Dong-Il “Dan” Choocw.snu.ac.kr/sites/default/files/NOTE/2457.pdf · 2018-01-30 · The...
Transcript of Dong-Il “Dan” Choocw.snu.ac.kr/sites/default/files/NOTE/2457.pdf · 2018-01-30 · The...
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Lecture 5:
Comb resonator design (1)
Dong-Il “Dan” Cho
S h l f El i l E i i d C S i School of Electrical Engineering and Computer Science, Seoul National University
Nano/Micro Systems & Controls Laboratory
Email: [email protected]: http://nml.snu.ac.kr
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Why is comb drive type ?
• Principle :Interlacing comb fingers create large capacitor area;Interlacing comb fingers create large capacitor area;electrostatically actuated suspended microstructures
• Features :– Fairly linear relationship between
it d di l tcapacitance and displacement– Higher surface area/ capacitance
than parallel plate capacitor– Electrostatic actuation: low power
(no DC current)
Electrostatic Comb Drive Type
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What is comb drive resonator ? (cont’d)
• Comb drives combine mechanical and electrostatic issues :issues :– Mechanical issues
• ElasticityS d i• Stress and strain
• Resonance (natural frequency)• Dampingp g
– Electrical issues• Capacitance• Capacitance• Electrostatic forces• Electrostatic work and energy
Tang, Nguyen and HoweJMEMS 1989.
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JMEMS 1989.
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Normal Stress and Strain
• Stress: force applied to surface /F Aσ =
measured in N/m2 or Pa,compressive or tensile
/F Aσ =
• Strain: ratio of deformation to length/l lε = ∆
measured in %, ppm,or microstrain
/l lε = ∆
Young’s Modulus:/E σ ε
Hooke’s Law:/E σ ε=
/ /K F l EA l= ∆ =
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Shear Stress and Strain
• Shear Stress: force applied parallel to surface
measured in N/m2 or Pa,
/F Aτ =
• Strain: ratio of deformation to length
/l lγ ∆ /l lγ = ∆
Shear Modulus:/G τ γ=
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Poisson’s Ratio
Tensile stress in x direction results in compressive stress in y and z direction (object becomes longer stress in y and z direction (object becomes longer and thinner)
P i ’ R ti< Materials >• Homogeneous• Poisson’s Ratio:
ν ε ε= − /
t t i / l it di l t i
y x
• Homogeneous– Isotropic– Anisotropic
• Heterogeneous= − transverse strain / longitudinal strain • Heterogeneous
Metals: Rubbers: C k
0.3ν ≈0.5ν ≈
0ν ≈Cork: 0ν ≈
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Mechanical Property for Poly Crystalline SiliconSilicon
• Young’s modulus for different deposition processes
Doping Conditions
PSG diffusion doping POCl3 diffusion doping
Deposition temp
1000℃60min
1050℃30min
1000 ℃120min
850℃180min
850℃210min
850 ℃240min
950℃60min
950℃90min
950 ℃120min
1000℃60min
1000 ℃
90min
Young’s
625℃
605
154.3±2.3
155.1±1.8
151.1±3
149.6±2.6
150.7±2.5
153.1±2.1
149±1.7
151.1±1.1
150.4±1.1
152.6±1.4
148.6±2
159 3 160 5 161 8 162 6 161 9 163 1 161 2 155 6 161 8 157 1 156modulus(GPa)
605℃
585
159.3±3.4
160.5±1.7
161.8±1.4
162.6±0.6
161.9±2.3
163.1±0.4
161.2±2
155.6±5.6
161.8±1.4
157.1±3.7
156±5.9
170.1 169.4 162.4 168.5 162.2 163.5 159 165.6 160.6 159.8 156.6585℃ ±2.2 ±6.2 ±1.6 ±1.9 ±2.6 ±3.1 ±3.3 ±1.4 ±1 ±1 ±2
[Ref] S Lee et al IOP Journal of Micromechanics and Microengineering vol 8 no 4 pp 330-337 Dec 1998
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[Ref] S. Lee et al., IOP Journal of Micromechanics and Microengineering, vol. 8, no. 4, pp. 330 337, Dec. 1998.
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Mechanical Property for Single Crystal Silicon
• Silicon Crystallography 면 ( ) { }
방향 < > [ ]
Silicon (100) Silicon (110) Silicon (111)
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Young’s modulus
Silicon (100) Silicon (110) Silicon (111)Silicon (100) Silicon (110) Silicon (111)
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Shear modulus
Silicon (100) Silicon (110) Silicon (111)Silicon (100) Silicon (110) Silicon (111)
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Poisson’s ratio
Silicon (100) Silicon (110) Silicon (111)Silicon (100) Silicon (110) Silicon (111)
[Ref] J. Kim et al., Proceedings of Transducers 2001, pp. 662-665, Munich, Germany, June 2001.
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Torsional stiffness (1)– Schematic of torsional springSchematic of torsional spring
Tz
xy θy θ
The dimension splits of torsional springs
- thickness(t) : 10 ㎛, 20 ㎛
- length(l) : 40 ㎛
- width(w) : 2 ㎛, 4 ㎛
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Torsional stiffness (2)– Silicon (110)Silicon (110)
The torsional stiffness on silicon (110) varies from -31.2 % to -26.6 %
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The torsional stiffness on silicon (110) varies from 31.2 % to 26.6 %
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Torsional stiffness (3)– Silicon (100)Silicon (100)
The torsional stiffness on silicon (100) varies from -1.6 % to 9.6 %.
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( )
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Torsional stiffness (4)– Silicon (111)Silicon (111)
The torsional stiffness on silicon (111) varies from 1.7 % to 2.3 %.
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( )
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Mechanical Property for Single Crystal Silicon (cont’d)(cont d)
• Torsional stiffness (5)– Summary for resultsy
The average for the normalized maximum torsional stiffness variation ratio (%)
[Ref] D. Kwak et al., Proceedings of IMECE 2003, Washington D.C, USA, November 2003.
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[ ] , g , g , ,
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Electrostatic Forces
• Parallel Plate Capacitor:• Capacitance:
0
012
/ /; dielectric constant of free space
rC Q V A dε εε= =
St d
12 ( 8.854 10 / ); dielectric permittivityr
F mε
−≈ ×
• Stored energy:2
21 1 : 2 2
ε= = =
Q AW CV CC d
• Electrostatic force between plates:0 =ε ε ε r
22
2
1 12 2x
W A CVF Vx d d
ε∂= − = − =
∂
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Electrostatic Actuation
• Positioning of capacitor plate:1 ε ε= − <
= − <
2 20
0
1/ 0
2( ) 0
el r
S
F AV x
F K x d
0where :distance at rest
(no applied voltage)
d
• Stable equilibrium when el SF F=
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Pull-In Point
The higher V, the closer the plate is pulled in. Fel → ∞ when d → 0. What is the closest stable distance xmin?min
• Fel and FS must be tangential:3 2
0
2 30
/ , so
/r
r
A x V K
V Kx A
ε ε
ε ε
=
=
• Substitute into Fel=FS to get=min 02 /3x d
can control x only from 2/3d0
to d0
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Electrostatic Comb Drive
ε ε= 0 /rC A d• Capacitance is approximately:
(L>>d)
ε ε= 0 2 /rn lh d(ignore fringing field effect)
• Change in capacitance whenmoving by △x :
/C A dε ε∆ = ∆0
0
/
2 /r
r
C A d
n xh d
ε εε ε
∆ = ∆
= ∆
• Electrostatic force :ε ε= =2 2
01/2 / /el rF V dC dx n h dV Electric field distribution in com-finger gaps
Note: Fel is independent of △x over wide range (fringing field), and quadratic in V.
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), q
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Comb Drive Design
[U i i ] [F ld d i t ][U spring suspension type] [Folded suspension type]
[Serpentine suspension type] [Fishhook suspension type]
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Comb Drive Design (cont’d)
[Bent beam serpentine suspension type] [Spiral spring suspension type]
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Comb Drive Failure Modes
• Comb drives require low stiffness in x direction but high stiffness in y, z direction as well as but high stiffness in y, z direction as well as rotations.
Note: comb fingers are in unstable equilibrium with respect to the g q py direction.
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Summary of Translation motion
• Acceleration, velocity, distancea v x
• Force, momentum= =a v x
F
• Kinetic energy
= =p mv Ft
1
• Dynamic (spring damper mass)
= 212
E mv
• Dynamic (spring, damper, mass)
• Oscillation (assume b=0)
= + +F Kx bx mx
( )
ωπ
=1
( ) =2
K Kf Hz
m m
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Summary of Rotation motion
• Angular acceleration, angular velocity, angleα ω φ= =
• Torque, angular momentumα ω φ= =
T r F= ×
• Kinetic energyL r p Iω= × =
1
• Dynamic (moment of inertia)
ω= 212
E I
• Dynamic (moment of inertia)
• Oscillation (assume b=0)
φ βφ φ= + +T K I
( )
π=
12
Kf
I
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Lumped-parameter Model
• The dynamic of motion is based on the lumped-parameter model– Input: external fore F, Output: displacement x
+ + = 0( ) ( ) ( ) sin( )mx t bx t Kx t F wt0( ) ( ) ( ) ( )
:mass, :damping, :stiffnesswhere m b K
Schematic microdevice Lumped-parameter model
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Lumped-parameter Model (cont’d)
• Transfer function:
2
1
( )x mH s
b KF s s= =
+ +2s sm m
+ +
Schematic microdevice Lumped-parameter model
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Resonators
• Analogy between mechanical and electrical system:system:- Mass m – Inductivity L- Spring K – Capacitance 1/C- Damping b – Resistance R
(depending where R is placed in circuit)S l ti t 2nd d diff ti l ti• Solution to 2nd order differential equation:
ω ωω ξω ωω
=+ ++ +
2 20 0
2222 0 0 0
( ) or 2
H ss ss s ξω
ω π
+ +
=
0 00
0 2 :natural frequencyo
s sQ
where f
ω ω
ξ ξ
= =
=
0 0
1 :mechanical system, : electrical system
: quality factor 1 /2 ( : )
Km LC
Q Q damping ratioDong-Il “Dan” Cho Nano/Micro Systems & Controls Lab.This material is intended for students in 4541.844 class in the Spring of 2008. Any other usage and possession is in violation of copyright laws
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ξ ξ= : quality factor 1 /2 ( : )Q Q damping ratio
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Mechanical Resonators
• Frequency and phase shift under damping:
τ ω ϕ
τ
−= +
=
/21( ) cos( )
: damping time
tx t Ae t
mwhere τ
ω ω ω ω ξ ω= − = − = − =2
21 0 0 02 2
: damping time
1 1 1 1
4 4 d
whereb
bKmω τ
ϕ
2 204 4
: phase shift
Km
• Energy dissipation: /0( ) tE t E e τ−=
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Quality Factor
• Definition: Quality factor (Q factor)– Ratio of stored energy and lost energy:Ratio of stored energy and lost energy:
02 2E
QE T
τπ π ω τ= = =∆
– Mechanical system: 0
m KmQ
b bω= =
– Similar for electric systems: (a) 0
1L LQ
R R Cω= =
(b)
R R C
0
CQ RC R
Lω= =
L
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Quality Factor (cont’d)
• How fast does energy dissipate?Q m
• What is the maximum amplitude for a given 0
(mechanical)Q m
bτ
ω= =
p gfrequency?: At resonance, amplitude is Q times the DC responseresponse
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Summary: Mechanical/Electrical Resonator
• Mechanicalresonator :
( ) ( ) ( ) 0: mass : damping : stiffness
mx t bx t Kx tm b K
+ + =resonator :
0
: mass, : damping, : stiffness
natural frequency / (for small b)
m b K
K mω =
• Torsionalresonator :
( ) ( ) ( ) 0: moment of inertia, : damping, : stiffness
I t b t k tI b kθ θ θ+ + =
• Electrical
0natural frequency /k Iω =
1• Electrical resonator :
1( ) ( ) ( ) 0
: inductivity, : resistance, : capacitance
Lq t Rq t q tC
L R C
+ + =
0natural frequency 1/LCω =
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Measurement
: DC bias
d i i l t PV
V
0
: drive signal at
: output current /d d
P
V
I V dC dt
ω∝
Feedbackvia transimpedance amplifier
Vc carrier signal at cω
Output signal:frequency spectrum includes but also ω ω ω ω±frequency spectrum includes , , but also
basis for frequency transferd c c dω ω ω ω±
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Application of Comb Drive
Accelerometer (NML SNU) Gyroscope (NML SNU)
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Application of Comb Drive
X-Y stage (MiSA SNU)Micromirror (UC Davis & Stanford) g ( )( )
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Reference
• Lee, S. W., Cho, C. H., Kim, J. P., Park, S. J., Yi, S. W., Kim, J. J., and Cho, D. I., "The Effects of Post-deposition Processes on Polysilicon Young's Modulus, IOP Journal of Micromechanics and Microengineering vol 8 no 4 pp 330 337 Dec Journal of Micromechanics and Microengineering, vol. 8, no. 4, pp. 330-337, Dec. 1998.
• Kim, J. P., Cho, D. I., and Muller, R. S., "Why is (111) Silicon a Better Mechanical Material for MEMS," Proceedings of Transducers 2001: 11th International Conference on Solid State Sensors and Actuators pp 662 665 International Conference on Solid State Sensors and Actuators, pp. 662-665, Munich, Germany, June 10-14, 2001.
• Kwak, D. H., Kim, J. P., Park, S. J., Ko, H. H., and Cho, D. I., "Why is (111) Silicon a better Mechanical Material for MEMS: Torsion Case," 2003 ASME International Mechanical Engineering Congress (IMECE 2003) Washington D C International Mechanical Engineering Congress (IMECE 2003), Washington D.C., USA, Nov. 15-21, 2003.
• Lee, S. W., Park, S. J., Yi, S. W., Lee, S. C., Cho, D. I., Ha, B. J., Oh, Y. S., and Song, C. M., "Electrostatic Actuation of Surface/Bulk Micromachined Single-crystal Silicon Microresonators " Proceedings of IEEE/RSJ International crystal Silicon Microresonators, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 2, pp. 1057-1062, Kyeongju, Korea, Oct. 17-21, 1999.
• Ko, H. H., Kim, J. P., Park, S. J., Kwak, D. H., Song, T. Y., Setiadi, D., Carr, W., B J d Ch D I "A Hi h f X/Y i Mi l t Buss, J., and Cho, D. I., "A High-performance X/Y-axis Microaccelerometer Fabricated on SOI Wafer without Footing Using the Sacrificial Bulk Micromachining (SBM) Process," 2003 International Conference on Control, Automation and Systems (ICCAS 2003), pp. 2187-2191, Gyeongju, Korea, Oct. 22 25 2003
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22-25, 2003.
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Reference (cont’d)
• Kim, C. H., Jeong, H. M., Jeon, J. U., Kim, Y. K., “Silicon micro XY-stage with a large area shuttle and no-etching holes for SPM-based data storage”, Journal of Microelectromechanical Systems vol 12 issue 4 pp 470 478 Aug 2003 Microelectromechanical Systems, vol. 12 , issue 4 , pp. 470 - 478, Aug 2003.
• Chang Liu, “Foundations of MEMS”, Pearson, 2006.• Nicolae O. Lobontiu, “Mechanical design of microresonators”, McGraw-Hill, 2006.
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