Ningning Zhou, Jason Clark, Kristofer Pister, Sunil Bhave, BSAC
David Bindel, James Demmel, Depart. of CS, UC Berkeley
Sanjay Govindjee, Depart. of CEE, UC Berkeley
Zhaojun Bai, Depart. of CS, UC Davis
Ming Gu, Jianlin Xia, Depart. of Mathematics, UC Berkeley
January, 2001

• Background
• SUGAR introduction
• Netlist input
• Algorithms with examples
• Element models
• More examples
• Conclusion

Current simulation approaches for MEMS devices:
• FEM, BEM MEMCAD, AutoBEM, ANSYS etc.
– Device/Process oriented;
– Not well integrated with other domains such as circuits;
– Poorly suited to do higher level design and optimization.
•
System level simulation NODAS, SUGAR
Netlist simulator
SUGAR SPICE

• Simulation package for MEMS devices implemented in MATLAB.
• Using Modified Nodal Analysis method modeled on SPICE.
• Ability to perform simulation in multi-energy domains such as electrical circuits, mechanical, thermal etc.
• Implemented static(DC), steady state (SS), modal frequency, transient and sensitivity analysis in different versions of SUGAR.

• Four versions released free on the web since June
1998. http://www-bsac.eecs.berkeley.edu/~cfm
• Hundreds of downloads from all over the world.
For example, in the period of 02/2000 ~ 04/2000,
121 downloads from universities(~40%), industries(10~20%), research labs(5~10%) etc..
• Active interaction with users.

Release time
2D(DC, SS, Modal, TA)
3D(DC, SS, Modal, TA)
Mechanical (beams, anchors, gaps)
V0.5 V1.01 V1.1 V2.0
06/98 11/99 07/00 Now
Simple electrical elements
Open framework for new models,
New netlist input allowing subnets
Sensitivity analysis

Netlist Input
Simulation
Engine
(Static, Transient,
Steady State)
Process Files
ODE Element Models

• Elements:
Beams Anchors Plate mass
Electrostatic gaps
Circuits elements (resistor, voltage source)
……
• Models:
Beam Linear mechanical model
Nonlinear mechanical model
Mechanical-electrical model etc.
Gap Nonlinear electro-mechanical model
Anchor Mechanical model
Electro-mechanical model
……

v1 n1 a1 b1 n2 a2 n4 g1 n5 n3 a3 g uses mumps.net
v1 Vsrc * [ n1 g ] [V=10] e1 eground * [ g ] [] a1 anchor p1 [ n1 ] [l=5e-6 w=10e-6 oz=180 R=100] b1 beam2de p1 [ n1 n2 ] [l=1e-4 w=2e-6 oz=0 R=1000] g1 gap2de p1 [ n2 n3 n4 n5 ] [l=1e-4 w1=1e-5 w2=2e-6 … gap=2e-6 R1=100 R2=100 oz=0] a2 anchor p1 [ n4 ] [l=5e-6 w=1e-5 oz=-90 R=100] e2 eground * [ n4 ] [] a3 anchor p1 [ n5 ] [l=5e-6 w=1e-5 oz=-90 R=100] e3 eground * [ n5 ] []

uses mumps.net
subnet XSusp [B] [susp_len=* angle=*][ a1 anchor parent [A] [l=10u w=10u h=6u oz=90+angle] b1 beam3d parent [A a1] [l=susp_len w=2u h=6u oz=0+angle] b2 beam3d parent [a1 a2] [l=10u w=2u h=6u oz=-90+angle] b3 beam3d parent [a2 B] [l=susp_len w=2u h=6u oz=180+angle] b4 beam3d parent [A a3] [l=susp_len w=2u h=6u oz=180+angle] b5 beam3d parent [a3 a4] [l=10u w=2u h=6u oz=-90+angle] b6 beam3d parent [a4 B] [l=susp_len w=2u h=6u oz=0+angle]
] subnet XMass [A B] [finger_len=*][ b1 beam3d parent [A b1] [l=25u w=50u h=6u oz=-90] b2 beam3d parent [b1 B] [l=25u w=50u h=6u oz=-90] b3 beam3d parent [b1 b2] [l=finger_len w=2u h=6u oz=0] b4 beam3d parent [b1 b3] [l=finger_len w=2u h=6u oz=180]
]
XSusp p1 [c(1)] [susp_len=200u angle=0] for k=1:10 [ mass(k) XMass p1 [c(k) c(k+1)] [finger_len=100u]
]
XSusp p1 [c(11)] [susp_len=200u angle=180]

Finding nodal variables (unknowns) by formulating and solving nodal equations at each node.
Nodal variables: mechanical displacements electrical potentials thermal temperatures……
Nodal equations at each node: sum of forces = 0 sum of currents = 0 sum of heat flux = 0 ……

• Finding the equilibrium point of the system
• SUGAR uses Newton-Raphson method solving nonlinear equation system f
 

0 x is the equilibrium nodal variables
Starting from an initial guess x
0
, iterates x n

1
 x n




 x f n



1 f ( x n
)
Until x n

1
 x n
 
(tolerance)

• Test structures are fabricated by MCNC;
• Beam: Nominal Lb=100um, w=2um, h=2um. Measured :
L=100um, w=1.74um, h=2.003um
• Gap plate: Lg=100um, w=10um, h=2.003um.
• Young’s Modulus: assume 165GPa.
• Simulation was done by considering fringing-field effects;
• Contact force model was used to get pull-in voltage;
V
+
-
1
0.8
0.6
0.4
0.2
0
6
1.8
1.6
1.4
1.2
6.5
7 7.5
8
Voltage V (v)
8.5
9 9.5
10
Lb
22
20
18
16
14
12
10
8
6
4
2
40 60
O Experimental results
Simulation results
80 100 120 140 160 180 200 220 240
Length of the beam L (um)
6

•
Finding the sinusoidal response of the system
• Linearizing the system at a DC equilibrium point, solving linear ODE system y


Ax

Cx

Bu
Du where u = sinusoidal excitation y = system output response
C = output matrix
D = feed forward matrix
Modal frequencies and modal shapes can be found by solving for system eigenvalues and eigenvectors.

• Simulation of a linear multiple mode resonator by Reid
Brennen. Sugar results matches his measurements within 5%.
100
0
-100
-200
10
2
The response of vertical displacement of mass
-5
-6
-7
-8
-9
-10
10
2
10
3
10
4
Frequency (Hz)
10
5
10
6
200
10
3
10
4
Frequency (Hz)
10
5
10
6
100
50
0
-50
-100
10
2
-11
The response of induced current in lower comb
-12
-13
-14
-15
10
2
10
3
10
4
Frequency (Hz)
10
5
10
6
10
3
10
4
Frequency (Hz)
10
5
10
6

Mode 1 at 15454 Hz
Mode 2 at 26983 Hz
Mode 3 at 31112 Hz
Mode 6 at 123010 Hz