Jason Vaughn Clark, Ningning Zhou, Kris Pister, Jim Demmel March 14-15, 2000 University of California at Berkeley 3-D Modeling in SUGAR 1.1

SUGAR topics to be covered Optimization Electrostatics Modal Analysis Thermal Expansion Residual stress Non-Inertial Frames Process Sensitivities

Spice-like environment Simulation Engine Analyses: Static,Transient, Steady-state,Sensitivity,Modal

Optimization Sparse solvers and other efficient algorithms from a collaboration with David Bindle, and Prof. Jim Demmel.

Device by: G. Fedder, R. Howe Working with Prof. James Demmel and David Bindle we’ve begun to apply more efficient linear algebraic methods in SUGAR. Here we show a plot of number of node versus simulation time required for the static analysis of a serpentine structure. Serpentine ( design by G. Fedder) Number of nodes More efficient linear algebraic methods

Nonlinear Electrostatics (1)Gap closing actuator with a time varying voltage input. (2)Microrelays with nonlinear eletrostatics and nonlinear beam stiffness.

Transient response of a gap-closing actuator. A) shows a plot of displacement as a function of time. The voltage ramps from 5V at t=5usec to 12V at t=500usec, and then releases. As the voltage increases linearly during this time interval, the space between the gap decreases at a nonlinear rate due to electrostatic forces; likewise, the period of oscillation decreases. The amplitude of oscillations decrease exponentially due to the viscous layer of air between the device and substrate. A) B) C) Transient response of a gap-closing actuator

Microrelays V. Milanovic, M. Maharbiz, A. Singh, B. Warneke, N. Zhou, H. K. Chan, and K. Pister Pull-down SwitchSee-saw Switch

Linear and Nonlinear Simulation See-saw switch Tilting (desired) Bending down (undesired) Gap distance (um) Pull-down voltage (V) V. Milanovic, M. Maharbiz, A. Singh, B. Warneke, N. Zhou, H. K. Chan, and K. Pister

Circuits: Induced Currents Multimode resonator.

This demonstrates steady-state analysis applied to a multi-mode resonator. The left figure shows the Bode and phase plot of the current induced on the sensing comb as a function of the frequency of the voltage at the driving comb of the figure on the left. The measured modes are with in 5% of experimental frequency given by Brennen et. al. Design by: R. Brennen Induced Current: Steady-state response produces induced current

3D Modal Analysis Accelerometer. Micromirror. Gyro.

Figure a) is the schematic design. Figure b) is the first mode shape corresponding to Hz from SUGAR, matches hand analysis 33 Hz within 18%. Figure c) is the third mode corresponding to Hz from SUGAR, matches hand analysis Hz within 0.5% Design provided by: Jonathan Simon, LLNL a) b) c) 3D mode analysis of an accelerometer

The mode shapes and frequencies of modes 1, 3, 4 and 6 are shown in Figures a) through d). Respectively they are 15.5kHz, 31.1kHz, 41.7kHz, and 123kHz. a) b) c) d) Three-dimensional mechanical mode analysis of a torsional mirror

Design by: Ashwin Seshia, R. T. Howe A. Seshia’s Gyro Layout

Mode Analysis of Gyro mode1 mode2 mode3 mode4

Out of Phase Mode Shape of Tuning Fork Resonant freqency (mode 3): Sugar : 4.3 kHz Hand calculation: ~3.6 kHz Experimental: ~3 kHz Process overetch: 2um width Resonant freqency, Fork: Sugar frequency: kHz Hand calculation: ~300 kHz Experimental: Attached to gyro: 240 kHz Plain resonator: 265 kHz

Thermal expansion Heatuator. Digital to Analog Converter.

Plot by: R. Conant, Muller UCB. Measurement by: B. Allan, HH98. The graph shows typical heat distribution along the hot and cold arm (R. Conant et al) by FEA. SUGAR is within 5.9% of measured deflection. The average beam temperature is used in SUGAR. T=600C T=150Cdy=4.82u T=600C T=150C Thermal expansion: The heatuator

MEM-DAC R. Yeh, Boris Murmann, K. Pister LSB MSB output

Linear and Nonlinear Analysis of MEM-DAC SUGARv1.1 SUGARv1.0

Residual strain Compressive / tensile stress, F=A . Strain gradients, M=EI .

Design by: Chi Pan, W. Hsu SUGARv1.1 simulation of residual stress. MEMS devices are often subject to residual stress effects which may affect device performance. Simulated deflection is within 0.59% of measured data. dy=8.35um Compressive residual strain 8.4um

Tensile/compressive micro strain gauge Liwei Lin, Albert Pisano, Roger Howe

SUGAR simulation showing out-of- plane residual strain. For beam lengths (tip z-deflection) 185um (1.5868um), 147um (1.00um), 117um (0.634um), 87um (0.3509um), 58um (0.1559um) show good agreement with measure data. Positive strain gradient

From Roger Howe’s course notes: Residual strain gradient

From Roger Howe’s course notes: Residual strain gradient

Close-up view of the residual strain effects. From Roger Howe’s course notes: Residual strain gradient

The strain gradient shown here is two orders of magnitude larger than that of the BiMEMS process in order to visually display the doming of the backbone and the bending down of the fingers. Simulation of negative strain gradient

Accelerating Frames All inertial forces are incorporated. Rotating substrate example.

The 4 Inertial Forces Total nodal F =  F electromagnetic +  F Inertial F Centrifugal F Coriolis F Transverse F transla = md 2 R /dt 2 R 0 F Translational

Transient performance of rotating Substrate 40us  t  rad/sec Ex(t) Ey(t) Y -X  E Z

Process sensitivities Process variations by Luca Schenato, Wei-Chung Wu Monte Carlo: performance possibilities, Ellipsoidal Calculus: worst case scenarios.

Example: process variation of stress

Process variation analysis in SUGAR By: Luca Schenato, Wei-Chung Wu [K nominal +  K] q = F static

Free on the web Transient, modal, steady-state, and static analysis Limited thermal model Residual stress/strain models Non-inertial frames Improved algorithms Preliminary process sensitivity modeling Conclusion