modeling of mems fabrication processes · 20% 70% 95% 1 81 test patterns position index pattern ......
TRANSCRIPT
Modeling of MEMS
Fabrication Processes
Prof. Duane Boning
Microsystems Technology Laboratories
Electrical Engineering and Computer Science
Massachusetts Institute of Technology
September 28, 2007
2
Spatial Variation in MEMS Processes
Wafer Scale Chip Scale Feature Scale
• Many MEMS processes face uniformity challenges due to:– Equipment limitations
– Layout or pattern dependencies
• Variations often highly systematic and thus can be modeled– Models can help improve process to minimize variation
– Models can help improve design to compensate for variation
3
Non-uniformity problems in MEMS
Silicon oxide
Silicon
Silicon
Plasma etch variation
turbine blades
Etch depth variation: imbalance in MIT microengine rotor
mask layout
~10 mm
e.g. A.H. Epstein et al., Proc. Transducers ‘97
4
Embossing for microfluidics manufacture
Non-uniformity problems in MEMS
Thermoplastic polymer
CoverSi stamp
Cover
Surface nonuniformity:failure to seal
Channel depth nonuniformity fromembossing polymer flow
5
Outline
• Background– spatial variation in MEMS fabrication processes
1. Deep reactive ion etch (DRIE)
2. Polymer hot embossing
• Conclusions
6
1. Deep-Reactive Ion Etching (DRIE)
Background: the DRIE process
Sources of manufacturing nonuniformity
Characterizing tool performance
Semi-physical non-uniformity model
Integrating the model into a design tool
Extending the model
7
Inductively-coupled plasma in DRIE chamber
exhaust
independent control of ions’
acceleration towards wafer
wafer
chuck
Xplasma
~
gas inlet
~
vacuum chamber
~10-100 mTorr
to wafer ‘load lock’
cross-section
r.f. supply
to excite plasma
8
(10 – 15s)(6 – 11s)
1. mask 2. SF6 etch
4. SF6 etch3. C4F8
passivation
flow rate
time
C4F8
SF6
••+
++++ eFFSFSeSF yxyx6SF6 dissociates:
nSiFnFSi +
•Ion-assisted chemical etching:
Journal of The Electrochemical Society, 146 (1) 339-349 (1999);
Robert Bosch GmbH, Pat. 4,855,017 and 4,784,720 (USA) and 4241045C1 (Germany) (1994)
Time-multiplexed ‘Bosch’ processing
9
wafer in
cross-sectiondevice/‘die’
spatial variation
wafer/chamber-scaleinter- and intra-
device
ion and
radical flux
distribution
competition for
reactants; diffusion
aspect ratio-
dependent
etching (ARDE)
wafer-
level
‘loading’
feature-scale
Non-uniformity at three length scales
FX
10
Approach: Characterization using familyof test wafer designs
(a) Symmetrical loading
(b) 5% average loading
(c) 95% average loading
11
Observed wafer/chamber-scale variation
H.K. Taylor et al., J. Electrochem. Soc., May 2006
1%
5%
20%
70%
95%
1
81
test patterns
position
index
pattern
density
12
Observed pattern-dependent variation
Average pattern density 5% throughout
Localized to differing extents
H.K. Taylor et al., J. Electrochem. Soc., May 2006.
13
Ji
Silicon
Neutral
Jn
Adsorbed neutrals
niiJvSJkER0
111+=
ion F neutral flux,
Jn
ion flux,
Ji
silicon silicon
R: etch rate
: surface coverage
kEi: activity constant for ions
vS0: activity constant for radicals
iiJEkR =
( )nJvSR = 1
0
Modeling basis:Ion-neutral synergism at silicon surface
adsorbed neutrals
Models for etching rate
• Mogab (1977)1: etch ratevaries inversely withloading
• Gottscho (1992)2: etchrate set by ion-neutralsynergism
1 J. Electrochem. Soc. 124 p1262 (1977). 2 J. Vac. Sci. Tech. B, 10, 2133 (1992)
14
Concentration equilibrium above wafer surface
Solving for concentration of F neutrals in steady state at (x, y):
‘loading’/pattern densityselectivityrate constant
Ji(x, y)
Consumption: Jn(x, y)
lateral transport generation,
recombination
C(x, y)
mask
silicon
C, Ce: fluorine concentration
G: fluorine generation rate
ave: wafer-average pattern density
Neglecting lateral
transport( ) ( )[ ] ( )
( )0
,,1,
21=+
yxCyxCyxG e
eaveave
( )( )( )[ ] 11
,,
21++
=aveave
e
yxGyxC
T.F. Hill, H. Sun, H.K. Taylor, and D.S. Boning, Proc. MEMS 2005
15
Ion-neutral synergism plus Mogab model
niiJvSJkER0
111+=
Equilibrium fluorine concentration
( ) ( )yxCuyxJ zn ,ˆ, =
Ion-neutral synergism
( )( )[ ] ( )[ ]
( )[ ] ( )[ ]{ } ( )[ ]yxGuvSyxJkE
yxGuvSyxJkEyxR
zaveaveii
zii
,ˆ11,
,ˆ,,
021
0
+++=
( )( ) ( )
( ) ( )[ ]{ } ( )yxByxA
yxByxAyxR
aveave ,11,
,,,
21+++
=
( )( )( )[ ] 11
,,
21++
=aveave
e
yxGyxC
T.F. Hill, H. Sun, H.K. Taylor, and D.S. Boning, Proc. MEMS 2005
17
wafer in
cross-sectiondevice/‘die’
spatial variation
wafer/chamber-scaleinter- and intra-
device
ion and
radical flux
distribution
competition for
reactants; diffusion
aspect ratio-
dependent
etching (ARDE)
wafer-
level
‘loading’
feature-scale
Non-uniformity at three length scales
F
19
Around every location with non-average pattern density, there is aperturbation of F concentration
An integrated wafer- and die-scale model
20
Assuming that the present 1 mm2 location is the only one on the waferwith non-average pattern density, re-write concentration equilibrium,and, element-wise, obtain the map Cisol(x,y):
An integrated wafer- and die-scale model
( ) ( ) ( )[ ]{ } ( )( ) ( ) ( )
0
ln
,,2,,,1,,
0
2
0
21 ++
r
r
yxCyxC
r
DyxCyxCyxyxyxG
c
isoleisol
isol
Lateral
transport termGeneration Consumption Recombination
21
Map of ‘surplus’ fluorine concentration defined as Cisol(x, y) Ce(x, y)
Superpose these perturbations of concentration via discrete 2-D
convolution of surplus concentration with diffusion ‘filter’, E
Filter contains ‘fovea’ to deal with microloading
An integrated wafer- and die-scale model
22
Integrated model fits witherror 0.8% 4.5% r.m.s. per wafer
Substitute modified C(x,y) into
wafer-level model, using maps
A(x,y) and B(x,y)
Obtain etch rate prediction
R(x,y)
23
A two-level model, tuned for each tool + recipe
A
B
+ 2 scalar
variables
two-level
model
T.F. Hill et al., Proc. MEMS 05 + H.K. Taylor et al., accepted for publication, J. Electrochem. Soc.
radial
distance
filter magnitude
characterization
wafers
characterization
wafers
24
Characterizing other tool-recipe combinations
STS2 at MTL
(25 mTorr)
STS Pegasus
(86 mTorr)
Etch rate
(μm/min)
25
Putting two-level model into action
two-level model
takes a few seconds to run
discretized
mask design + scalar
constants
drafting software
highlight
problems
on-screen
refine
mask
design
26
CAD tool for nonuniformity prediction
Ali Farahanchi
Discretized
mask design
Die-scale
variation
Chamber-scale
variation
Combined
prediction
27
DRIE Modeling Contributions
Understanding of uniformity’s dependence on patterndensity and localization
Observed pattern interactions over ~30 mm
Semi-physical model for non-uniformity caused bytool designpattern design
Ability to predict non-uniformity on 1-mm lateral gridfor any etched pattern
28
2. Polymer Hot Embossing
Background
Simulations of uniformity
Characterization experiments for uniformity
29
Background: Hot Embossing
• Goal:
– Formation of surface structures inpolymer or other materials
– Microfluidics & other applications
• Key Issue:
– Embossing requires flow ofdisplaced material: patterndependencies
Hot Embossing
3030
Hot Micro- and Nano-Embossing
tload thold
load
temperature
Glass-transition
temperature
• To choose an optimal process, we need toassign values to
• Heat
• Time
• Our load and temperature are constrained by
• Equipment
• Stamp and substrate properties
time
3333
PMMA in compression
using model of N.M. Ames, Ph.D. thesis, MIT, 2007
Compare this ratio, P/Q, to the Deborah number, tmaterial/tload
3434
Starting point: linear-elastic material model
• Embossing done at high temperature, with low elastic modulus
• Deformation ‘frozen’ in place by cooling before unloading
• Wish to compute deformation of a layer when embossed with anarbitrarily patterned stamp
• Take discretized representations of stamp and substrate
E(T)
3535
Response of material tounit pressure at one location
( )
( ) ( )dd
yx
p
Eyxw
+=
22
2 ,1),(
( ) ( ) ( ) ( )[ ]11122122
2
,,,,,
1yxfyxfyxfyxf
EF ji +=
radius, r
w
( ) ( ) ( )2222lnln, yxyxyxxyyxf +++++=
General load response:
Response to unit pressure in a single element of the mesh:
x1,y1
x2,y2
Unit pressure here
Fi,j defined here
Point load response
wr = constant
load
3636
1-D verification of approach for PMMA at 130 °C
Extracted Young’s
modulus ~ 5 MPa at 130 °C
• Iteratively find distributionof pressure consistentwith stamp remainingrigid while polymerdeforms
• Fit elastic modulus that isconsistent with observeddeformations
3737
2-D linear-elastic modelsucceeds with PMMA at 125 °C
1
3
5
7
2
4
6
8
Lateral position (mm) Lateral position (mm)
To
po
gra
ph
y (
mic
ron
)
0
15 μm
1 2 3 4 5 6 7 8
Simulation
Thick, linear-elastic material model
Experimental data
1 mm
Si stamp
cavity
protrusion
4343
Yielding at 110 °C
Simple estimates of strain rate:
N.M. Ames, Ph.D. Thesis, MIT, 2007
stamp
polymer
w
penetration
holdt2
w
npenetratio 10-3 to 10-1 during loading
10-4 to 10-3 during hold
Local contact pressure
at feature corners > 8 MPa
4444
Modeling combinedelastic/plastic behavior
Compressive
stress
Compressive
strain0.4
Yield stress
Plastic flow
De << 1 De >> 1De ~ 1
Consider
plastic
deformation
instantaneous
Consider flow to be
measurable but not to modify
the pressure distribution
substantially during hold
Deborah number
De = tmaterial/tload, hold
4545
Modeling combinedelastic/plastic behavior
( ) ( ) ( ) ( ) ( ) ( )x,ypf
holdBtA
yieldpx,ypx,y
efx,ypx,yw ++=
Plastic flow
De << 1 De >> 1De ~ 1
fp
radius
Tuned to represent cases from
capillary filling to
non-slip Poiseuille flow
Existing
linear-elastic
component
Material
compressed
Elastic: E(T)
Plastic flow
fe
radius
Volume
conserved
4646
Status and future directions –polymer hot emboss modeling
• The merits of a linear-elastic embossing polymer modelhave been probed
• This simulation approach completes an 800x800-elementsimulation in:• ~ 45 s (without filling)
• ~ 4 min (with some filling)
• Our computational approach can be extended to capture yieldingand plastic flow
• Is a single pressure distribution solution sufficient to modelvisco-elasto-plastic behaviour?
• Abstract further: mesh elements containing many features
47
Conclusions
• Spatial variation a concern in MEMS fabricationprocesses
• Semi-empirical modeling approach developed:
– Physical model basis
– Process characterization for tool/layoutdependencies
• Applications:
– Deep reactive ion etch (DRIE)
– Chemical-mechanical polishing (CMP) [not shown]
– Current focus: Polymer hot embossing
48
Acknowledgements
• Singapore-MIT Alliance (SMA)
• Surface Technology Systems Ltd.
• Hongwei Sun, Tyrone Hill, Ali Farahanchi (MIT)
• Nici Ames, Matthew Dirckx, David Hardt,and Lallit Anand (MIT); Yee Cheong Lam (NTU)
• Ciprian Iliescu and Bangtao Chen (Institute ofBioengineering and Nanotechnology, Singapore)