EE C245 - ME C218
Introduction to MEMS Design
Fall 2003
Roger Howe and Thara Srinivasan
Lecture 11
Electrostatic Actuators II
EE C245 – ME C218 Fall 2003 Lecture 11
Today’s Lecture
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Linear (vs. displacement) electrostatic actuation:
vary overlap area: electrostatic comb drive
Electrostatic springs: positive (comb levitation)
Second-order effects in electrostatic actuators:
charged dielectrics, work functions, depletion, and Casimir
Reading:
1. W. C. Tang, M. G. Lim, and R. T. Howe, “Electrostatic comb drive
levitation and control method,” Journal of Microelectromechanical
Systems, 1, 170-178 (1992).
2. B. D. Jensen, S. Mutlu, S. Miller, K. Kurabayashi, and J. J. Allen,
“Shaped comb fingers for tailored electromechanical restoring
force,” Journal of Microelectromechanical Systems, 12, 373-383
(2003).
3. Kudrle, T. D., et al, “Pull-in suppression and torque magnification in
parallel plate electrostatic actuators with side electrodes,” 12th Int.
Conf. on Solid-State Sensors, Actuators, and Microsystems
(Transducers ’03), Boston, Mass., June 8-12, 2003, pp. 360-363.
EE C245 – ME C218 Fall 2003 Lecture 11
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Interdigitated Comb Drive
Common bias:
DC offset VP connected
to shuttle through poly0
“ground plane”
William Tang, Ph.D. EECS Dept., 1990
(this device by Clark Nguyen, Ph.D. 1994)
EE C245 – ME C218 Fall 2003 Lecture 11
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Electrostatic Force: a First Pass*
t
g
stator (fixed electrode)
rotor (not … but moving)
gap = g, thickness = t
L = finger length
x = overlap length
L
x
W. C. Tang, Ph.D. EECS Dept., 1990
EE C245 – ME C218 Fall 2003 Lecture 11
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First-Pass Electrostatic Force (Cont.)
• Neglect fringing fields
• Parallel-plate capacitance between stator and rotor
Vrs
Vrs
⎛
ε xt ⎞
ε xt ⎞ 2
1⎛
W ′( x,Vrs ) = ∫ q( x,V ′)dV ′ = ∫ ⎜⎜ 2 N o ⎟⎟V ′dV ′ = ⎜⎜ 2 N o ⎟⎟Vrs
2⎝
g ⎠
g ⎠
0
0⎝
Fe =
ε t⎞
∂W ′ 1 2 dCrs 1 2 ⎛
= Vrs
= Vrs ⎜⎜ 2 N o ⎟⎟
2
2
∂x
dx
g ⎠
⎝
independent of x!
• Have we forgotten anything? The substrate!
EE C245 – ME C218 Fall 2003 Lecture 11
5
Comb Drive Force: a Second Pass
• Energy must include capacitance between the stator
and rotor and the underlying ground plane, which is
typically biased at the stator voltage Vs … why?
t
g
L
zo
x
+
-
EE C245 – ME C218 Fall 2003 Lecture 11
Vs
+
-
Vr
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Comb-Drive Force with
Ground Plane Correction
• Finger displacement changes capacitances from stator and rotor
to the ground plane modifies the electrostatic energy
Fe , x =
∂W ′ 1 dCsp 2 1 dCrp 2 1 dCrs
=
Vs +
Vr +
(Vs − Vr )2
∂x
2 dx
2 dx
2 dx
Gary Fedder, Ph.D.,
pp. 119-122, 1994
EE C245 – ME C218 Fall 2003 Lecture 11
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Capacitance Expressions
• Consider case where Vr = Vp = 0 V
• Csp depends on whether or not fingers are engaged
′ ,u ( L − x)]
Csp = N [Csp′ ,e x + Csp
Crs = NCrs′ x
EE C245 – ME C218 Fall 2003 Lecture 11
Gary Fedder, Ph.D.,
pp. 119-122, 1994
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Simulation (2D Finite Element)
Fe , x =
N
2
(C rs′ + Csp′ ,e − Csp′ ,u )Vs
2
20-40% reduction of Fe
EE C245 – ME C218 Fall 2003 Lecture 11
Gary Fedder, Ph.D., p. 123, 1994
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Vertical Force (Levitation)
Fe , z =
∂W ′ 1 dCsp 2 1 dCrp 2 1 dCrs
(Vs − Vr )2
Vs +
=
Vr +
2 dz
2 dz
2 dz
∂z
Consider Vr = 0 V as shown:
Fe, z =
EE C245 – ME C218 Fall 2003 Lecture 11
1 ⎡ d (Csp′ ,e + C rs′ )⎤ 2
Nx
⎥Vs
2 ⎢⎣
dz
⎦
W. C. Tang, JMEMS, 1992 (reader)
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Levitation Force
“electrical spring const.”
Fe, z ≅ k e (Δze − Δz )
constant
Levitation force adds to the
mechanical spring constant in
the z direction increases
the resonant frequency
EE C245 – ME C218 Fall 2003 Lecture 11
Gary Fedder, Ph.D., p. 122, 1994
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Vertical Resonant Frequency
Must account for electrical
springs in finding MEMS
resonant frequencies
comb (x-axis) ke = 0
comb (z-axis) ke > 0
parallel plate ke < 0
EE C245 – ME C218 Fall 2003 Lecture 11
W. C. Tang, JMEMS, 1992 (reader)
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Relative Forces for Surface Microstructures
x
Comb drive (x-direction)
y
Vr = 0 V
L
x
V1
(V1 = V2 = Vs = 1V)
Fe ,x =
ε ot 2
Vs
g
Differential || plate (y-direction)
V2
gap = g = 1 μm,
thickness = t = 2 μm
finger length = L =100 μm
overlap length x = 75 μm
(V1 = 0 V, V2 = 1V)
Fe , y =
Fe, y
Fe , x
1 ε o tx 2
V2
2 g2
ε otx 2
V2
2
1 x
g
=
=
2ε ot 2 2 g
Vs
g
|| plate wins
big … for
surface MEMS
EE C245 – ME C218 Fall 2003 Lecture 11
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Levitation Suppression
Pattern Poly0 into differentially biased electrodes to minimize
field lines terminating on top of comb
Penalty: x-axis force is reduced
EE C245 – ME C218 Fall 2003 Lecture 11
W. C. Tang, JMEMS, 1992 (reader)
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Experimental Measurements
Shuttle is pulled down
(toward the substrate)
with zero applied voltage
Why?
EE C245 – ME C218 Fall 2003 Lecture 11
W. C. Tang, JMEMS, 1992 (reader)
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Charged Dielectrics:
No Applied Voltage Needed!
Minimize
exposed
dielectrics!
Nitride charge inferred from deflection and simulated field distribution is
consistent with typical values
EE C245 – ME C218 Fall 2003 Lecture 11
W. C. Tang, JMEMS, 1992 (reader)
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Work Function Differences
Example: p+ structure over n+ poly0 electrode
z
p+ poly-Si
-------------------n+ poly-Si
++++++++++++++++++++
Equilibrium band diagram
p+ poly-Si
EF
n+ poly-Si
How is charge
exchanged to
reach equilibrium?
Answer: mobile
charge on dielectric
z
surfaces
EE C245 – ME C218 Fall 2003 Lecture 11
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Depletion Effects in Silicon
ρ(x) Q = +qN X
d d
n type silicon (SOI structure)
--------------------
++++++++++++++++++++
depletion
region
+qNd
g
x
-Xd
x
-Q
E(x)
Eair=(εSi / εo)ESi
-Xd
+ -
V
Q = f(V)
EE C245 – ME C218 Fall 2003 Lecture 11
g
x
Nonlinear charge-storage affects electrostatic force
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