A monolithic spatial light modulator chip was developed consisting of a large number of micrometer-scale mirror cells which can be rotated through an angle by application of an electrostatic field. The field is generated by electronics integral to the chip. The chip has application in photoreceptor based non-impact printing technologies. Chips containing over 16000 cells were fabricated, and were tested to several billions of cycles. Finite Element Analysis (FEA) of the device was used to model both the electrical and mechanical characteristics

Hornbeck, L. J.

Nelson, W. E.

Sheerer, T. J.

Welsh 

NASA Technical Reports Server

. 
N89-22953 
F I N I T E  ELEMENT ANALYSIS OF A MICROMECHANICAL 
DEFORMABLE M I R R O R  DEVICE 
T .  J. SHEERER, W .  E .  NELSON AND L .  J .  HORNBECK 
TEXAS INSTRUMENTS INCORPORATED, DALLAS T X .  
ABSTRACT: 
T e x a s  I n s t r u m e n t s  h a s  developed a mono l i th i c  s p a t i a l  l i g h t  
m o d u l a t o r  ch ip  c o n s i s t i n g  o f  a l a r g e  number o f  
m i c r o m e t e r - s c a l e  mirror  c e l l s  w h i c h  can  be r o t a t e d  t h r o u g h  a n  
a n g l e  by a p p l i c a t i o n  o f  a n  e l e c t r o s t a t i c  f i e l d .  The f i e l d  i s  
g e n e r a t e d  by electronics i n t e g r a l  t o  t h e  c h i p .  The c h i p  h a s  
a p p l i c a t i o n  i n  p h o t o r e c e p t o r  b a s e d  n o n - i m p a c t  p r i n t i n g  
t e c h n o l o g i e s .  Chips  c o n t a i n i n g  over 16000 ce l l s  h a v e  been 
f a b r i c a t e d ,  and h a v e  been tes ted t o  s e v e r a l  b i l l i o n s  o f  cycles.  
F i n i t e  E l e m e n t  A n a l y s i s  ( F E A )  of t h e  device h a s  been u s e d  t o  
m o d e l  bo th  the  e l e c t r i c a l  and  m e c h a n i c a l  c h a r a c t e r i s t i c s .  
INTRODUCTION: 
T h e  very h i g h  component d e n s i t y  a c h i e v e d  i n  i n t e g r a t e d  
c i r cu i t s  i s  w e l l - k n o w n .  U s i n g  the same p r o c e s s i n g  t e c h n i q u e s  i t  
is a l s o  poss ib le  t o  p r o d u c e  m i c r o m e c h a n i s m s  on a s i m i l a r  s c a l e .  
Petersen (1 )  h a s  described t h e  m a n u f a c t u r e  and t e s t i n g  o f  
extremely s m a l l  s i l i c o n  c a n t i l e v e r s  and a l s o  l i s t s  s e v e r a l  
commercia l  a p p l i c a t i o n s  o f  m i c r o m e c h a n i c a l  devices. 
T h e  d e f o r m a b l e  m i r r o r  device ( D M D )  consis ts  o f  a c h i p  
c o n t a i n i n g  a l a r g e  number o f  m i r r o r  ce l l s  a s  shown i n  F i g .  (1). 
T h e  ce l l  consis ts  o f  a 19um x 19 um aluminum m i r r o r  p i v o t e d  a t  
t w o  corners by  4.5 um b e a m s .  The m i r r o r  and i t s  s u p p o r t  
s t r u c t u r e  a r e  0.375um i n  thickness  w h i l e  the  beam i s  o f  
n o m i n a l  w i d t h  1 . 0  um and n o m i n a l  thickness 0.08 um. 2.3 um 
b e l o w  t h e  s t r u c t u r e  a r e  t w o  a d d r e s s  electrodes and t w o  l a n d i n g  
electrodes.  T o  r o t a t e  the m i r r o r  t h r o u g h  a n  a n g l e ,  , a b i a s  
p o t e n t i a l , & ,  i s  a p p l i e d  t o  the u p p e r  s t r u c t u r e  and l a n d i n g  
e lectrodes w h i l e  a p p r o p r i a t e  a d d r e s s  p o t e n t i a l s  a r e  a p p l i e d  t o  
t he  a d d r e s s  e lectrodes a s  shown i n  F i g .  ( 2 ) .  T h e  ce l l  may be 
u s e d  a s  a n  o p t i c a l  lever t o  d e f l e c t  a l i g h t  beam i n  and o u t  o f  
the  f i e l d  o f  a p r o j e c t i o n  l ens  a s  shown i n  F i g .  ( 3 ) .  T h e  e f f e c t  
is t h u s  of b e i n g  a b l e  t o  a c t i v a t e  and d e a c t i v a t e  a "pixel"  a t  
the  very h i g h  speed o f  the  DMD c e l l .  The DMD i s  m o r e  f u l l y  
d e s c r i b e d  i n  ( 2 )  and ( 3 ) .  In  a n  e lectronic  p r i n t e r  u s i n g  a 
x e r o g r a p h i c  t y p e  process t h e  l i g h t  i s  u s e d  t o  se lec t ive ly  
d i s s i p a t e  c h a r g e  on a t r a n s f e r  medium,  p r i o r  t o  the c h a r g e  
b e i n g  t r a n s f e r e d  t o  p a p e r , a l l o w i n g  the  selective depos i t i on  o f  
"toner" m a t e r i a l  on t h e  p a p e r .  T h e  m o s t  common t y p e  of p r i n t e r  
u s i n g  t h i s  process i s  the  l a s e r  p r i n t e r ,  w h i c h  r e q u i r e s  i n  
a d d i t i o n  t o  t h e  laser  a complex o p t i c a l  system and a 
m e c h a n i c a l l y  r o t a t i n g  m i r r o r  t o  ' s c a n '  t he  t r a n s f e r  medium.  U s e  
o f  the DMD ( F i g .  ( 4 ) )  i s  s imp ler ,  a l l o w i n g  u s e  of a c o n v e n t i o n a l  
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I 
incandescent s o u r c e .  A d d i t i o n a l l y ,  by v a r i a t i o n  o f  the "on '' 
t i m e  o f  the  c e l l ,  the  DMD a l l o w s  u s e  o f  g r e y - s c a l e  p r i n t i n g .  
Mechanical  a n a l y s i s  of s u c h  devices by FEA i s  not d i f f e r e n t  
f r o m  a n a l y s i s  o f  l a r g e  s t r u c t u r e s ,  the o n l y  r e q u i r e m e n t  b e i n g  
j u d i c i o u s  choice o f  u n i t s  f o r  d imens ions  a n d  m e c h a n i c a l  
p r o p e r t i e s  t o  a v o i d  f a u l t s  d u e  t o  a r i t h m e t i c  over f low or 
u n d e r f l o w .  E l e c t r o s t a t i c  a n a l y s i s  i s  a l s o  r e l a t i v e l y  s i m p l e ,  a s  
the  e q u a t i o n s  o f  e l e c t r o s t a t i c s  a r e  i d e n t i c a l  w i t h  those o f  
h e a t  t r a n s f e r .  B y  u s e  o f  COSMIC NASTRAN i n  these a p p l i c a t i o n s  
i t  i s  p o s s i b l e  t o  a c c u r a t e l y  model  the  b e h a v i o r  o f  a DMD and 
a s s e s s  the  e f f e c t s  o f  d e s i g n  m o d i f i c a t i o n s  w i t h o u t  the  very 
c o n s i d e r a b l e  expense o f  a p r o d u c t i o n  r u n .  
EQUATIONS OF ELECTROSTATICS A N D  THEIR ANALOGS: 
The a n a l o g y  between e l e c t r o s t a t i c s  and  s t e a d y - s t a t e  
h e a t  t r a n s f e r  i s  e x a c t ,  and N A S T R A " s  h e a t  t r a n s f e r  
c a p a b i l i t i e s  c a n  be u s e d  i n  the s o l u t i o n  o f  e l e c t r o s t a t i c  f i e l d  
p rob lems  w i t h o u t  m o d i f i c a t i o n .  In  h e a t  t r a n s f e r  w e  h a v e :  
w h e r e a s  i n  e l e c t r o s t a t i c s  the  p o l a r i z a t i o n ,  D ,  i s  g i v e n  by: 
Heat  s o u r c e s  a n d  s i n k s  a r e  e q u i v a l e n t  t o  p o i n t  c h a r g e s  and  
f i x e d  t e m p e r a t u r e s  a r e  e q u i v a l e n t  t o  f i x e d  p o t e n t i a l s .  In  
e l e c t r o s t a t i c s  the  p o t e n t i a l  g r a d i e n t  i s  refered t o  a s  the  
f i e l d ,  E .  There i s  no d i r e c t l y  a n a l o g o u s  t e r m  i n  h e a t  t r a n s f e r .  
The a n a l o g y  between the  d i f f e r e n t  t e r m s  i s  l i s t e d  i n  T a b l e  (1) 
b e l o w :  
TABLE 1 :  ANALOGY B E m E N  ELECTROSTATICS AND HEAT TRANSFER 
I HEAT TRANSFER I ELECTROSTATICS 
TEMPERATURE 
TEMPERATURE GRADIENT 
HEAT FLCX 
HEAT SOURC% 
CONDUCTIVITY 
POTENTIAL V 
ELECTRIC FIELD E 
POLARIZATION D 
POINT CHARGE q 
PERMITTIVITY E 
29 1 
The force exerted on a c h a r g e  by an  e lec tr ic  f i e l d  i s  g i v e n  by: 
F =  q . D  ( 3 )  
where D i s  the e l ec t r i c  p o l a r i z a t i o n  and q t h e  c h a r g e .  As  the 
c h a r g e  i t s e l f  contr ibutes  t o  the f i e l d  i t  i s  n e c e s s a r y  t o  
i n t e g r a t e  the  a b o v e  e x p r e s s i o n  t o  o b t a i n  a u s e f u l  expression i n  
t e r m s  o f  E,  g i v i n g  n o r m a l i z e d  force,  or p r e s s u r e ,  i n  vacuo:  
F = 0 . 5 G E . E  ( 4 )  
where i s  the  p e r m i t t i v i t y  o f  f r e e  s p a c e .  T h i s  expression 
a l l o w s  c a l c u l a t i o n  o f  the p r e s s u r e  exerted a t  an element f a c e  
d i rec t l y  f r o m  the  o u t p u t  o f  the  f i n i t e  element a n a l y s i s .  T h e  
c h a r g e  d i s t r i b u t i o n  r e s u l t i n g  f r o m  a p o t e n t i a l  d i s t r i b u t i o n  i s  
o b t a i n a b l e  f r o m  the s i n g l e  p o i n t  c o n s t r a i n t  forces w h i c h  i n  the 
c a s e  o f  h e a t  t r a n s f e r  i n d i c a t e  h e a t  s o u r c e s  and s inks .  O n e  
i m p o r t a n t  d i f ference  i n  b e h a v i o u r  between the e l e c t r o s t a t i c  
c a s e  and the t h e r m a l  c a s e  i s  t h a t  a c o n d u c t i n g  m a t e r i a l  a c t s  i n  
e l e c t r o s t a t i c s  i n  a manner  e q u i v a l e n t  t o  an  i n f i n i t e l y  
conductive m a t e r i a l  in h e a t  t r a n s f e r ,  so t h a t  a l l  p o i n t s  on the 
c o n d u c t o r  a r e  a t  the  same p o t e n t i a l .  T h i s  can  be m o d e l l e d  by 
a p p r o p r i a t e  c o n s t r a i n t s  i n  the NASTRAN i n p u t  deck. 
ELECTROSTATIC MODEL OF THE DMD : 
A 3-0 m o d e l  o f  the  DMD ce l l  was c o n s t r u c t e d  u s i n g  s o l i d  
e l e m e n t s  a s  shown i n  F i g .  (5). F i g .  ( 6 )  p l o t s  the  z-component  o f  
the e lec tr ic  f i e l d  , E z ,  vs. x - c o o r d i n a t e  a t  the  center and e d g e  
o f  the model  f o r  a b i a s  p o t e n t i a l  o f  -12V and a d d r e s s  
p o t e n t i a l s  of +5V and z e r o .  F i g .  ( 7 )  i s  a f r i n g e  p l o t  o f  E z  
super imposed  on the  m o d e l .  O n  the  b a s i s  o f  the  r e s u l t s  o b t a i n e d  
i t  was d e t e r m i n e d  t h a t  f r i n g e i n g  e f f e c t s  w e r e  of r e l a t i v e l y  
s m a l l  s i g n i f i c a n c e  and a series of 2-0 m o d e l s  w e r e  made w i t h  
m i r r o r  r o t a t i o n  a n g l e s  f o r m  1 t o  9 d e g r e e s  i n  the X Y p l a n e .  
R o t a t i o n  o f  a p p r o x i m a t e l y  9 . 2  d e g r e e s  was s u f f i c i e n t  t o  b r i n g  
t he  m i r r o r  i n  c o n t a c t  w i t h  the  l a n d i n g  e lectrode.  From e a c h  of 
these m o d e l s  The u s e f u l  component  o f  e lec tr ic  f i e l d ,  Ey, was 
o b t a i n e d  a s  a f u n c t i o n  o f  l o c a t i o n  on t h e  m i r r o r .  In  increments 
c o r r e s p o n d i n g  t o  element s i z e  on the  m o d e l ,  the  force exerted 
on the mirror  was c a l c u l a t e d  a s  a f u n c t i o n  o f  l o c a t i o n  u s i n g  Ey 
and the mirror’s w i d t h  a s  a f u n c t i o n  o f  1 o c a t i o n . F i g s .  (8-11)  
a r e  f r i n g e  p l o t s  o f  p o t e n t i a l  V and f i e l d  Ey f o r  t w o  d i f f e r e n t  
r o t a t i o n  a n g l e s ,  F i g s .  (12-13) p l o t  Ey vs. l o c a t i o n  f o r  
d i f f e r e n t  r o t a t i o n  a n g l e s  f o r  b i a s  p o t e n t i a l s  o f  -12V and -16V, 
w i t h  an  a d d r e s s  p o t e n t i a l  o f  +5V. T h e  choice o f  +5V a l l o w s  u s e  
o f  s t a n d a r d  ( T T L  or  CMOS) l o g i c  electronics t o  control the DMD, 
w i t h  the  b i a s  p o t e n t i a l  b e i n g  g e n e r a t e d  s e p a r a t e l y .  
292 
MECHANICAL MODEL OF THE DMD: 
The m e c h a n i c a l  model  o f  the DMD was c o n s t r u c t e d  i n  order t o  
ver i fy  the  a s s u m p t i o n  t h a t  the device a c t e d  a s  a r i g i d  body 
( the  m i r r o r )  m o u n t e d  on to r s ion  rods .  U s i n g  the m e a s u r e d  
dimensions of the device a m o d e l  was c o n s t r u c t e d  u s i n g  p l a t e  
and b a r  elements and s u b j e c t  t o  modal  and s t a t i c  a n a l y s i s .  I t  
was d e t e r m i n e d  t h a t  d e f o r m a t i o n  o f  the mirror  p l a t e  was 
n e g l i g i b l e ,  and  t h a t  the  to r s ion  b a r s  had  three d e f o r m a t i o n  
modes  o f  in teres t  w i t h  f r e q u e n c i e s  o f  56.2, 192 and 5 4 5  kHz. a s  
p l o t t e d  i n  F i g .  (14). T h e  h i g h e r  m o d e s  involved p l a t e  
d e f o r m a t i o n  a t  very h i g h  f r e q u e n c i e s .  These r e s u l t s  i n d i c a t e d  
t h a t  the e l a s t i c  c o n s t a n t s  u s e d  w e r e  v a l i d  ( e x p e r i m e n t a l l y  the  
f i r s t  mode was a t  a p p r o x i m a t e l y  54 kHz) and t h a t  i n  a p p l y i n g  
e l e c t r o s t a t i c  forces t o  the m i r r o r  i t  was v a l i d  t o  u s e  a s i n g l e  
lumped force a t  a n  a r b i t r a r y  node, and not  n e c e s s a r y  t o  a p p l y  
p r e s s u r e s  over e a c h  e l e m e n t  o f  the DMD m o d e l .  I t  was a l s o  
d e t e r m i n e d  f r o m  t h i s  m o d e l  t h a t  a force o f  1.74 m i l l i d y n e s  
a p p l i e d  a t  the  DMD t i p  was r e q u i r e d  t o  f u l l y  r o t a t e d  the  m i r r o r .  
RECONCILIATION OF MODELS A N D  EXPERIMENTAL DATA: 
U s i n g  the  e l e c t r o s t a t i c  DMD m o d e l  i t  i s  possible  t o  p l o t  
force or t o r q u e  on the DMD a g a i n s t  a n g l e  f o r  d i f f e r e n t  b i a s  and 
a d d r e s s  p o t e n t i a l s .  I t  s h o u l d  be possible  t o  p r e d i c t  the  
p o t e n t i a l s  a t  w h i c h  the  device c a n  f u l l y  r o t a t e .  I n  t h i s  
c i r c u m s t a n c e  the e l e c t r o s t a t i c  force w i l l  exceed the  m e c h a n i c a l  
r e s t o r i n g  force a t  a l l  a n g l e s  o f  r o t a t i o n ,  and w i l l  
t a n g e n t i a l l y  a p p r o a c h  the r e s t o r i n g  force c u r v e  a t  a c r i t i c a l  
a n g l e  w h i c h  i s  r e a d i l y  d e t e r m i n e d  e x p e r i m e n t a l l y .  
E x p e r i m e n t a l l y ,  f o r  a n  a d d r e s s  p o t e n t i a l  o f  S V ,  a b i a s  
p o t e n t i a l  o f  -16V i s  f o u n d  n e c e s s a r y ,  w i t h  a c r i t i c a l  a n g l e  
a r o u n d  5.5 d e g r e e s .  F i g .  (15)  shows e x p e r i m e n t a l  d a t a  p l o t t i n g  
b i a s  v o l t a g e  vs. a d d r e s s  v o l t a g e  f o r  s p o n t a n e o u s  r o t a t i o n .  
F i g .  (16) p l o t s  the ne t  force on the DMD f r o m  e l e c t r o s t a t i c  
f i e l d s  vs. a n g l e  for a n  a d d r e s s  p o t e n t i a l  of +SV and b i a s  
p o t e n t i a l  o f  -12V and -16V. Also  p l o t t e d  i s  the m e c h a n i c a l  
r e s t o r i n g  force b a s e d  on the f i n i t e  element s t r u c t u r a l  m o d e l  of 
the  DMD. The c u r v e  f o r  -16V does a p p e a r  t o  t a n g e n t i a l l y  
a p p r o a c h  the m e c h a n i c a l  force c u r v e  a t  around 5 d e g r e e s ,  and 
t h u s  i s  i n  excellent a g r e e m e n t  w i t h  the  e x p e r i m e n t a l  d a t a .  
CONCLUSIONS: 
The u s e  o f  very s imp le  2-0 m o d e l s  o f  e l e c t r o s t a t i c  f i e l d s  h a s  
proven t o  g i v e  a c c u r a t e  v a l u e s  o f  e l e c t r o s t a t i c  forces  i n  a 
m i c r o m e c h a n i c a l  d e v i c e .  I t  a p p e a r s  l i k e l y  t h a t  a n y  desired 
a c c u r a c y  i s  a t t a i n a b l e  i f  a s u f f i c i e n t l y  complex m o d e l  i s  u s e d .  
The response t i m e  o f  the  device, a l t h o u g h  f a s t  f o r  a mechanism,  
i s  much less  t h a n  the r ise  t i m e  o f  the electrode p o t e n t i a l s .  
T h u s  the d a t a  on force vs. r o t a t i o n  c o u l d  a l s o  be u s e d  i n  a 
t r a n s f e r  f u n c t i o n  or n o n l i n e a r  s p r i n g  i n p u t  t o  a l l o w  d y n a m i c  
m o d e l l i n g  o f  the device. 
293 
REFERENCES : 
( 1 )  K u r t  E. Petersen, " S i l i c o n  a s  a Mechanical  M a t e r i a l " ,  P r o c .  
IEEE, 70-5, 420, (1982)  
( 2 )  L. J. Hornbeck, "128x128 D e f o r m a b l e  Mirror Device", I E E E  
T r a n s .  E lec tron  Devices ED-30 ( S ) ,  539, (1983)  
( 3 )  D . R .  Pape and L.J. Hornbeck, " C h a r a c t e r i s t i c s  of t h e  
Deformable  M i r r o r  Device f o r  O p t i c a l  I n f o r m a t i o n  P r o c e s s i n g " ,  
O p t i c a l  E n g i n e e r i n g ,  22 ( 6 ) ,  675 (1983)  
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Finite element analysis of a micromechanical deformable mirror device

Abstract

Similar works

Full text

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