Modeling of MEMS Fabrication Processes

Modeling of MEMS Fabrication Processes Prof. Duane Boning Microsystems Technology Laboratories Electrical Engineering and Computer Science Massachusetts Institute of Technology September 28, 2007

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 2

Non-uniformity problems in MEMS Plasma etch variation Silicon Silicon oxide Silicon Etch depth variation: imbalance in MIT microengine rotor ~10 mm turbine blades mask layout e.g. A.H. Epstein et al., Proc. Transducers 97 3

Non-uniformity problems in MEMS Embossing for microfluidics manufacture Si stamp Thermoplastic polymer Cover Surface nonuniformity: failure to seal Cover Channel depth nonuniformity from embossing polymer flow 4

Outline Background spatial variation in MEMS fabrication processes 1. Deep reactive ion etch (DRIE) 2. Polymer hot embossing Conclusions 5

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 6

Inductively-coupled plasma in DRIE chamber cross-section vacuum chamber ~10-100 mtorr gas inlet wafer plasma X ~ r.f. supply to excite plasma to wafer load lock chuck exhaust ~ independent control of ions acceleration towards wafer 7

Time-multiplexed Bosch processing flow rate 1. mask 2. SF 6 etch SF 6 C 4 F 8 3. C 4 F 8 passivation 4. SF 6 etch (6 11s) (10 15s) time SF 6 dissociates: SF 6 + + e S xfy + S xfy + F + e Ion-assisted chemical etching: Si + nf SiF n 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) 8

Non-uniformity at three length scales device/ die spatial variation wafer in cross-section feature-scale inter- and intradevice wafer/chamber-scale aspect ratiodependent etching (ARDE) competition for reactants; diffusion ion and radical flux distribution waferlevel loading F X 9

Approach: Characterization using family of test wafer designs (a) Symmetrical loading (b) 5% average loading (c) 95% average loading 10

Observed wafer/chamber-scale variation 1% 5% 1 20% 81 position index 70% 95% pattern density test patterns H.K. Taylor et al., J. Electrochem. Soc., May 2006 11

Observed pattern-dependent variation Average pattern density 5% throughout Localized to differing extents H.K. Taylor et al., J. Electrochem. Soc., May 2006. 12

Modeling basis: Ion-neutral synergism at silicon surface Models for etching rate Mogab (1977) 1 : etch rate varies inversely with loading Gottscho (1992) 2 : etch rate set by ion-neutral synergism R: etch rate : surface coverage ke i : activity constant for ions vs 0 : activity constant for radicals ion flux, ion J i R R 1 J i = Silicon 1 R kei Ji vs0 F Neutral neutral flux, J n silicon silicon J n adsorbed Adsorbed neutrals neutrals = k E J i i = vs ( ) 0 1 J n + 1 J n 1 J. Electrochem. Soc. 124 p1262 (1977). 2 J. Vac. Sci. Tech. B, 10, 2133 (1992) 13

Concentration equilibrium above wafer surface J i (x, y) lateral transport C(x, y) generation, recombination Consumption: J n (x, y) mask silicon C, C e : fluorine concentration G: fluorine generation rate ave : wafer-average pattern density Solving for concentration of F neutrals in steady state at (x, y): G ( x y) [ + ( 1 )] C ( x, y) ( x, y) e, 1 ave 2 ave e = C e ( x, y) = C G( x, y) [ + ( 1 )] 1 1 ave 2 ave + 0 Neglecting lateral transport rate constant selectivity loading /pattern density T.F. Hill, H. Sun, H.K. Taylor, and D.S. Boning, Proc. MEMS 2005 14

Ion-neutral synergism plus Mogab model Ion-neutral synergism Equilibrium fluorine concentration 1 = 1 + R kei Ji vs0 1 J n C e ( x, y) = G( x, y) [ + ( 1 )] 1 1 ave 2 ave + J ( x, y) = uˆ C( x y) n z, R ( x, y) = [ kei J i ( x, y) ][ vs uˆ 0 zg( x, y) ] [ ke J ( x, y) ]{ [ + ( 1 )] + 1} + [ vs uˆ G( x, y) ] R i i ( x, y) = A 1 ave 2 ave A( x, y) B( x, y) ( x, y) { [ + ( 1 )] + 1} B( x y) 1 ave 2 ave +, 0 z T.F. Hill, H. Sun, H.K. Taylor, and D.S. Boning, Proc. MEMS 2005 15

Tuning chamber model to uniform-pattern data Position on wafer 81 81 A(x,y) B(x,y) 1 1 16

Non-uniformity at three length scales device/ die spatial variation wafer in cross-section feature-scale inter- and intradevice wafer/chamber-scale aspect ratiodependent etching (ARDE) competition for reactants; diffusion ion and radical flux distribution waferlevel loading F 17

Measurement points experience a local effective density 18

An integrated wafer- and die-scale model Around every location with non-average pattern density, there is a perturbation of F concentration 19

An integrated wafer- and die-scale model Assuming that the present 1 mm 2 location is the only one on the wafer with non-average pattern density, re-write concentration equilibrium, and, element-wise, obtain the map C isol (x,y): G ( x, y) { ( x, y) + [ 1 ( x, y) ]} C ( x, y) C ( x, y) C ( x, y) C ( x, y) 2D + 2 r0 isol e isol 1 2 isol Generation Consumption Recombination r ln r c 0 Lateral transport term 0 20

An integrated wafer- and die-scale model Map of surplus fluorine concentration defined as C isol (x, y) C e (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 21

Integrated model fits with error 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) 22

A two-level model, tuned for each tool + recipe characterization wafers characterization wafers A B filter magnitude radial distance + 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. 23

Characterizing other tool-recipe combinations STS2 at MTL (25 mtorr) STS Pegasus (86 mtorr) Etch rate (μm/min) 24

Putting two-level model into action discretized mask design + scalar constants two-level model takes a few seconds to run drafting software refine mask design highlight problems on-screen 25

CAD tool for nonuniformity prediction Die-scale variation Chamber-scale variation Combined prediction Discretized mask design Ali Farahanchi 26

DRIE Modeling Contributions Understanding of uniformity s dependence on pattern density and localization Observed pattern interactions over ~30 mm Semi-physical model for non-uniformity caused by tool design pattern design Ability to predict non-uniformity on 1-mm lateral grid for any etched pattern 27

2. Polymer Hot Embossing Background Simulations of uniformity Characterization experiments for uniformity 28

Background: Hot Embossing Hot Embossing Goal: Formation of surface structures in polymer or other materials Microfluidics & other applications Key Issue: Embossing requires flow of displaced material: pattern dependencies 29

Hot Micro- and Nano-Embossing Glass-transition temperature temperature load time t load t hold To choose an optimal process, we need to assign values to Heat Time Our load and temperature are constrained by Equipment Stamp and substrate properties 30

PMMA in compression N.M. Ames, Ph.D. thesis, MIT, 2007 31

PMMA in compression, 140 C using model of N.M. Ames, Ph.D. thesis, MIT, 2007 32

PMMA in compression Compare this ratio, P/Q, to the Deborah number, t material /t load using model of N.M. Ames, Ph.D. thesis, MIT, 2007 33

Starting point: linear-elastic material model E(T) 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 an arbitrarily patterned stamp Take discretized representations of stamp and substrate 34

Response of material to unit pressure at one location General load response: 1 w( x, y) = E 2 p(, ) 2 ( x ) + ( y ) 2 d d w load radius, r Point load response wr = constant Response to unit pressure in a single element of the mesh: 1 = E 2 F i, j 2 2 1 2 2 1 + 1, [ f ( x, y ) f ( x, y ) f ( x, y ) f ( x y )] ( ) ( ) ( ) x, y = y ln x + x 2 + y 2 + x ln y + x 2 y 2 f + 1 F i,j defined here x 1,y 1 x 2,y 2 Unit pressure here 35

1-D verification of approach for PMMA at 130 C Iteratively find distribution of pressure consistent with stamp remaining rigid while polymer deforms Fit elastic modulus that is consistent with observed deformations Extracted Young s modulus ~ 5 MPa at 130 C 36

2-D linear-elastic model succeeds with PMMA at 125 C Si stamp 1 2 cavity Simulation protrusion 1 mm 15 μm Topography (micron) 3 5 4 6 7 8 1 2 3 4 5 6 7 8 0 Lateral position (mm) Lateral position (mm) Thick, linear-elastic material model Experimental data 37

Linear-Elastic Model Succeeds at 125 C, p ave = 0.5 MPa stamp penetration w polymer p 38

Linear-Elastic Model Succeeds at 125 C, p ave = 1 MPa Features filled, 1MPa 39

Linear-elastic model succeeds below yielding at other temperatures 40

Extracted PMMA Young s moduli from 110 to 140 C 41

Material flows under an average pressure of 8 MPa at 110 C stamp polymer 42

Yielding at 110 C stamp penetration polymer w Simple estimates of strain rate: penetration w t hold 2 10-3 to 10-1 during loading 10-4 to 10-3 during hold Local contact pressure at feature corners > 8 MPa N.M. Ames, Ph.D. Thesis, MIT, 2007 43

Modeling combined elastic/plastic behavior Compressive stress Yield stress 0.4 Compressive strain Plastic flow Deborah number De = t material /t load, hold De << 1 De ~ 1 De >> 1 Consider plastic deformation instantaneous Consider flow to be measurable but not to modify the pressure distribution substantially during hold 44

Modeling combined elastic/plastic behavior Elastic: E(T) Plastic flow De << 1 De ~ 1 De >> 1 Plastic flow w + e yield ( ) ( ) ( ) ( ) ( ) x,y = p x,y f x,y + p x,y p A Bt f ( x,y) hold p Existing linear-elastic component Tuned to represent cases from capillary filling to non-slip Poiseuille flow f e Material compressed f p Volume conserved radius radius 45

Status and future directions polymer hot emboss modeling The merits of a linear-elastic embossing polymer model have been probed This simulation approach completes an 800x800-element simulation in: ~ 45 s (without filling) ~ 4 min (with some filling) Our computational approach can be extended to capture yielding and plastic flow Is a single pressure distribution solution sufficient to model visco-elasto-plastic behaviour? Abstract further: mesh elements containing many features 46

Conclusions Spatial variation a concern in MEMS fabrication processes Semi-empirical modeling approach developed: Physical model basis Process characterization for tool/layout dependencies Applications: Deep reactive ion etch (DRIE) Chemical-mechanical polishing (CMP) [not shown] Current focus: Polymer hot embossing 47

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 of Bioengineering and Nanotechnology, Singapore) 48