proportional to a derivative of the output capacitance Cout with
respect to displacement x and can be calculated from (3). The results of analytical calculation of
capacitance Cout and normalized electrostatic force 2Fe/(Vout)2 are presented in Figure 3b (dashed line).

PowerMEMS 2018
Journal of Physics: Conference Series 1407 (2019) 012087
IOP Publishing
doi:10.1088/1742-6596/1407/1/012087
4
2
0
2
0
/,if
0, if
/, if
out out out
e out
out out out
VN tg x L
FxL
VN tg x L
H
H
d
°
°

®
°t
°
¯
. (3)
2.2. COMSOL FEM model
As represented in Figure 3c, the full 3D model of the segment of symmetrical comb-drive actuator
with and without initial overlap is implemented in COMSOL Multiphysics® software. AC/DC
Module is used to define the capacitance and the electrostatic force affecting to the moving electrode.
The parameters of the model are the same as in the analytical calculation. Comparison of analytically
calculated (dashed line) capacitance Cout and normalized electrostatic force 2Fe/(Vout)2=dCout/dx with
FEM simulation results (solid line) for Lout = 2 um are shown in Figure 3b. FEM calculated
capacitance is higher than analytically calculated due to fringing field capacitance, which is ignored in
the parallel plate approximation. Then, we observe presence of the electrostatic force when |x|<|Lout|, as
Cout is not constant against displacement in this region. Figure 3d presents the capacitance of the
symmetrical comb-drive actuator as function of the moving electrode displacement for the different
overlap Lout.
2.3. Potential energy of the system
To study the stability of the symmetrical comb-drive actuator during logic state transfer, we analyze
the potential energy distribution as function of the output voltage Vout and initial overlap Lout. Equation
(4) defines the potential energy U(x, Vout):

22
( ) (0)
(, ) 22
out out
out out
CxC
kx
UxV V


, (4)
There are two contributions to potential function: a positive parabolic term, representing the linear
mechanical spring, and a negative parabolic term, which accounts the electromechanical coupling.
Plotting U(x, Vout) versus x provides an interpretation of the non-linear dynamical behavior in terms of
a potential well (c.f. Figure 4a-b). For calculation we have used both analytically (Figure 4a) and FEM
(Figure 4b) calculated capacitance Cout for Lout = 2 um.
Figure 4. a-b) The potential energy distribution U(x, V
out
) as function of the applied voltage Vout
(
Lout = 2 um) for a) analytical and b) FEM calculated Cout. The parameter k
=2.39 N/m is taken from
[15]. c) Normalized Vf0/Vact and Vf1/Vact as function of the normalized initial overlap Lout/g.
Within parallel plate approximation, electrostatic force is zero, when |x|<|Lout|. Consequently, the
output voltage increasing do not change potential energy in this displacement range. Thus, "0" state is
stable, and cannot be switched by the output. On the other hand, as presented in Figure 1d, "1" state is
not maintained if the output voltage less than 11.6 V. Let us denote this voltage as Vf1. The potential
curve for FEM model, which accounts the fringing field, is more complicated. Firstly, the false "1"
voltage also exist and equals 21.2 V. Secondly, due to presence the electrostatic force in |x|<|Lout|
region, the voltage increasing can destroy equilibrium at x = 0 (c.f. Figure 4b). We denote the minimal

PowerMEMS 2018
Journal of Physics: Conference Series 1407 (2019) 012087
IOP Publishing
doi:10.1088/1742-6596/1407/1/012087
5
required output voltage, which eliminates stable minimum at x = 0, as Vf0. The analysis of potential
curves, based on FEM simulation, allows us to identify these two voltages for different initial overlaps
Lout. These voltages are critical to using of the symmetrical comb-drive actuator as the output of the
CAL MEMS buffer. The false "0" voltage Vf0 is the upper bound of the output voltage, which
guarantee the absence of false switching from "0" to "1" (c.f. Figure 2d). The false "1" voltage Vf1 is
the lower bound of the output voltage, which guarantee the maintaining of the high state during
decreasing of the input, i.e. false switching from "1" to "0" (c.f. Figure 2d). When the overlap is
negative or equal to zero these voltages are equal and there is no possibility to build a MEMS buffer
logic gate. As presented in Figure 4c, the overlap increasing allows to split these voltages and create
the working region for MEMS buffer. In the graph this voltage is normalized to Vact, equals to the
voltage required to cause the displacement equal to Lout for unsymmetrical comb-drive transducer with
the same number of electrodes (Nout).
The received results will be used for design of recently proposed contactless four-terminal MEMS
buffer element for capacitive adiabatic logic.
3. Conclusion
The stability problem of symmetrical comb-drive transducer in presence of fringing field is
investigated analytically and numerically in COMSOL Multiphysics® software and applied to the
sizing of MEMS variable capacitance for CAL. The instability in the intended travel direction is
observed. The approach for definition of the potential energy of the system is proposed. The using of
the symmetrical comb-drive transducer with positive overlap allows to create a logic gate capable to
transfer high and low logic states.
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