Lec23p.ElectricalSti..

EE C245 – ME C218
Introduction to MEMS Design
Fall 2007
Prof Clark T
Prof.
T.-C.
C Nguyen
Dept of Electrical Engineering & Computer Sciences
Dept.
University of California at Berkeley
Berkeley,
y CA 94720
L t
Lecture
23 Electrical
23:
El t i l Stiffness
Stiff
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
1
Lecture Outline
• Reading:
g Senturia, Chpt.
p 5, Chpt.
p 6
• Lecture Topics:
ª Energy Conserving Transducers
(Charge Control
(Voltage Control
p
Transducers
ª Parallel-Plate Capacitive
(Linearizing Capacitive Actuators
(Electrical Stiffness
ª Electrostatic Comb-Drive
Comb Drive
(1st Order Analysis
(2nd Order Analysis
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
2
Linearizing the Voltage-to-Force
T
Transfer
f F
Function
ti
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
3
Linearizing the Voltage-to-Force T.F.
• Apply a DC bias (or polarization) voltage VP together with
the intended input (or drive) voltage vi(t), where VP >> vi(t)
v(t)
(t) = VP + vi(t)
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
4
Differential Capacitive Transducer
• The net force on the
suspended
d d center
electrode is
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
5
Remaining Nonlinearity
(El t i l Stiff
(Electrical
Stiffness))
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
6
Parallel-Plate Capacitive Nonlinearity
• Example: clamped-clamped laterally driven
beam with balanced electrodes
• Nomenclature:
Va or vA
Conductive
Structure
Electrode
km
va=|va|cosωt
d1
VA
m
x
t
Va or vA = VA + va
AC or Signal
g
Component
p
Total Value
(lower case variable; v1
DC Component lower case subscript)
(upper case variable;
V1
upper case subscript)
EE C245: Introduction to MEMS Design
Lecture 23
Fd1
VP
C. Nguyen
11/20/08
7
Parallel-Plate Capacitive Nonlinearity
• Example: clamped-clamped laterally driven
beam with balanced electrodes
• Expression for ∂C/∂x:
Conductive
Structure
Electrode
km
d1
m
Fd1
v1
V1
EE C245: Introduction to MEMS Design
Lecture 23
x
VP
C. Nguyen
11/20/08
8
Parallel-Plate Capacitive Nonlinearity
• Thus, the expression for force
from the left side becomes:
Conductive
Structure
Electrode
km
d1
m
Fd1
v1
V1
EE C245: Introduction to MEMS Design
Lecture 23
x
VP
C. Nguyen
11/20/08
9
Parallel-Plate Capacitive Nonlinearity
• Retaining only terms at the drive frequency:
Fd 1 ω
o
Co1
2 Co1
= VP1
| v1 | cos ωot + VP1 2 | x | sin
i ωo t
d1
d1
Drive force arising from the
input excitation voltage at
the frequency of this voltage
• These
Th
two
t
together
t
th mean
m
that
th t this
thi
Proportional
P
i
l to
displacement
force acts against the spring
g force!
restoring
ª A negative spring constant
ª Since it derives from VP, we call it
the electrical stiffness,
stiffness given by:
EE C245: Introduction to MEMS Design
Lecture 23
90o phase-shifted from
drive, so in phase with
displacement
ke = VP21
C. Nguyen
Co1
2 εA
= VP1 3
2
d1
d1
11/20/08
10
Electrical Stiffness, ke
• The electrical stiffness ke
behaves like any other stiffness
• It affects resonance frequency:
Conductive
Structure
Electrode
k m − ke
k
ωo =
=
m
m
′
km
d1
12
k m ⎛ ke ⎞
⎜⎜1 − ⎟⎟
=
m ⎝ km ⎠
m
12
⎛
ε
A⎞
′
⎟
ωo = ωo ⎜⎜1 −
3⎟
k
d
m
1 ⎠
⎝
2
VP1
Fd1
v1
Frequency is now a
function of dc-bias VP1
EE C245: Introduction to MEMS Design
x
Lecture 23
V1
VP
C. Nguyen
11/20/08
11
Voltage-Controllable Center Frequency
Gap
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
12
Microresonator Thermal Stability
−1.7ppm/oC
Poly-Si μresonator pp oC
17ppm/
• Thermal
h
l stability
bl
of
f poly-Si
l
micromechanical
h
l resonator is
10X worse than the worst case of AT-cut quartz crystal
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
13
Geometric-Stress Compensation
• Use a temperature dependent mechanical stiffness to null
frequency shifts due to Young’s modulus thermal dep.
[Hsu et al
al, IEDM
IEDM’00]
00]
[W.-T. Hsu, et al., IEDM’00]
• Problems:
ª stress relaxation
ª compromised design
flexibility
EE C245: Introduction to MEMS Design
Lecture 23
[Hsu et al IEDM 2000]
C. Nguyen
11/20/08
14
Voltage-Controllable Center Frequency
Gap
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
15
Excellent Temperature Stability
R
Resonator
t
Top Metal
Electrode
[Hsu et al MEMS’02]
Top Electrode-toResonator Gap
p×
−1.7ppm/
pp oC
Elect. Stiffness:
ke ~ 1/d3 Ø
Elect.-Stiffness
Compensation
−0.24ppm/oC
AT-cut
Quartz
Q
Crystal at
Various
Cut
Angles
Frequency:
fo ~ (km - ke)0.5 ×
Counteracts
reduction in
frequency due to
Young’s modulus
temp. dependence
EE C245: Introduction to MEMS Design
Uncompensated
μresonator
resonator
Lecture 23
On par with
quartz!
C. Nguyen
100
[Ref:
Hafner]
11/20/08
16
Measured Δf/f vs. T for kemp
μResonators
μ
Compensated
VP-VC
1500
16V
1000
14V
12V
10V
9V
8V
7V
6V
4V
2V
0V
CC
500
0
-500
Design/Performance:
fo =10MHz,
=10MHz Q=4,000
Q=4 000
VP=8V, he=4μm
do=1000Å, h=2μm
Wr=8μm,
=8μm Lr=40μm
[Hsu et al MEMS’02]
-1000
1000
−0.24ppm/oC
-1500
300
320
340
360
Temperature [K]
380
• Slits
Slit help
h l tto release
l
th
the stress
t
generated
t d b
by lateral
l t
l
thermal expansion Ö linear TCf curves Ö –0.24ppm/oC!!!
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
17
Can One Cancel ke w/ Two Electrodes?
• What if we don’t like the
x
Fd1
d1
o
Ele
ectrode
e 1
Subtracts from the
Fd1 term, as expected
Fd 2 ω
Co 2
= −VP 2
| v2 | cos ωot
d2
2
+ VP 2
Co 2
| x | sin ωot
2
d2
d2
m
v1
v2
V
add, 1
V2
Adds to the quadrature term → ke’s
no matter the electrode configuration!
EE C245: Introduction to MEMS Design
km
Fd2
Electrod
E
de 2
dependence of frequency on VP?
• Can
C
we cancell ke via
i a diff
differential
i l
input electrode configuration?
• If we do a similar analysis for Fd2
at Electrode 2:
Lecture 23
VP
C. Nguyen
11/20/08
18
Problems With Parallel-Plate C Drive
• Nonlinear voltage-to-force
EE C245: Introduction to MEMS Design
Lecture 23
x
d1
d2
m
Electrod
E
de 2
Fd1
km
Fd2
Ele
ectrode
e 1
transfer function
ª Resonance frequency becomes
dependent on parameters (e.g.,
bias voltage VP)
ª Output current will also take on
nonlinear characteristics as
amplitude grows (i.e., as x
approaches
h do)
ª Noise can alias due to
nonlinearity
y
• Range of motion is small
ª For larger motion, need larger
gap … but larger gap weakens
the electrostatic force
ª Large motion is often needed
(e g by gyroscopes,
(e.g.,
gyroscopes
vibromotors, optical MEMS)
v1
v2
V1
V2
VP
C. Nguyen
11/20/08
19
Electrostatic Comb Drive
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
20
Electrostatic Comb Drive
• Use of comb-capacitive tranducers brings many benefits
ª Linearizes voltage-generated input forces
ª (Ideally) eliminates dependence of frequency on dc-bias
dc bias
y
ª Allows a large range of motion
Stator
Rotor
z
x
Comb-Driven Folded Beam Actuator
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
21
Comb-Drive Force Equation (1st Pass)
x
T View
Top
Vi
y
z
x
Lf
Side View
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
22
Lateral Comb-Drive Electrical Stiffness
x
T View
Top
Vi
y
z
x
Lf
Side View
2 Nεhx
∂C 2 Nεh
• Again:
g
C (x ) =
→
=
d
∂x
d
• No (∂C/∂x) x-dependence
p
→ no electrical stiffness: ke = 0!
• Frequency immune to changes in VP or gap spacing!
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
23
Typical Drive & Sense Configuration
y
z
x
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
24
Comb-Drive Force Equation (2nd Pass)
• In our 1st pass, we neglected
ª Fringing fields
ª Parallel-plate
P
ll l l
capacitance
i
between
b
stator and
d rotor
ª Capacitance to the substrate
• All of these capacitors must be included when evaluating the
energy expression!
Stator
Rotor
Ground
Plane
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
25
Comb-Drive Force With
Ground Plane Correction
• Finger displacement changes not only the capacitance
between stator and rotor, but also between these structures
and
d the
h ground
d plane
l
→ modifies
difi the
h capacitive
i i energy
[Gary Fedder,
Fedder Ph.D.,
Ph D
UC Berkeley, 1994]
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
26
Capacitance Expressions
• Case: Vr = VP = 0V
• Cspp depends
p
on whether or not
fingers are engaged
Capacitance per
unit length
l
h
Region 2
Region 3
[Gary Fedder, Ph.D., UC Berkeley, 1994]
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
27
Comb-Drive Force With
Ground Plane Correction
• Finger displacement changes not only the capacitance
between stator and rotor, but also between these structures
and
d the
h ground
d plane
l
→ modifies
difi the
h capacitive
i i energy
[Gary Fedder, Ph.D., UC Berkeley, 1994]
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
28
Simulate to Get Capacitors → Force
• Below: 2D finite element simulation
20-40% reduction of Fe,x
EE C245: Introduction to MEMS Design
Lecture 23
C. Nguyen
11/20/08
29