Asymmetric fluid-structure dynamics in nanoscale imprint lithography

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corrugations on the flow behavior of parallel plate squeeze films have beenstudied from a theoretical and numerical perspective [Freeland 2000]. Freelanddeveloped analytic and numerical solutions for two and three-dimensionalgeometries for flows found in both SFIL and nanoimprint lithography. The noninertialsinkage of a flat, inclined plate has been thoroughly studied. However,the effect of the asymmetry in the pressure distribution across the plate wasneglected in computing the sinkage rate [Moore 1964]. Moore proceeded withthe assumption of a pressure distribution, which is a parabola for any sectionperpendicular to the directions spanning the plate. This assumption neglects thecorner effects, but is useful in approximating the three-dimensional pressure dueto a specified load condition. In this thesis, the effect of a non-symmetricpressure variation across a smooth template is treated analytically and applied to anumerical simulation of the equations of motion for the SFIL machine.In this chapter, the Reynolds equation has been used to study the case ofa squeeze film flow between a flat, quartz template and flat, rigid wafer substrate.First a derivation of the incompressible Reynolds equation is given. Next, areduced form of the Reynolds equation is considered. The two-dimensionalReynolds equation can be applied to flow geometries where side leakage can beneglected in one of the lateral directions; the squeeze film in the y direction canbe considered infinite. The case of an infinite, flat surface that is parallel to asubstrate is presented. This is extended to the case of an infinite, flat surface thatis inclined relative to a substrate. Applying specific boundary conditions,analytical solutions for the pressure, force, and torque are obtained from the twodimensionalReynolds equation.The analytical solutions to the three-dimensional problem (finite plategeometry) are reviewed as used as a benchmark for comparing the twodimensionalsolutions. The three-dimensional solution for the case of a parallel19
square plate with fluid completely filling the gap is considered. Finally, theclosed-form solution to the case of parallel circular plates is presented. Theaxisymmetric case has a readily available solution to the problem of a growingfluid boundary layer.2.2 THE INCOMPRESSIBLE REYNOLDS EQUATIONThe Reynolds equation is the basic equation for fluid lubrication. Itprovides a relationship between the thickness of a fluid film and the pressure.Since the problem of interest is not a traditional lubrication flow, it is aprerequisite that careful consideration be given to the assumptions used inmodeling the squeeze film effect of the etch barrier liquid. The characteristiclength for the expulsion step in the SFIL process is on the order of hundreds ofnanometers. A desired final base layer thickness of 100 - 200 nm is virtually atthe limit of contacting plates in most traditional engineering contexts. For a fluidlayer at these thicknesses, does the assumption of the fluid as a continuum remainvalid? An empirical criterion such as the Stribeck Curve 2 shows that for filmsabove about 10 nm, the lubricant film behavior can be described by bulk,continuum properties [Bhushan 1995]. Also, at this scale of motion, the effectof van der Waals forces could become important as compared with the pressureforces from the bulk fluid and the external forces. The pressures created by thevan der Waals forces are proportional toPVWA∝ or havg∝ 3AπP[2.1]36πhavg6 applied2 Refer to page 292 of the Handbook of Micro/Nanotribology by Bharat Bhushan.20
Page 3 and 4: Asymmetric Fluid-StructureDynamics
Page 5 and 6: AcknowledgementsFirst of all, I wou
Page 7 and 8: Table of ContentsList of Tables ...
Page 9 and 10: 6.2.3 Control Software ............
Page 11 and 12: List of FiguresFigure 1.1 Optical m
Page 13 and 14: Figure 6.9 Force due to fluid press
Page 15 and 16: The exponential escalation in the c
Page 17 and 18: 1.2.1 Optical Lithography Process O
Page 19 and 20: patterns to the desired specificati
Page 21 and 22: 1.3 STEP AND FLASH IMPRINT LITHOGRA
Page 23 and 24: transferlayerwaferStep 1: Spin-coat
Page 25 and 26: the etch barrier fluid has an infin
Page 27 and 28: Researchers at the University of Te
Page 29 and 30: stage development. Design requireme
Page 31: minimum and maximum base layer thic
Page 35 and 36: 7. The fluid is assumed incompressi
Page 37 and 38: ⎛ ∂ ⎞⎜τdz ⎟dxdy⎝+ τ
Page 39 and 40: where V 1 and V 2 correspond to U 1
Page 41 and 42: ddx⎛⎜h⎝3dp ⎞ ∂h⎟ = 12µ
Page 43 and 44: dhdtLzhθh αh βxx αx = 0 x βFig
Page 45 and 46: C2hhtanθ2 21= αxαxβxαxβ( ) (
Page 47 and 48: ⎛⎜ hD⎜⎝6C ⎞1tanθ⎟24µh
Page 49 and 50: at the plate edges. Freeland obtain
Page 51 and 52: For the case of 10 psi of pressure,
Page 53 and 54: are the same as the width of the tr
Page 55 and 56: orientation stages relative to the
Page 57 and 58: Wafer ChuckAir SolenoidMounting Pla
Page 59 and 60: The one degree-of-freedom template
Page 61 and 62: Figure 3.7 Initial and final desire
Page 63 and 64: 7645321Figure 3.8 Active stage prot
Page 65 and 66: Chapter 4: Real-Time Gap Sensing Vi
Page 67 and 68: R( λ)( 4πnd/ λ)2 2 −2αd−αd
Page 69 and 70: Rate Monitoring and is adapted for
Page 71 and 72: 10090807060PSD504030201000 500 1000
Page 73 and 74: the multi-imprint and active stage
Page 75 and 76: 5.2.2 Composite Stiffness and Dampi
Page 77 and 78: ( )2πR−1.5clamping length of 1.5
Page 79 and 80: 65actuator height, micron432100 0.1
Page 81 and 82: time marching is required to minimi
Page 83 and 84:
5.4 SQUEEZE FILM DYNAMICS OF INCLIN
Page 85 and 86:
force, lb504540353025201510500 0.05
Page 87 and 88:
improve upon these results by tunin
Page 89 and 90:
Again, Figure 5.12 shows that a bas
Page 91 and 92:
optically flat quartz plate coated
Page 93 and 94:
optic probe was illuminated with a
Page 95 and 96:
Figure 6.3 Screenshot of control so
Page 97 and 98:
6.3 EXPERIMENTAL PROCEDUREExperimen
Page 99 and 100:
film thickness, nm40003800360034003
Page 101 and 102:
Figure 6.6 shows that the force due
Page 103 and 104:
similar. In Figures 6.7 and 6.8, th
Page 105 and 106:
500045004000film thickness, nm35003
Page 107 and 108:
Figure 6.13 shows the force-time pl
Page 109 and 110:
⎛ n ⎞in the FFT that is not ind
Page 111 and 112:
when measuring thin layers in the r
Page 113 and 114:
is a major milestone. The next step
Page 115 and 116:
7.2.4 Control schemeThe ultimate go
Page 117 and 118:
Assuming the liquid perfectly wets
Page 119 and 120:
ase h (for a semi-circular notch,h
Page 121 and 122:
Chou, S. Y., P .R. Krauss, and P. J
Page 123 and 124:
Tripp, J. H. 1983. Surface roughnes
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Asymmetric fluid-structure dynamics in nanoscale imprint lithography

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