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

16

Tuning chamber model to uniform-pattern data

A(x,y)

1

81

1

81

Position on wafer

B(x,y)

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

18

Measurement points experiencea local ‘effective’ density

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

3131

PMMA in compression

N.M. Ames, Ph.D. thesis, MIT, 2007

3232

PMMA in compression, 140 °C

using model of N.M. Ames, Ph.D. thesis, MIT, 2007

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

3838

Linear-Elastic Model Succeedsat 125 °C, pave = 0.5 MPa

stamp

penetration

w

p

polymer

3939

Linear-Elastic Model Succeedsat 125 °C, pave = 1 MPa

Features filled,

1MPa

4040

Linear-elastic model succeedsbelow yielding at other temperatures

4141

Extracted PMMA Young’smoduli from 110 to 140 °C

4242

Material flows under an averagepressure of 8 MPa at 110 °C

stamp

polymer

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)