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Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Analysis and simulation of MOS
capacitor
feedback for stabilizing electrostatically actuated
mechanical devices
J.I. Seeger, S.B. Crary
Center for Integrated Sensors and Circuits, University of Michigan,
Email: joseeger@eecs. wnich. edu
Email: crary(a)umich.edu
Abstract
We review our earlier work that explained how a series capacitance implements negative
feedback and stabilizes electrostatically actuated MEMS devices against the well known
"pull-in" instability. We demonstrate through analysis and computer simulation that stability can be achieved using a series MOS capacitor. We also show that using MOS capacitor feedback can reduce the supply voltage required to deflect the mechanical members
and can introduce both an electrostatic deflection limit as well as hysteresis.
1 Introduction
One of the difficulties inherent in the use of electrostatically actuated MEMS devices; such as pressure sensors, accelerometers, or comb-drive actuators; is an instability known as "pull-in" that causes the oppositely charged mechanical members to snap together. This instability effectively prohibits stable operation of
many devices over their full range of mechanical deflection.
In a previous paper by Seeger [1], we showed that simply placing a capacitive element in series with the electromechanical device provides negative feedback that extends the stable operating region, with a concomitant increase in the
''pull-in voltage." For a device with linear restoring forces, addition of a series
capacitance less than one-half that of the nominal device capacitance stabilizes
the device over the entire gap.
We now present theory and results of computer simulations showing how a
series MOS capacitor can be used effectively for stabilization.
2 Device Model
Throughout the analysis, we use the model of a parallel-plate capacitor with one
free electrode and one fixed electrode, shown in Figure 1. Assuming stiff elec-
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Microstructures
trodes and ignoring fringingfields,the capacitance between the electrodes is approximated as C = ^r , where e is the absolute permittivity of the material
between the electrodes, A is the
area of the capacitive plates and x is
the distance between the electrodes.
The undeflected gap size is d.
The free electrode is subject
to an electrostatic attractive force,
approximated by fe(x) = ^p-,
where V is the voltage between
the electrodes. The free electrode
is subject to an external force /^,
due, for example, to a pressure
differential or an accelerating reference frame. The electrode is also
subject to a linear restoring force,
Figure 1: Device Model. Parallel
K(X) = kd - kx, and a damping
plate capacitor with gap x.
force, ()(x) - bx.
3 Constant- Voltage Operation
It is well-known that for a constant voltage applied across the device, electrostatic actuation can cause the free electrode to deflect no more than one-third of
the nominal gap. Applied voltages larger than the pull-in voltage, %,,- = 7g&,
where 7 = \/~^- and XQ = d — f^xt/k is the gap-size due solely to the external
force, cause the free electrode to move beyond the stable range and snap against
the fixed electrode. These results follow from the equilibrium position and the
potential energy of the device.
For a device in equilibrium, the relationship between applied voltage and
equilibrium position, %*, is
Figure 2 is a plot of this equation which shows that the system has two equilibria
for voltages less than pull-in and no equilibria beyond pull-in.
Figure 3 shows plots of the potential energy for the device under constant
applied voltages. For voltages less than %,,, there is only one local minimum, or
stable equilibrium position; the local maximum corresponds to the other equilibrium position, which is unstable. An equilibrium position corresponds to a minimum in potential energy if
dx
dx
37*3
0.
(2)
This equation combined with equation (1) results in the stability condition, ~ >
|. Thus, using constant-voltage, electrostatic actuation the free electrode cannot
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Microstructures
0.15
0.10
0.05
^0.00
JC
stable
equilibria
-0.05
-0.10
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x/d
V/y
Figure 2: Equilibrium Position vs. Figure 3: Normalized Potential Energy
vs. Normalized Gap
Device Voltage
come to rest at any point between X/XQ = 0 and X/XQ = 2/3. Any attempt to
violate this restriction results in the electrodes snapping together.
4 Controlled- Voltage Operation
4.1 Capacitor Feedback
A capacitor, C/, in series with the device (Figure 4) provides stabilizing, negative
feedback. Since the circuit is a voltage divider, the voltage across the device is
v;
(3)
1 + ^7
is the zerowhere K = §°-and Co =
voltage capacitance of the device. If the free
Figure 4: Device with Capacelectrode begins to snap, the capacitance of
itor Feedback
the device increases which tends to reduce
the voltage across the device. This in turn, causes the electrodes to separate, opposing the original motion.
The relationship between supply voltage and equilibrium position is found
by combining equations (1) and (3):
(4)
The stability condition becomes
(5)
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Microstructures
which combined with equation (4) proves that equilibrium gaps ~ > ^- are
stable. Thus for K > 2 or equivalently C/ < Co/2, all operating states are stable.
For K < 2, an unstable region exists, but the stable operating range is extended.
Figure 5 shows that the potential energy for the device with the feedback
capacitor (K — 2) has stable equilibria over the entire gap.
0.1 0.2 0.3 0.4 0.5 0.6 07 0.8 0.9
x/d
Figure 5: Capacitor Feedback (K — 2): Normalized Potential Energy vs. Normalized Gap
With capacitor feedback, the critical pull-in voltage is
f A'7
K >2
(6)
which for K = 2, is a factor of 3\/3 greater than the pull-in voltage for the device without feedback. In general, the device with feedback requires larger supply voltages to reach any equilibrium point (see Figure 8).
4.2 MOS Capacitor Feedback
A MOS capacitor operated in depletion
(Figure 6) can use lower supply voltages
to stabilize an electrostatically actuated
device because the MOS capacitor increases
the amount of negative feedback. As with
capacitor feedback, if the free electrode
starts to snap, the voltage on the device
drops and the voltage on the feedback device
increases. When the gate voltage on the
MOS capacitor increases, the depletion
region extends further. The effective MOS
capacitance decreases which tends to further
reduce the voltage on the device.
Figure 6: Device with MOS
Capacitor Feedback. I/ is
chosen to operate the MOS
capacitor in deep depletion.
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Simulation and Design of Microsystems and Microstructures
203
However, if the gate voltage is increased beyond the MOS threshold voltage, %, the MOS capacitor operates in the inversion region, which has the undesired effect of increasing the capacitance of the device. To avoid this situation,
the capacitor must be operated in deep depletion by applying a sufficiently large
positive voltage, % « % , to the source and drain of a MOSFET, see reference [2] .
4.2.1 Capacitance Model
Using the delta-depletion model of a MOS capacitor, the small-signal capacitance
is
Cmo
accumulation [Vg - V/b < 0]
depletion
inversion [%,>(% + %•)]
where V/&, the flat-band voltage, can be changed during processing through ion
implantation; the capacitance of the gate oxide is
Ag is the gate area of the MOS capacitor; t^ is the thickness of the gate oxide;
NA is the substrate doping concentration; eo is the permittivity of free-space; q is
the electronic charge constant; esi and c^ are the relative permittivities of silicon
and SiO], respectively, see reference [3].
In order to find the equilibria of the device with MOS capacitor feedback,
we need to examine the large-signal capacitance. Solving equation (7) for gate
charge, Q, and dividing by gate voltage, I/, gives the following large signal capacitance:
(8)
The capacitance in inversion has been left out for brevity. Both the large-signal
capacitance and small-signal capacitance are plotted in Figure 7.
The gate-to-source and gate-to-drain capacitances are neglected in the following discussion. For large gate areas, these parasitics can be neglected because
the gate-to-substrate capacitance dominates.
4.2.2 Equilibrium
With the MOS capacitor as the feedback device, the relationship between the supply voltage and the device voltage is found from the voltage divider equation,
Vs = V(l -f- ;£-), which can be rewritten in terms of V and x:
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Micro structures
Gate Voltage
Figure 7: Small-Signal (dashed line) and Large-Signal (solid line) Capacitance
of a MOS Capacitor
where Km — -jf-> Substituting the equilibrium relationship for V and x* from
equation (1) into equation (9) results in the equilibrium equation for the device
with MOS capacitor feedback.
4.2.3 Zero Flat-Band Voltage Behavior
If V/b = 0, the MOS capacitor always operates in depletion. The voltage across
the device is found from equation (9):
V=
(10)
Using this equation for the device voltage, the electrostatic force can be written as a function of supply voltage and gap-size, which leads to the value of Km
that meets the stability criterion and the corresponding pull-in voltage:
(11)
(12)
It can be shown that at K = Vpi^ , the small-signal MOS capacitance, ^- , is equal
to one-half the nominal device capacitance, Co- For 14/7 < 0.01, the critical
value of Km and the corresponding pull-in voltage are:
This implies that for a gate area, Ag < ^^ ^.- , the device will be stable.
The supply voltage needed to close the gap, Vpi^, is one-half of that required for
the device with capacitor feedback. Typical values of the capacitance ratio, Km,
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Micro structures
205
are less than'two, the critical value for capacitor feedback.
Figure 8 compares the equilibrium position versus voltage curves for the device with and without feedback. The curves corresponding to feedback each use
the appropriate critical value of the capacitance ratios. For the device with MOS
capacitor feedback several equilibrium curves are shown corresponding to different values of Vsh-
Figure 8: Equilibrium Position vs. Supply Voltage. Feedback devices are critically sized. Solid lines are for V£/7 = 0, 0.01, 0.1, and 1.0
4.2.4 Non-Zero Flat-Band Voltage Behavior
If the flat-band voltage is greater than zero, the equilibria and stability of the device with feedback can be changed significantly. If the voltage on the MOS capacitor is less than the flat-band voltage, the MOS capacitor is in accumulation and
has a constant capacitance. Thus, the device follows the capacitor feedback behavior. Since the value of Km is typically less than two, operation with the MOS
capacitor in accumulation has an unstable region. If the voltage on the MOS capacitor increases beyond theflat-bandvoltage, the depletion region grows and the
capacitance decreases, which increases the amount of stabilizing feedback.
By properly choosing V/& and Km, the MOS capacitor behavior can be used
to further reduce the supply voltage required to deflect the free electrode. The
flat-band voltage should be chosen so that the MOS capacitor switches from accumulation to depletion where the device would normally snap. Using the voltage divider relationship for the gate voltage and the pull-in voltage in equation
(6) leads to the choice for the flat-band voltage:
(13)
Km should be chosen so that the device is stabilized when the MOS capacitor
moves into depletion. Similar to the situation with capacitor feedback, this is
achieved if at zero gap, the ratio of the zero-voltage, device capacitance to the
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206
Simulation and Design of Microsystems and Microstructures
small-signal capacitance of the MOS
value is found according to
capacitor is greater than two. The critical
5v,
^_
^9 X_„
= 0 = Co/2 =» /C^l + \W/V, - ^,/V, = 2,
where
I^/%
=W
+ ^Tr (l - %) + (%)'
•
(14)
(15)
Equation (15), the voltage of the MOS capacitor at zero gap, the pull-in voltage,
was found by evaluating the equilibrium equation (9) at x = £* = 0.
The critical values of Km, Vjb, and Vp^^ can be determined by simultaneously solving equations (13), (14), and (15). There are exactly two values of Km
that give stability over the entire gap in this design. One solution, Km = 2, is exactly the same design as capacitor feedback. The other solution depends on V& and
7 and requires a numerical solution. This design requires lower supply voltages
than for capacitor feedback. Using this design, the device will have an unstable
region if the external force increases, so the design should be done assuming XQ
due to the largest anticipated force. Figure 9 shows the equilibrium curve for a
device with MOS capacitor feedback designed using the latter solution.
Different equilibrium behavior can be achieved by varying the values of V$,
Km, and V/&. Figures 11 and 13 illustrate two possible equilibrium curves.
4.2.5 Simulations
Using MATLAB [4], the behavior of a device was simulated with MOS feedback
designed according to the three previous equations. The lower plot in Figure 10
shows that the normalized supply voltage consisted of a constant voltage plus a
sawtooth waveform with amplitude 37.4mV/V and period 400^s. The upper plot
shows the normalized deflection of the free electrode. For the initial constant voltage of 585mV/V, which biases the MOS capacitor in depletion, the slight voltage
ramp causes the free electrode to move close to half of the gap distance. For comparison, after 800//6, the constant voltage is reduced to 400mV/V, which biases
the MOS capacitor in accumulation. Here, the same voltage ramp hardly affects
the position of the free electrode. This demonstrates that using a MOS capacitor
can significantly reduce the supply voltage required to move the free electrode.
The simulated device is nominally underdamped. When the feedback device
acts as a capacitor, the device remains underdamped. However, these simulations
show that for this type of design, when the MOS capacitor is biased in depletion,
the device becomes overdamped.
Next, the device was simulated with feedback that effectively prevents the
free electrode from coming in contact with the fixed electrode under electrostatic
actuation. The lower plot in Figure 12 shows that the supply voltage was a triangle waveform with 800/^s period. As the supply voltage increases, the device becomes unstable and snaps. This causes the voltage on the MOS device to increase
and drives it into depletion. Once the MOS capacitor is in depletion, increasing
or decreasing the supply voltage has little effect on the position of the free electrode. When the supply voltage is lowered sufficiently, the free electrode snaps
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Microstructures
1.0r
0.9
0.8
0.7
0.6-< 0.5
0.4]
0.31
0.4
0.1 !•
0L
0.5I
0.5
0.6
-0.4
0.2
0
0.5
1.0
1.5
time (ms)
Figure 9: Equilibrium Position vs. Figure 10: Transient Simulation using
Supply Voltage (solid line).
The MOS feedback design from Figure
dash-dotted line shows the corre- 9. The upper plot shows normalized
sponding voltage across the device. deflection. The lower plot shows norThe dashed line extends the equilib- malized supply voltage (solid), device
rium curve for capacitor feedback voltage (dash-dotted), and MOS gate
with # = #m = 0.32. 14/7 = 0.01; voltage (dashed).
away from the fixed electrode as the MOS device returns to accumulation. Thus
with this type of feedback design, the device exhibits hysteresis.
Finally, the device was simulated with feedback that creates an instability in
both accumulation and depletion. The lower plot in Figure 14 shows the the supply voltage was a tabletop waveform with amplitude 50mV/V and 800//s period
plus a constant 420mV/V. The results show that a waveform with 50m V/V amplitude is sufficient to deflect the free electrode by close to 70% of the nominal gap
size; however, there is a region in the middle of the gap where the free electrode
cannot come to rest.
5 Conclusion
Through theory and simulation, we have shown that MOS capacitors can be used
to stabilize electrostatically actuated devices and can do so while requiring lower
supply voltages than capacitor feedback.
Acknowledgments
This research was funded in part by contracts from DARPA (contract #DABT63C-0111) and Tanner Research.
Transactions on the Built Environment vol 31, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509
Simulation and Design of Microsystems and Microstructures
f
c
1.0
1.0
0.9
0.8
0.7
0.6
0.4
0.3
0.2
0.1
0
>; 0.5
0.5
1.5
;0.5
0.4
0.6
0.8
1.0
V/Y and Vg/Y
Figure 11: Equilibrium Plot. Refer
to Figure 9 for an explanation of
the plot. %/7 = ID-*; ^ = 0.2;
1.0
0.9
0.8
0.7
0.6
?
c
0.4
0.3
0.2
0.1
0
1.0
0.2
0.5
1.0
1.5
time (ms)
Figure 12: Transient Simulation using
design from Figure 11. Refer to Figure
10 for an explanation of the plot.
0.5
0.5
0.4
0.6
V/Y and V^/i
Figure 13: Equilibrium Plot. Refer
to Figure 9 for an explanation of the
plot. T4/7 = 10-3; Km = 0.138;
0.2
0.4
0.3
0.2
0.1
n
>' V<
•/•
• i \/
1.0
•
1.5
»/ "x- :/»
\/'
0.5
1.5
1.0
time (ms)
Figure 14: Transient Simulation using
design from Figure 13. Refer to Figure
10 for an explanation of the plot.
References
[1] Seeger, J.I. & Crary, S.B. Stabilization of Electrostatically Actuated Mechanical Devices, to be published in The Ninth International Conference
on Solid-State Sensors and Actuators (Transducers '97), Chicago, Illinois,
USA, 1997.
[2] Schroder, O.K. Advanced MOS Devices, Addison-Wesley, Massachusetts,
USA, 1990.
[3] Pierret, R.F. Field Effect Devices, Addison-Wesley, Massachusetts, USA,
1990.
[4] The Mathworks Inc., Natick, MA, USA.