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Pattern Dependent Modeling of Polymer Microfluidics Manufacturing Processes Duane Boning and Hayden Taylor Microsystems Technology Laboratories Electrical Engineering and Computer Science Massachusetts Institute of Technology October 24, 2007 Spatial Variation in MEMS Processes Wafer Scale Chip/Part 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 Previous Work: Variation in DRIE wafer in cross-section device/‘die’ spatial variation feature-scale aspect ratiodependent etching (ARDE) wafer/chamber-scale inter- and intradevice competition for reactants; diffusion F ion and radical flux distribution waferlevel ‘loading’ X 3 Current Focus: 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 4 Hot Micro- and Nano-Embossing Glass transition temperature temperature stamp load polymer tload thold time • To choose an optimal process, we need to assign values to – Heat – Time • Load and temperature are constrained by – Equipment – Stamp and substrate properties 5 Hot Micro-Embossing Compared to Nanoimprint Lithography stamp stamp polymer polymer typ. 10 micron stiff substrate typ. 100 nm Sub-micrometer features Chou et al Appl. Phys. Lett. 67(21): 3114 (1995) Micro-embossing Nano-imprint 6 PMMA: A Typical Embossing Material • Near Tg: elasto-plastic • ~125-170 C: rubbery • Higher temperatures: viscous fluid Stress-strain curves for PMMA at various temperatures, strain rate 0.001/s (solid line is data; dashed line is model) Ames et al. Proc. ICOMM 2006 7 PMMA in Compression N.M. Ames, Ph.D. thesis, MIT, 2007 8 PMMA in Compression, 140 °C Unload below TG using model of N.M. Ames, Ph.D. thesis, MIT, 2007 Unload above TG 9 PMMA in Compression Compare this ratio, P/Q, to the Deborah number, tmaterial/tload using model of N.M. Ames, Ph.D. thesis, MIT, 2007 10 10 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 11 Response of Material to Unit Pressure at One Location load radius, r General load response: 1 2 w( x, y ) E p , x y 2 2 dd Point load response wr = constant w Response to unit pressure in a single element of the mesh: 1 f x2 , y2 f x1 , y2 f x2 , y1 f x1 , y1 Fi , j E f x, y y ln x x 2 y 2 x ln y x 2 y 2 2 Fi,j defined here x2,y2 x1,y1 Unit pressure here 12 12 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 13 13 2-D Characterization Test Pattern stamp cavity protrusion 14 14 2-D Linear-Elastic Model Succeeds with PMMA at 125 °C Si stamp 1 2 3 4 5 6 7 8 Lateral position (mm) Lateral position (mm) protrusion 1 mm Simulation 15 μm 0 1 2 3 4 5 6 7 8 Topography (micron) cavity Thick, linear-elastic material model Experimental data 15 15 Linear-Elastic Model Succeeds at 125 °C, pave = 0.5 MPa stamp penetration polymer w p 16 16 Linear-Elastic Model Succeeds at 125 °C, pave = 1 MPa Features filled, 1MPa 17 17 Linear-Elastic Model Succeeds Below Yielding at Other Temperatures 18 18 Extracted PMMA Young’s Moduli from 110 to 140 °C 19 19 Material Flows Under an Average Pressure of 8 MPa at 110 °C stamp polymer 20 20 Scaling Point Load Response Function for Flow 21 21 Time-Stepped Simulation of Flow stamp stamp stamp stamp stamp polymer polymer polymer polymer polymer t=0 t = Δt t = 2Δt t = 3Δt t = nΔt z p [i, j,0] 0 i, j z[i, j, t ] z0[t ] z p [i, j, t ] ze [i, j, t ] z[i, j, t ] z p [i, j, t t ] (1 t )Fe [i, j] p[i, j, t ] 22 22 2-D Characterization Test Pattern cavity protrusion 23 23 Scaling Point Load Response Function for Flow 24 24 Scaling Point Load Response Function for Flow 25 25 Scaling Point Load Response Function for Flow 26 26 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 27 27 Modeling Combined Elastic/Plastic Behavior Compressive stress Yield stress Compressive strain 0.4 Plastic flow Deborah number De = tmaterial/tload, hold De << 1 Consider plastic deformation instantaneous De ~ 1 De >> 1 Consider flow to be measurable but not to modify the pressure distribution substantially during hold 28 28 Modeling Combined Elastic/Plastic Behavior Plastic flow Elastic: E(T) De << 1 De ~ 1 De >> 1 Plastic flow A Bt wx,y px,y f x,y px,y p f x,y e yield hold p Existing linear-elastic component fe Tuned to represent cases from capillary filling to non-slip Poiseuille flow Material compressed radius fp Volume conserved radius 29 29 Embossing Simulation: Thin Layers Rowland et al., JVST B 23 p.2958 30 30 Status and Future Directions – Polymer Hot Embossing Model • The merits of a linear-elastic embossing polymer model have been probed • This simulation approach completes an 800x800element 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 31 31 Conclusions • Spatial variation a concern in MEMS fabrication processes • Semi-empirical modeling approach developed: – Physical model basis – Process characterization for tool/layout dependencies • Applications: – Current focus: Polymer hot embossing – Deep reactive ion etch (DRIE) – Chemical-mechanical polishing (CMP) 32 Acknowledgements • Singapore-MIT Alliance (SMA) • Ciprian Iliescu and Bangtao Chen (Institute of Bioengineering and Nanotechnology, Singapore • Nici Ames, Matthew Dirckx, David Hardt, and Lallit Anand (MIT); Yee Cheong Lam (NTU) 33