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EE C245 – ME C218Introduction to MEMS Design
Fall 2007Fall 2007
Prof Clark T C NguyenProf. Clark T.-C. Nguyen
Dept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley
Berkeley, CA 94720y
L t 23 El t i l Stiff
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 1
Lecture 23: Electrical Stiffness
Lecture Outline
• Reading: Senturia, Chpt. 5, Chpt. 6g p p• Lecture Topics:
Energy Conserving TransducersCharge ControlCharge ControlVoltage Control
Parallel-Plate Capacitive TransducerspLinearizing Capacitive ActuatorsElectrical Stiffness
Electrostatic Comb DriveElectrostatic Comb-Drive1st Order Analysis2nd Order Analysis
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 2
Linearizing the Voltage-to-Force T f F tiTransfer Function
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)
(t) V (t)EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 4
v(t) = VP + vi(t)
Differential Capacitive Transducer
• The net force on the d d suspended 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 kElectrode
Conductive Structure
•Nomenclature: kmElectrode
Va or vAva=|va|cosωt
VAd1
xt
V V +
m
Fd1
Va or vA = VA + va
Total Value AC or Signal Component
V
v1Total Value
DC Component(upper case variable;
g p(lower case variable; lower case subscript)
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 7
V1 VP
(upper case variable; upper case subscript)
Parallel-Plate Capacitive Nonlinearity
• Example: clamped-clamped laterally driven beam with balanced electrodes kElectrode
Conductive Structure
• Expression for ∂C/∂x: kmElectrode
d1
xm
Fd1
V
v1
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 8
V1 VP
Parallel-Plate Capacitive Nonlinearity
• Thus, the expression for force from the left side becomes: kElectrode
Conductive Structure
kmElectrode
d1
xm
Fd1
V
v1
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 9
V1 VP
Parallel-Plate Capacitive Nonlinearity
• Retaining only terms at the drive frequency:CVCVF oo i|||| 121 txd
Vtvd
VF oo
Poo
Pdo
ωωω sin||cos|| 21
1211
1
111 +=
P i l Drive force arising from the input excitation voltage at
the frequency of this voltage
Proportional to displacement
90o phase-shifted from the frequency of this voltage 90 phase-shifted from drive, so in phase with
displacement• Th t t th m th t thi • These two together mean that this force acts against the spring restoring force!g
A negative spring constantSince it derives from VP, we call it the electrical stiffness given by: 212 AVCVk o ε
==
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 10
the electrical stiffness, given by: 31
121
1 dV
dVk PPe ==
Electrical Stiffness, ke
• The electrical stiffness kebehaves like any other stiffness kElectrode
Conductive Structure
• It affects resonance frequency: kmElectrode
−′ kkk em
21⎞⎛
==mm
emoω d1
x1 ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
kk
mk
m
em m
Fd1
21
3
211 ⎟⎟
⎞⎜⎜⎛−=′
AkVP
ooεωω
V
v131⎟⎠
⎜⎝ dkm
oo
Frequency is now a
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 11
V1 VPFrequency is now a
function of dc-bias VP1
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 -17ppm/oC
h l b l f l h l
pp
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 13
• Thermal stability of poly-Si micromechanical resonator is 10X worse than the worst case of AT-cut quartz crystal
Geometric-Stress Compensation• Use a temperature dependent mechanical stiffness to null frequency shifts due to Young’s modulus thermal dep.
[Hsu et al IEDM’00] [W.-T. Hsu, et al., IEDM’00][Hsu et al, IEDM 00]
• Problems:stress relaxationcompromised design flexibility
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 14
flexibility
[Hsu et al IEDM 2000]
Voltage-Controllable Center Frequency
Gap
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 15
Excellent Temperature StabilityR t Top Metal
ElectrodeResonator
[Hsu[Hsu et alet al MEMS’02]MEMS’02]
−1.7ppm/oCElect.-StiffnessCompensation
Top Electrode-to-Resonator Gap
[Hsu [Hsu et alet al MEMS’02]MEMS’02]
pp Compensation−0.24ppm/oC
AT-cut Quartz
p
Elect. Stiffness: ke ~ 1/d3 Q
Crystal at Various
Cut Angles
Frequency:fo ~ (km - ke)0.5
Uncompensated resonator
Angles
Counteracts reduction in
frequency due to On par with
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 16
μresonator100
[Ref: Hafner]
frequency due to Young’s modulus temp. dependence
quartz!
Measured Δf/f vs. T for ke-Compensated μResonators
150016V
VP-VC
mp μ
500
100016V
14V12V10V
0
500
7V8V9V10V
-1000
-500 6V4V2V0V
Design/Performance:f =10MHz Q=4 000
-1500
1000
300 320 340 360 380
0VCC
fo =10MHz, Q=4,000VP=8V, he=4μm
do=1000Å, h=2μmW =8μm L =40μm
−0.24ppm/oC
300 320 340 360 380
• Slit h l t l th t t d b l t l
Temperature [K]Wr=8μm, Lr=40μm
[Hsu et al MEMS’02]
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 17
• Slits help to release the stress generated by lateral thermal expansion linear TCf curves –0.24ppm/oC!!!
Can One Cancel ke w/ Two Electrodes?
•What if we don’t like the dependence of frequency on VP?C l k i diff i l k
xF• Can we cancel ke via a differential
input electrode configuration?• If we do a similar analysis for F
kmFd1
Fd2
If we do a similar analysis for Fd2at Electrode 2: d1 d2
Subtracts from the de 2
e 1
tvCVF o ωcos||2=
Fd1 term, as expected m
Elec
trod
ectr
ode
C
tvd
VF oPdo
ωω cos||
22
22
22 −= E
Ele
V
v1
V
v2txdCV o
oP ωsin||2
2
222+
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 18
V1 VPV2Adds to the quadrature term → ke’s add,
no matter the electrode configuration!
Problems With Parallel-Plate C Drive
•Nonlinear voltage-to-force transfer function
Resonance frequency becomes kx
FResonance frequency becomes dependent on parameters (e.g., bias voltage VP)Output current will also take on
kmFd1
Fd2
Output current will also take on nonlinear characteristics as amplitude grows (i.e., as x
h d )
d1 d2
de 2
e 1
approaches do)Noise can alias due to nonlinearity
m
Elec
trod
ectr
ode
y• Range of motion is small
For larger motion, need larger gap but larger gap weakens
E
Ele
gap … but larger gap weakens the electrostatic forceLarge motion is often needed (e g by gyroscopes V
v1
V
v2
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 19
(e.g., by gyroscopes, vibromotors, optical MEMS)
V1 VPV2
Electrostatic Comb DriveElectrostatic 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 benefitsLinearizes voltage-generated input forces(Ideally) eliminates dependence of frequency on dc bias(Ideally) eliminates dependence of frequency on dc-biasAllows a large range of motion y Stator Rotor
xz
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 21
Comb-Driven Folded Beam Actuator
Comb-Drive Force Equation (1st Pass)
T Vi
x
Top View
y
Lfxz
Side View
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 22
Lateral Comb-Drive Electrical Stiffness
T Vi
x
Top View
y
Lfxz
Side View
• Again: ( ) hNChxNxC εε 22=
∂→=g
•No (∂C/∂x) x-dependence → no electrical stiffness: ke = 0!
( )dxd
xC∂
→
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 23
p e
• Frequency immune to changes in VP or gap spacing!
Typical Drive & Sense Configuration
y
xz
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 neglectedFringing fieldsP ll l l i b d Parallel-plate capacitance between stator and rotorCapacitance to the substrate
• All of these capacitors must be included when evaluating the All of these capacitors must be included when evaluating the energy expression!
Stator Rotor
Ground
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 25
Ground Plane
Comb-Drive Force With Ground Plane Correction
• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures
d h d l difi h i i and the ground plane → modifies the capacitive energy
[Gary Fedder Ph D
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 26
[Gary Fedder, Ph.D., UC Berkeley, 1994]
Capacitance Expressions
• Case: Vr = VP = 0V• Csp depends on whether or not p pfingers are engaged
Capacitance per l h
Region 2 Region 3
unit length
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 27
[Gary Fedder, Ph.D., UC Berkeley, 1994]
Comb-Drive Force With Ground Plane Correction
• Finger displacement changes not only the capacitance between stator and rotor, but also between these structures
d h d l difi h i i and the ground plane → modifies the capacitive energy
EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 28
[Gary Fedder, Ph.D., UC Berkeley, 1994]
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