The effect of droplet size, droplet placement and gas dissolution on throughput and defect rate in UV Nanoimprint Lithography

Transcription

1 The effect of droplet size, droplet placement and gas dissolution on throughput and defect rate in UV Nanoimprint Lithography Akhilesh Jain The University of Texas at Austin, McKetta Department of Chemical Engineering, 200 E. Dean Keeton St. Stop C0400, Austin, TX Andrew Spann The University of Texas at Austin, McKetta Department of Chemical Engineering, 200 E. Dean Keeton St. Stop C0400, Austin, TX Roger T. Bonnecaze a) The University of Texas at Austin, McKetta Department of Chemical Engineering, 200 E. Dean Keeton St. Stop C0400, Austin, TX a) Electronic mail: Simulation of multi-drop spreading in UV Nanoimprint Lithography is performed to study the effects of droplet size, droplet arrangement, droplet placement error and gas diffusion on filling-time and defects. Simulations are carried out for square, hexagonal and modified hexagonal arrangements of up to 1024 droplets ink-jetted on a substrate to determine the optimum arrangement for minimum imprint time. The effect of error in droplet placement by the inkjet dispenser on the imprint time for different droplet sizes is also investigated. The square droplet arrangement leads to the shortest fluid filling time for a flat template. The filling time increases significantly for droplet placement errors of more than 0.01% relative to the width of the substrate. A model is presented to study the diffusion of gas encapsulated between droplets into the resist. A dimensionless parameter α ~ D kh Ho measures the relative importance gas diffusion to hydrodynamics, where D is 1

2 the gas diffusion constant, k H is the Henry s law constant, μ is the resist viscosity and is the surface tension of the imprint resist. For small values of α, gas diffusion is slower than resist spreading and defect size is diffusion-controlled while for larger values, gas diffusion is faster than resist spreading and the defect size is hydrodynamically-controlled. Scaling laws are developed to extrapolate predictions on filling time, residual layer thickness and defects here for hundreds to a thousand droplets to tens and hundreds of thousands of droplets. I. INTRODUCTION UV nanoimprint lithography (UVNIL) is a low cost, high throughput process to replicate high resolution nano- and micro- patterns from a template onto a rigid substrate 1. UVNIL has been used for fabrication of many nano-enabled devices such as high density bit pattern media 2,3, photonics 4,5 and displays 6. Lithography techniques like immersion and EUV lithography also provide high resolution patterning; however, these techniques are more expensive with lower throughput 7. Pattern resolution in these techniques is limited by the wavelength of light and the numerical aperture of the optical system. Patterning sub-100 nm structures require complicated optical elements which increases the cost of exposure tools. UVNIL does not have these limitations and has the following competitive advantages over other lithography techniques such as immersion and EUV lithography: (i) pattern resolution in UVNIL is essentially only limited by the resolution of the template pattern 8,9 ; (ii) UVNIL has demonstrated sub-100 nm pattern replication for features with high aspect ratios 10 ; and (iii) the cost of ownership per wafer is significantly lower for UVNIL compared to photolithography and EUV 11,12. The process for UVNIL includes a transparent template (also called a mask or mold), a rigid (glass or silicon) substrate, an inkjet dispenser and a UV light source as shown in 2

3 Figure 1. The stepper template is patterned with micro- or nano- features of the same size as the final pattern 13. The nanoimprint process begins with the inkjet dispensing low viscosity photo-curable resist droplets on the substrate. A typical inkjet dispenser is capable of dispensing droplets of volume as low as one picoliter (10-15 m 3 ) at a variable pitch ranging from 85 m to 150 m 14,15. The droplet density and arrangement depend on the volume of resist required to fill the local cavities on the template and achieve a desired residual layer thickness (RLT). The stepper template is then lowered on the droplets until they merge together to form a uniform resist film. Once the fluid spreading is complete and the entire template is filled, UV light is irradiated on the resist through the transparent template. The UV light causes photo-polymerization of the monomers to create a solid resist film. The exposure dose and time depend on the resist material and RLT 16,17. Once the resist is completely cured, the template is peeled off from the resist through controlled movement of the template. The surface energy of the template is low compared to the substrate, which allows the resist to preferentially adhere to the substrate. This process is repeated multiple times on different portions of the substrate to imprint a large wafer. 3

4 FIG. 1. (Color online) Schematic showing the main steps for pattern replication in UV Nano Imprint Lithography (UVNIL). High throughput and low defectivity are critical for high volume manufacturing of devices such as flash memory using UVNIL 18,19. A recent cost of ownership calculations shows that a four-imprint tool system needs to produce 15 wafers per hour per imprint station at a defect density of 0.1 defects/cm 2 to compete with a single ArF self-aligned quadruple patterning tool 18. This cost advantage was calculated for 15 nm half pitch grating. With current UVNIL technology, five wafers of size 300 mm can be imprinted per hour per imprint station 19,20. This implies that a three-fold increase in process throughput is required. In UVNIL UV exposure and template/resist separation typically require 0.1 to 0.2 seconds for a 4

5 template of size 6 in x 6 in 19. UV exposure and template/resist separation are well understood and are not the throughput limiting step. The throughput is limited by fluid filling, which takes about one to five seconds depending on the droplet size and surface energy of the substrate and template 20. Fluid filling involves resist spreading over the entire wafer and the surrounding gas escaping from the features on the template as the resist fills these features. High throughput may lead to defects, namely unfilled regions, in the final pattern since the resist may not have sufficient time to fill the features or the gas may not escape the feature completely. By optimizing droplet size and allowed spreading time, defectivity of the imprint process can be improved. Khusnatdinov et al. conducted nanoimprinting with varying drop volume and spread times to demonstrate their effect on defect density 20. They demonstrated a ten-fold improvement in defect rate by reducing the drop volume from 1.5 pl to 0.9 pl when nanoimprinting with a 28 nm half pitch imprint mask. They also demonstrated that the defect density can be reduced from 30 defects/cm 2 to 0.1 defects/cm 2 by increasing the filling time from 0.7 seconds to 1 second. Thus, defect density needs to be considered in the context of throughput 19. Conducting experiments to identify optimum droplet size and arrangement for different template patterns can be expensive and laborious. Previously, modeling and simulation has been effective in studying fluid behavior and gas trapping during droplet spreading in UVNIL. Colburn et al. studied the spreading of droplets between a template and a substrate for UV imprint lithography under several template control schemes 21. They proposed models for pressure distribution and filling time for the droplets as the template is moved with a fixed velocity, fixed pressure or constant applied force. Reddy et al. presented a dynamic, multi-drop simulation based on lubrication theory and simulated spreading of up 5

6 to 49 drops with a flat template 22. Various mechanisms for interface advancement have been described to predict incomplete feature filling leading to air trapping. Local effect of sharp features on interface advancement with patterned template has been explored Chauhan et al. analyzed the diffusion of gas entrapped in the features through liquid imprint resist and found that the absolute time for diffusion of a gas, entrapped in features is very short and gas diffusion is not a cause for non-filling of features during the UVNIL 26. We simulate multi-drop spreading in UVNIL with a flat template to study the effect of various factors such as droplet size, droplet arrangement, droplet placement error, gas diffusion that influence filling time and defectivity in the UVNIL process. We explore three droplet arrangements: (a) square and (b) hexagonal and (c) modified hexagonal (as shown in Figure 3). We also study the effect of error in droplet placement by the inkjet dispenser on the filling time for different droplet sizes. For the same final RLT, if the size of the droplets is reduced, more droplets need to be dispensed in order to fill the template completely. We perform simulations with up to 1024 droplets. The objective is to understand the effect of droplet arrangement, droplet placement accuracy and drop size on throughput and identify the optimum droplet arrangement for minimum throughput. We also present a model to study the diffusion of gas encapsulated between droplets into the resist and identify parameters that determine whether the defect size and density are controlled by gas diffusion or hydrodynamic spreading. 6

7 II. SIMULATION METHOD FIG. 2. (Color online) (a) Droplets between a substrate and a template. R m is the radius of curvature of the liquid meniscus and R is the radius of a droplet. H is the gap between the top and bottom surface at any time. θ 1 and θ 2 are the contact angles of the liquid with the top and bottom surface. (b) Schematic shows one droplet on a substrate (top). The domain is discretized into cells and the fluid content in each cell is tracked based on a characteristic function f (bottom). f is 1 and 0 for cells completely filled liquid and gas respectively. For cells partially filled with liquid, f is between 0 and 1. A schematic of droplets with radius R in a gap of height H between a template and a substrate is shown in Figure 2(a). The template can be controlled to move with a constant velocity, constant pressure or applied force 21. As the template approaches the substrate, the liquid-air interface of the droplet exerts capillary force on the template. The fluid flow in the 7

8 gap results in viscous force. This viscous force in the fluid balances the capillary force due to the liquid-air interface and any externally applied force. The typical width of the template L used in UVNIL is cm 20,27. During UVNIL the gap H between the template and the substrate reduces from a few microns to finally a few nanometers. Thus, the size of the gap for fluid flow is much smaller than the transverse length scale. Reynolds lubrication theory is a simplified form of Navier-Stokes equations that can be used to describe this flow of thin fluid films 28,29. The governing equations describing the pressure and velocity fields in the drop using Reynolds lubrication theory are 3 H H P, (1) T 12 2 H U P, (2) 12 where H is the gap between substrate and template, P is the pressure in the droplet, U is the vertically averaged fluid velocity and μ is the viscosity of the imprint resist. The pressure distribution in the fluid is calculated using Equation 1. The capillary pressure at the droplet interface is used as boundary condition to solve the pressure field. The imprint material used in UVNIL is highly wetting and it makes small contact angles θ 1 and θ 2 with the template and the substrate as shown in Figure 2(a). The pressure at the liquid-air interface is given by P int erface 1 1 Patm, (3) R Rm where R m is the radius of curvature of the liquid meniscus, R is the radius of curvature of the droplet and is the surface tension of the imprint. R m can be written in terms of gap H as m 1 2 R H cos cos. In the imprinting process the gap H is much smaller than the radius 8

9 R, therefore R R 1 1 m radius of curvature R and can be written as. Therefore, the capillary pressure at the interface depends only on the P int erface where P atm is the atmospheric pressure, = cos cos 2ˆ Patm, (4) H and is the surface tension of the imprint resist. Initially, we assume that UVNIL is being carried out in vacuum or in a gas which is highly soluble in imprint material and so the gas release and gas trapping can be neglected. The rate of gas dissolution will be considered later in paper. The imprint process begins with the droplets making contact with both the substrate and the template. We assume that the feature filling is practically instantaneous as the feature fill time is negligible compared to the time required for the fluid to spread across the gap 26. The typical values of parameters in the UVNIL process are listed in Table 1. 9

10 Table I: Typical values of the parameters used for the UVNIL process Parameter Symbol Typical value Surface tension 30 dyne/cm Viscosity μ Pa.s Template width L 10 cm Substrate width L 10 cm Feature height H nm Initial gap H o 1 μm Final gap H f 5 nm Resist contact angle θ 1, ᴏ Initial drop height H drop 1 μm 5 mm (1 droplet, H drop = 1 μm) Initial drop radius R o 500 μm (100 droplets, H drop = 1 μm) 200 μm (1024 droplets, H drop = 1 μm) The governing equations and boundary condition [Equations (1-3)] are nondimensionalized using following characteristic values of the variables: P 2 ˆ H, Hc Ho, c o 2 2 ˆ, V ˆ H 6 L and U ˆ H 6 Lwhere H o is the initial gap. The x- and 2 Tc 6 L Ho c o c o y-coordinates are non-dimensionalized using the width of the template L. The dimensionless governing equations are then given by 3 h p v, (5) 10

11 2 u h p, (6) where, p = P/P c, h = H/H c, t = T/T c, v = V/V c = h t and u = U/U c. v is the dimensionless template velocity. The dimensionless pressure boundary condition is p int erface 1. (7) h The equations are solved using the Volume of Fluid (VOF) method 30. The domain is discretized into cells and the fluid content in each cell is tracked based on a characteristic function f. This function is defined as 1 for liquid and 0 for gas (as shown in Figure 2(b)). A value of f between 0 and 1 implies that the cell is partially filled and contains the droplet interface. Several interface reconstruction algorithms were explored to define the droplet interface efficiently and accurately Efficient least squares VOF interface reconstruction algorithm (ELVIRA) described by Pilliod et al. 31 is used here because it is second order accurate and fast. In order to solve for the pressure field, Equation (5) is discretized using a second order finite difference scheme. The typical grid size for one droplet in the simulation is 64 x 64 cells to ensure volume loss is less than 1%. The simulation is stable if the time step is chosen using the Courant-Friedrichs-Lewy (CFL) condition which states that a fluid particle may not travel further than one cell during one time step, i.e. u t < x. A guess for template velocity v is used to initiate the simulation. Generalized minimal residual method (GMRES) method, 35 an iterative method to solve non-symmetric linear systems, is used to numerically calculate the pressure field in Equation (5) at every time step. The total force on the template is calculated by integrating the pressure over the entire substrate. Then the template velocity is adjusted such that the viscous force balances the capillary force and any externally applied force. The pressure field is then recalculated. Once 11

12 the pressure field is known, the total fluid fluxes in the x- and y-directions are determined using Equation (6). Immersed boundary methods are used to extrapolate the pressure field and accurately determine the fluid flux at the interfacial cells 36. Once the velocity field is known, the fluid is advanced using the VOF method and the template is lowered. At every time step, the liquid-air interface is reconstructed based on f. The simulation is stopped once the desired gap is reached and the fluid fills the entire domain. Simulation of large number of droplets in UVNIL is computationally expensive and time consuming since the size of the domain scales as the number of droplets in the process. To accelerate the computations a parallel implementation of GMRES method is used for fast computation of the pressure field. The simulation also becomes computationally more expensive as the droplet size increases since droplet interface, velocity flux and pressure need to be calculated on a larger domain. For faster simulations without loss of accuracy, the domain is remeshed with one-fourth of the initial number of cells every time the radius of the droplet doubles. Mesh sizes and times steps were chosen to ensure no more than one-percent variation of volume. The simulations were performed using the High Performance Computing resources at Texas Advanced Computing Center (TACC) at The University of Texas at Austin. The computational time varied from six hours (for 100 drops) to a day (for 1024 drops) on 12 core Intel Xeon processor with 2.6 GHz clock speed and 64 GB RAM. Simulations were validated against the results by Reddy et al. 22 Simulations are carried out droplets dispensed in square, hexagonal and modified hexagonal arrangements. The droplet placement in a square arrangement is shown in Figure 3(a). The shaded region in Figure 3(a) shows the droplet-free region between the droplets and the edge of the substrate. This region is 16% of the total imprint area for 100 drops in square arrangement. Figure 3(b) shows droplets dispensed in a hexagonal arrangement where 12

13 l l 32(=0.866). The figure illustrates that the droplet-free region (shaded) is 26% of h b the total imprint area. Figure 3(c) shows droplets dispensed in a modified hexagonal * * arrangement where l l In this arrangement the droplet-free region at the edge is h b reduced by increasing the distance between the droplets. This droplet arrangement generally results in unfilled region at the corners and edges. Placing additional droplets at the edge region and corners is necessary to cover the unfilled region and fill the mold edge region completely. Figure 3(c) shows 105 droplets placed in a modified hexagonal arrangement. The droplet-free region (shaded) at the edge reduces from 26% of the substrate in hexagonal arrangement to 16% in the modified hexagonal arrangement. In all the simulations, the gap between the substrate and the template closes from a gap of h = 1 to a final gap of h = The total volume of droplets in all arrangements is equal to the volume required to fill the gap between the template and the substrate to a desired final gap of h = h f =

14 FIG. 3. (Color online) Schematic showing top view of 100 droplets dispensed on a substrate. The droplets are dispensed in a (a) square, (b) hexagonal and (c) modified hexagonal arrangement. The shaded region represents the droplet-free region between the droplets and the edge of the substrate. The droplet-free region at the edge is 16%, 26% and 16% of the substrate in the square, hexagonal and modified hexagonal arrangement respectively. The droplet placement in the different arrangements is shown in the right column. The value of lh lb and l * * h lb is and respectively. 14

15 III. RESULTS Figure 4 shows the spreading of 9 droplets, including the location of the interface and the pressure field within the droplet. The droplets spread on the substrate as a flat template approaches the substrate. The template is driven by the capillary forces in the droplet and has no external force acting on it. The pressure is negative at the liquid-air interface because of the capillary pressure and positive at the center because of the viscous component of the pressure. The capillary pressure is very low towards the end when the gap is very small. At all times the viscous force balances the capillary force creating a net zero force on the template. 15

16 FIG. 4. (Color online) Contour map showing pressure field for a drop spreading on a substrate at h = 0.11, h = and h = The template is flat and has net zero force acting on it. 16

17 FIG. 5. (Color online) Simulation of UVNIL with about 100 droplets dispensed in (a) square arrangement (b) hexagonal arrangement and (c) modified hexagonal arrangement. The droplets reach the edge of the substrate towards the end of the process creating unfilled edges at small gaps. The gap is completely filled with resist at h = The simulation is repeated for multi-drop UVNIL with flat template with zero force acting on it. Figure 5 shows spreading of about 100 droplets that are dispensed in (a) square, (b) hexagonal and (c) modified hexagonal arrangement. The desired final gap is h = The droplets spread as the gap between the template and the substrate closes. At very small gaps, 17

18 the neighboring droplets merge together and spread to the edge of the substrate forming unfilled edge regions. Figure 5 (a-c) show these unfilled edges for the three arrangements. The unfilled edge in the hexagonal arrangement is significantly larger than the unfilled edge in the square or modified hexagonal arrangement. At h = 0.01 the gap is completely filled with the resist. A. Throughput for square, hexagonal and modified hexagonal droplet arrangement Figure 6(a) shows the gap height h as a function of time t for UVNIL with multiple droplets in a square arrangement. For larger number of droplets, the overall filling time is shorter. Figure 6(b) shows that plots for different number of droplets collapse into a single curve when the x-axis is scaled with number of droplets N. An analytical expression for the gap height h as a function of time t was previously derived for multiple droplets spreading on a substrate with net zero force on a flat template 21,22, and is given by h f 1 t 16 N h 1 2. (8) The gap h from the analytical solution is shown in Figure 6(b) by dashed line. The simulation results (solid line) match the analytical solution until the droplets come in contact. The analytical expression is only valid for droplets spreading before merging. As the droplets merge, they form a single a large droplet which spreads slower than multiple droplets 22. Thus, after merging, the simulated filling time starts to deviate from the analytical solution. 18

19 FIG. 6. (Color online) (a) Figure shows gap height h with time t for multiple droplets dispensed in a square arrangement. The template is flat and has net zero force acting on it. (b) The plot collapses into a single curve when the x- axis is scaled with number of droplets N. This shows that filling time scales as 1/N. The inset shows the simulation result starts to deviate from the analytical solution (dashed line) after droplet merging. 19

20 FIG. 7. (Color online) (a) Figure shows gap height h with time t for multiple droplets dispensed in a hexagonal arrangement. The template is flat and has net zero force acting on it. The filling time for droplets arranged in a square arrangement is shown in dashed line. (b) The plot shows the filling time close to the time of droplet merging. The filling time for hexagonal arrangement is longer than that for square arrangement. Figure 7 shows the gap height h as a function of filling time t for hexagonal droplet arrangement. The filling time for about the same number of droplets in square arrangement is 20

21 shown in dashed lines. Similar to square arrangement, the time scales as 1/Nh 2 except at the end of the process when the gap is very small (as shown in Figure 7(b)). For UVNIL with substrate length L = 10 cm, initial gap height of H o = 1 m, resist viscosity = Pa.s and surface tension = 70 dyne/cm, characteristic time T c = 857 seconds. The simulated filling time for 100 droplets in square arrangement is 52 seconds while the filling time for 101 droplets in hexagonal arrangement is 420 seconds. Therefore, for about the same number of droplets, the overall filling time for droplets in hexagonal arrangement is longer than droplets in square arrangement. This is attributed to the different size of unfilled edge regions in square arrangement and hexagonal arrangement. Before droplet spreading begins, the area of droplet-free region between the droplets and the edge of the substrate in square arrangement is 16% of the total imprint area as compared to 26% in hexagonal arrangement (Figure 3). A plot of the unfilled edge area as function of h for 100 droplets in square and hexagonal droplet arrangement is shown in Figure 8(a). The unfilled area is larger in hexagonal arrangement compared to square arrangement. Since the resist spreads slowly after droplet merging, larger unfilled edge region takes longer to fill resulting in longer overall filling time for hexagonal droplet arrangement. 21

22 FIG. 8. (Color online) (a) The area of unfilled edge region for square, hexagonal and modified arrangement of droplets at different gap heights. The hexagonal arrangement has the largest unfilled edge area while square arrangement has the smallest area. (b) The gap height h as a function of filling time t for droplets dispensed in square (100 drops, solid), hexagonal (101 drops, dash) and modified hexagonal (105 drops, dash-dot) arrangements. The inset plot shows the filling time close to the time of droplet merging. The modified hexagonal arrangement provides significant reduction in filling time compared to hexagonal arrangement. The overall filling time is least for droplets dispensed in a square arrangement. 22

23 A modified hexagonal arrangement is proposed in which the size of the unfilled region is reduced by spreading out the droplets in hexagonal arrangement towards the edges and adding more droplets at the corner as shown in Figure 3(c). Before droplet spreading begins, the area of droplet-free region in modified hexagonal arrangement is same as square arrangement (as shown in Figure 3). Figure 8(a) shows that after droplet merging, the unfilled region in modified hexagonal arrangement is larger than that in square arrangement. Figure 8(b) shows the filling time for UVNIL with about 100 droplets dispensed in square, hexagonal and modified hexagonal arrangement. The filling time required to close from a gap height of h = 1 to h = is almost the same for all droplet arrangements as shown in Figure 8(b). However, the filling time required to close from h = to h = 0.01 is considerably longer in hexagonal and modified arrangement compared to square arrangement as shown in Figure 8(b) inset. The filling time for 105 droplets in modified hexagonal arrangement is 143 seconds. This is a significant improvement in filling time compared to 420 seconds required for 101 droplets in hexagonal arrangement. However, imprinting with droplets in square arrangement is still the fastest which requires 52 seconds. Thus the unfilled edge region and droplet arrangement has a significant impact on the overall filling time. For flat templates, the square droplet arrangement is found to be the optimum droplet dispensing pattern to achieve fast droplet spreading. B. Effect of droplet placement error There is error in droplet placement during dispensing based on the size of the droplet and distance between the imprint head from the substrate. This error can manifest itself in droplet spreading and overall filling time. The normalized placement error is defined as 23

24 E, (9) L where L is the substrate width and E is the distance between the target droplet position and the actual droplet position. For a substrate width of 10 cm and a dispensing error of 10 m, is The filling time required for the gap height to close from h = 1 to h = 0.01 is simulated for droplets dispensed in a square arrangement with different placement error. Figure 9 shows spreading of 81 droplets dispensed in a square arrangement with = Unlike square or hexagonal arrangements, the unfilled regions are randomly spread over the substrate due to the droplet placement error. Droplets merge to create larger droplets of different sizes. The resist completely fills the gap between the template and the substrate at h =

25 FIG. 9. (Color online) Simulation of UVNIL with 81 droplets dispensed in a square arrangement with a droplet placement error = 0.02 at (a) t = 2.0x10-4, (b) t = 0.01, (c) t = and (d) t = The gap closes from h = 1 to h = Figure 10 shows gap height as a function of time for different values of droplet placement error. Figure 11(a) shows the overall filling time (t i ) required for the gap to close 25

26 from h = 1 to h = 0.01 for different values of and number of droplets N. For < 10-4, increase in has negligible effect on filling time. However for > 10-4, the filling time increases significantly as increases. As the placement error increases, the droplets start merging sooner creating larger droplets of different sizes. Since larger droplets lead to longer filling time, early droplet merging leads to longer filling time. Figure 11(b) shows the ratio t i, /t i,0 as function of where t i, is the filling time with droplet error placement. t i,0 is the filling time with no placement error. This ratio is a measure of sensitivity of filling time to placement error. We find that for a fixed, t i, /t i,0 is larger for more number of droplets. For example, for = 0.01, t i, /t i,0 is 1.5 for 9 droplets and 4 for 100 droplets. This indicated that the filling time is more sensitive to when droplet size is smaller. 26

27 FIG. 10. (Color online) (a) Gap height h as a function of filling time t for multiple droplets dispensed in a square arrangement with droplet placement error = 0, and (b) The plot shows the filling time close to the time of droplet merging. Larger error in droplet placement leads to longer filling time. 27

28 FIG. 11. (Color online) (a) The total filling time t i required for the gap height to reduce from h = 1 to h = 0.01 as a function of normalized droplet placement error. The total filling time increases as increases. (b) Figure shows the ratio t i, /t i,0 as function of where t i, is the filling time required for the gap to close from h = 1 to h = 0.01 with droplet error placement. t i,0 is the filling time with = 0. 28

29 C. Gas Dissolution When droplets make contact, unfilled regions or gas-filled pockets are formed, as seen, for example in Figure 7. Large gas-pockets will remain after UV curing and result in defects in the imprint process. However, small gas-pockets will dissolve away into the photocurable monomer 26,37. If the gas dissolution is fast relative to hydrodynamic motion of the interfaces, then one may ignore the gas in modeling. If the gas dissolution rate is relatively slower, then waiting times are necessary to remove the voids 20. The simulations presented so far assume that gas dissolution is fast enough that gas dissolution can be ignored. Here, we develop a model to predict the rate of dissolution of gas-pockets, determine the waiting times and the conditions under which it is fast enough to ignore. FIG. 12. (Color online) (a) Schematic showing a gas-pocket trapped between droplets. (b) The gaspocket is modeled as a cylinder of radius R d and height of H. The gas concentration is C d inside the pocket and C i at the gas-liquid interface. Figure 12 shows a schematic for a gas-pocket formed during the UVNIL process. The pocket can be modeled as a cylinder of radius R d and height H. C d is the gas concentration inside the gas-pocket, m is the number of moles and C i is the gas concentration at the gasliquid interface. The gas-liquid interfacial area A d and pocket volume S d are given by 29

30 A d 2 RH (10) d and S d m R H 2 d (11) Cd For a final gap height H f and substrate width L, S d can be written as LH ns LH (12) 2 2 d d f where, n d is the number of gas-pockets and L 2 H f is the total liquid volume in the gap at any time. Rearranging, we get 2 R H H d L n H d f (13) The number of pockets, their initial pocket volume and time of their formation can be derived from the simulations. The rate at which moles of gas deplete from the pocket can be written as dm dc DA dt dx, d (14) gas liquid int erface where D is the diffusion constant. The concentration gradient profile in the resist can be written as 38 dc dx C C i bulk, (15) gas liquid int erface Rd where C bulk is the gas concentration in the bulk of the liquid. C bulk can be assumed to be negligible. C i can be determined using Henry s law 26 : RT C C k s i, d (16) H 30

31 where R is the gas constant, T s is the surrounding temperature and k H is the Henry s law constant. Applying Equations (16) and (15) into (14), we find dm m RT dt S R k s DAd (17) d d H Applying Equation (10) in Equation (17), we find dm mh DRT dt S k s 2 (18) d H Using the ideal gas law, the gas pressure P gas is given by P gas mrt S s (19) d We can non-dimensionalize Equations (11), (13), (18) and (19) using r d = R d /L, h = H/H c, p gas = P gas /P c, t = T/T c, and s d = S d /L 2 H o. The characteristic values of the variables H c, P c and T c 2 are given by H c H o, P 2 ˆ H and T 6 L ˆ H given by c o c o. The dimensionless equations are r h h (20) 2 f d nd h s d r h (21) 2 d dm dt mh (22) s d p gas m RTs 2 s 2ˆ L (23) d where 31

32 12 D RTs (24) ˆ H k o The dimensionless force f app applied on the template taking into account gas pressure is given by 22 H 1 h vr 8h 2 2 fapp N r p 3 gas. (25) f app = 0 when there is no force acting on the template. Since the total volume of the droplets q is equal to the volume of resist required to fill the gap at h = h f, 2 q N r h h f. (26) Assuming net zero force on the template and using Equation (26), we find dh h h v 8N 1 pgas dt h f h 3 2 f. (27) 32

33 FIG. 13. (Color online) Figure shows the plot for (a) diameter of the gas-pocket (in microns) and (b) H-H f as function of time T (in seconds) for number of drops N = 10,000, substrate width L = 2.5 cm and final gap height H f = 10 nm. The results indicate that the pockets shrink faster for larger values of α. The unknowns r d, h, s d, m and p gas are solved numerically using Equations (20-24) and (27). For UVNIL, the typical value of R = Pa.m 3 K -1 mol -1 ; T s = 300 K; D = m 2 s -1 and k H = Pa.m 3 mol -1 26,39. For these values, α ranges from If the 33

34 substrate width is 2.5 cm and desired final RLT is 10 nm, the initial pocket size is about 120 m when 10,000 droplets are dispensed. The diameter of the pockets at different times as it shrinks is shown in Figure 13. The initial pocket filling is orders of magnitude faster compared to the remainder of the pocket filling process. Initially, the gas pressure inside the pocket is about the same as the atmospheric pressure and gas diffusion is slow. As the template lowers, the pocket size reduces and the gas pressure and concentration increase. The increase in gas pressure slows down the template and the gap height becomes closes very slowly as shown in Figure 13(b). The high gas concentration leads to a higher gas diffusion rate. As the gas diffuses into the resist, the pocket slowly fills with the resist. We find that, as α increases, the pockets fill faster due to faster gas diffusion. The black line shows simulation results in which gas diffusion is neglected and the pockets fill only by hydrodynamic droplet spreading. For values of α 10-4, gas diffuses faster than the time required to fill the pocket by hydrodynamic motion of the interface. This implies that for α 10-4, the non-filling defects in UVNIL are only a result of hydrodynamic non-filling and not gas diffusion. However for α < 10-4, the gas diffusion is the rate limiting process for the smallest defects. Thus, for low values of α, defect size is diffusion-controlled while for high values, it is hydrodynamically-controlled. Figure 14 shows the time at which the droplets come in contact T contact, time required for gas to diffuse T diff and the total filling time T fill for different number of droplets and values of α. T fill is a sum of T contact and T diff. T contact depends only on the number of droplets and is independent of α. T diff reduces as α increases. The figure suggests that for achieving filling time of less than a second on 2.5 cm x 2.5 cm substrate, α needs to be greater than For α = 10-4, T contact and T diff become comparable suggesting that the time required for the droplets to spread and come in contact is about the same as the time required for the entrapped gas to diffuse. 34

35 FIG. 14. (Color online) Plot shows time at which the droplets come in contact T contact (solid line), time required for gas to diffuse T diff (dotted line) and the total filling time T fill (dashed line) for different number of droplets N and values of α. T fill is a sum of T contact and T diff. IV. CONCLUSIONS Multi-drop spreading is simulated for UVNIL with flat template and droplets inkjetted in square, hexagonal and modified hexagonal arrangements. Lubrication theory is used to describe the pressure and velocity field for the fluid flow. VOF method is used to track fluid interface. No external force is applied on the template during the imprint process. We find that droplet size, droplet arrangement and droplet placement accuracy are critical to achieving high throughput in UVNIL. Large unfilled edge regions increase the filling time for hexagonal and modified hexagonal droplet arrangement. For the same size of droplets, the square arrangement is found to be the optimum arrangement as it provides lower filling time than both hexagonal and modified hexagonal droplet arrangement. The effect of droplet placement error by inkjet system on the filling time is observed. For >10-4, the filling time increases significantly droplet placement error. Filling time for UVNIL with smaller droplets 35

36 is more sensitive to droplet placement error. We also propose a model to study the diffusion of gas encapsulated between droplets into the resist. We find that the defect size can be diffusion controlled or hydrodynamics-controlled based on a parameter α which scales as ~ D kh Ho. For values of α < 10-4, gas diffusion is slow and defect size is diffusioncontrolled while for higher values, it is hydrodynamically-controlled. For achieving filling time of less than a second on 2.5 cm x 2.5 cm substrate, α needs to be greater than Droplet size, droplet arrangement, droplet placement accuracy and non-filling are some of the most critical factors affecting throughput and defect rate. We have successfully presented the effect of these factors on UVNIL process. Designing the UVNIL process based on the proposed models can significantly improve the throughput and the defect rate for the process. ACKNOWLEDGMENTS The authors thank Drs. S. V. Sreenivasan, Randall Schunk and Shrawan Singhal for their helpful comments on this work. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing High Performance Computing resources that have contributed to the research results reported within this paper. Funding for this project was provided by NASCENT Engineering Research Center supported by the National Science Foundation under Cooperative Agreement No. EEC M. Colburn, S. Johnson, M. Stewart, S. Damle, T. Bailey, B. Choi, M. Wedlake, T. Michaelson, S. V. Sreenivasan, J. Ekerdt, and C. G. Willson, P. Soc. Photo-Opt. Ins. 3676, 379 (1999). 2 Z. M. Ye, R. Ramos, C. Brooks, L. Simpson, J. Fretwell, S. Carden, P. Hellebrekers, D. LaBrake, D. J. Resnick, and S. V. Sreenivasan, Proc. SPIE 7970 (2011). 36

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