Physical and Reduced Order Dynamic Analysis of MEMS

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Physical and Reduced Order Dynamic Analysis of
MEMS
S. K. De and N. R. AluruBeckman Institute for
Advanced Science TechnologyUniversity of
Illinois at Urbana-Champaign
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Outline

• Introduction MEMS Devices and MEMS-CAD
• Applications and Design Issues
• Simulation Methods
• Full Simulation (Lagrangian Dynamics)
• Full Simulation Results
• ROM -Reduced Order Modeling (KL decomposition)
• ROM Results
• Characteristics of MEMS Dynamics
• Resonant Frequency Stiffness Hardening and
  Softening
• Second Super-harmonic Resonance
• Switching Time/Power
• Conclusions

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Introduction MEMS Devices
MEMS Micro-electro-mechanical Systems.
(TI)
Texas Instruments-Micro-mirror
Analog Devices-Accelerometer
Electrical Domain
Energy exchange
Mechanical Domain
Onix Eight-Optical Fiber Switching
Analog Devices-Gyroscope
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RF-MEMS Applications

• Switches in radar systems, wireless
  communications (cellular phones)
• Tunable capacitors and inductors
• Passband/multi-band Filters/Oscillators/Mixers

Advantages of RF-MEMS

• Low insertion loss, high isolation and low return
  loss
• Low cost, low power consumption (only during
  actuation), large down and up state capacitive
  ratio (wireless communications)
• Versatility to be fabricated on almost any
  silicon substrate
• On-chip RF-filters small size, huge numbers in
  parallel

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Introduction MEMS-CAD
Device level analysis vs. system level analysis
Physical/Device Level
System Level
Full Simulation (Lagrangian Dynamics)
Compact Models
Reduced Order Models
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MEMS Dynamics
Electrical Domain Electrostatic Analysis
Mechanical Domain Structural Analysis

• Applied potential difference
• Relative position
• Electrostatic charge
• Electrostatic Pressure

• External Acceleration
• Structural Stiffness
• Structural inertia
• Structural Damping
• Electrostatic Pressure
• Deformation and position

Time evolution is the concern
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Design Issues in Dynamics

• Switching speed
• Switch down (pull-in for capacitive switches
  during high applied voltages)
• Switch up (after applied voltage is removed)
• Depends on resonant frequency, applied voltage,
  squeeze film damping
• Main limitation for capacitive switches

Switch Down
Switch Up
Figures J. M. Huang, et. al., Sensors and
Actuators A, Vol. 93, 273-285, 2001
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Design Issues in Dynamics

• Q-factor
• Measurement of power absorption bandwidth
• Factor by which displacement is amplified at
  resonance
• For CMOS compatibility, high Q-factor needed as
  it influences dynamic range of circuits
• Depends on squeeze film damping

Q-factor from frequency response curve
Figure Y. C. Loke, et. al., Sensors and
Actuators A, Vol. 96, 67-77, 2002.
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Design Issues in Dynamics

• Resonant Frequency
• Depends on applied voltage, deformation
• High resonant frequency desired for fast
  switching, high Q-factor
• Increased through structural modification,
  residual stresses
• Residual stresses increase pull-in voltage

• Phase Shifts
• Phase shifting and time delaying circuits, e.g.,
  phased array radar communication antennas

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Full Simulation Lagrangian Dynamics

• Coupled electro-mechanical analysis
• Linear/Non-linear Elasticity for mechanics
• Boundary Integral equation for exterior
  electrostatics
• Full Lagrangian description (both mechanical and
  electrostatic analysis done on undeformed
  geometry) presented in ICCAD 2002 for statics
  (G. Li and N. R. Aluru)
• Relaxation method for consistency
• Process repeated in each time step
• Accurate
• Costly in terms of time and memory

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Full Lagrangian Description

• Mechanical and Electrostatic Analysis done on
  undeformed geometry
• Eliminates
• Surface re-discretization
• Re-computation of interpolation functions

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Governing Equations Non-linear elasticity
Lagrangian Elasticity for mechanical deformation
Newmark Scheme with implicit-trapezoidal rule
for dynamics
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Governing Equations Lagrangian Electrostatics
Lagrangian Boundary Integral Equation for
exterior electrostatics
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Full Simulation (Semi-Lagrangian)
Discretize mechanical domain
Initialization
Compute FEM/FCM interpolation functions
Time Step n0,1,2,..,T
Discretize electrical domain
Compute BEM/BCM interpolation functions
Electrostatic Analysis Electrostatic Pressure
Structural Analysis Mechanical Deformation
Update and go to next time step
Solution Converged?
No
Yes
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Full Simulation (Lagrangian)
Discretize mechanical domain
Initialization
Compute FEM/FCM interpolation functions
Faster and Efficient
Discretize electrical domain
Compute BEM/BCM interpolation functions
Time Step n0,1,2,..,T
Electrostatic Analysis Electrostatic Pressure
Structural Analysis Mechanical Deformation
Update and go to next time step
Solution Converged?
No
Yes
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Full Simulation MEM Cantilever Beam Dynamics
Simulations for 80µm by 0.5µm Si MEM cantilever
beam, gap 0.7 µm
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Full Simulation MEM Fixed-fixed Beam Dynamics
Simulations for 80µm by 0.5µm Si MEM fixed-fixed
beam, gap 0.7 µm
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Transverse Comb Drive Dynamics
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Lateral Comb Drive Dynamics
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Reduced Order Models/Macro-Models
Levels of modeling MEMS in simulation

• System Level
• Requires low-order behavioral models of all the
  components of the system
• Represented as a network of lumped elements
  analogous to electric circuit elements
• Small set of coupled ODEs that can be easily
  integrated

SYSTEM LEVEL Lumped Networks ODEs
MACROMODELS
Macro-models Link between the Physical level and
the System level

• Physical Level
• Full behavior of the continuum
• Simultaneous solution of coupled PDEs
• Generally non-linear (except for small motions)
• Computationally very expensive

PHYSICAL LEVEL 2D/3D Simulation PDEs
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Reduced Order Modeling KL Decomposition

• Method of Snapshots
• KL decomposition and SVD
• Basis functions generated for mechanical
  displacement (u) and electrostatic charge density
  (s)
• Basis functions used in simulation instead of
  interpolation functions
• Collocation and method of normal equations for
  solving
• Order of problem (say 5000) reduces as very few
  basis needed (1-3 basis)
• Faster and cheaper
• Can be integrated with circuit simulators

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Karhunen Loève Decomposition
Dynamical System
Run full-simulation
Take snapshots ui
Minimize this by
Approximate
Collocation formulation
Method of normal equations for solving
Reduces order of system from N (5000) to n(3)
(nltltN)
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Basis Generation

• Snapshots taken for u, s at different time
  instants t1,t2,,tn
• Homogeneous boundary conditions satisfied by
• subtracting steady state S(x,y) for mechanics
• SVD done of the ensemble matrix
• V is the matrix of basis functions

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Reduced Order Modeling Basis Functions
Shapes of Electrical Basis Functions
Shapes of Mechanical Basis Functions
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Reduced Order Modeling Dynamics
Applied Voltage 40.5V (Pull-in) (Lin and Non-Lin
are different)
Applied Voltage 10V (Lin and Non-Lin are same)
Simulations for 80µm by 1.0µm Si fixed-fixed
beam, gap 0.7 µm
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Characteristics Resonant Frequency
Resonant Frequency variation of the fixed-fixed
beam for different gaps
Resonant Frequency variation of the cantilever
beam for different gaps
Spring-hardening Mechanical
Spring-softening Electrical
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Second Super-harmonic Resonance
10V DC
30V DC

• Primary peak at resonant frequency of MEM device
  (DC signal)
• Secondary peak at half of resonant freq. (AC
  signal)
• Important design Issue (electrical cross talk,
  passband/multi-band filters/oscillators)

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Switching time DC signal

• Time taken for pull-in for applied voltages above
  the Pull-in voltage
• Decreases with increase in voltage non-linearly

Simulations for 80µm by 1.0µm Si fixed-fixed
beam, gap 0.7 µm
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Switching time Power Consumption
Power analysis from RC series circuit analogy of
MEM device DC bias with AC signal at resonant
frequency/half of resonant freq. gives faster
switching at less peak power than a pure DC
signal
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Conclusions

• MEMS dynamics simulations needed for design
  issues
• Full Simulation accurate but costly
• Reduced Order Models Faster and Cheaper
• Spring hardening or softening occurs depending on
  applied voltage and geometry
• Second Super-harmonic Resonance Electrical
  Cross-talk, multi-band/pass-band filter
• Faster switching by DC bias and AC signal at
  resonant freq. less power required