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Modeling of Capillary Force-Induced Mold Deformation for Sub-10 nm UV Nanoimprint Lithography

A model has been developed to predict the dynamic behavior of the filling process and to simulate the capillary force-induced defamation of nanostructures on the imprint mold during ultraviolet nanoimprint lithography (UV-NIL) at sub-10 nm resolution. The dynamic behavior of resist liquid filling under different conditions was studied using hydrodynamics, and the capillary force-induced deformation of mold structures was modeled using beam bending mechanics for both wetting and non-wetting mold structures. A critical aspect ratio of mold structures were proposed and calculated based on the modeling results. The theoretical and simulated results provide possible guide for optimization of imprint mold for high-fidelity UV-NIL at sub-10 nm resolution.


Introduction
Nanoimprint lithography (NIL) is a promising technology that enables cost-effective and highthroughput fabrication for micro-and nanopatterns. One of the most advanced NIL techniques under intensive study is the ultraviolet NIL (UV-NIL). During UV-NIL process, a UV-curable resist is spin-coated or dispensed on a substrate in the atmospheric environment. Then the resist is covered by a mold with nanopatterns and the resist liquid will fill the nanocavities due to capillary force. The resist is then cured under ultraviolet exposure afterward, leaving solidified nanoimprinted patterns on the substrate. Fabrications of nanostructures with high transfer fidelity using the UV-NIL have been widely reported, and the patterning resolution could achieve at 10 nm when a helium ion beam lithography (HIBL) patterned hydrogen silsesquioxane (HSQ) mold was used. [1][2][3] However, pattern deformation and structure collapse arose as essential issues when advancing the UV-NIL toward sub-10 nm resolution. 1 Therefore, the general understanding of the filling process and the deformation of mold structures during the UV-NIL is highly demanded for the optimization of the UV-NIL process and design of HSQ mold structures for the use in sub-10 nm resolution. Concurrently, numerous efforts have been made to investigate the filling process [4][5][6] and bubble formation and dissolution process [7][8][9] for the UV-NIL using hydromechanics simulations, as well as molecular dynamics studies. 10 However, with the further reduce of structure dimensions, the structures of the imprint mold would suffer from deformations due to the dramatically increased capillary force 11 and thus limits the further application of UV-NIL.
Although the resist deformation for NIL has been investigated using solid mechanics 12 and molecular dynamics methods, 13 and the mold deformation for thermal NIL has been studied, 14 to our best knowledge, the general model of the capillary force induced pattern deformation on the mold for UV-NIL towards sub-10 nm resolution has not been reported yet. Herein, we have proposed a theoretical model, which incorporates both the hydrodynamics and solid mechanics, to simulate the stress distribution in the model structures during the UV-NIL process through modeling the dynamic filling process of the resist flow for both wetting and nonwetting mold materials and compare the theoretical modeling with the results obtained from a numerical simulation using the finite-element method (FEM). Moreover, the capillary forceinduced deformation of mold structures in the sub-10 nm UV-NIL was theoretically investigated.
Based on the analysis, the critical aspect ratio of mold structures applicable was proposed and calculated for typical mold made of materials with different yield strengths.

Model and simulation aspect
The model discussed here has been applied to a system in which two parallel beams of width w are separated by a space of gap d, the depth of the beams is L. The models for the wetting and nonwetting imprinting process are schematically demonstrated in Figure 1a and 1b, respectively. In a typical UV-NIL process with a wetting mold, when the mold approaches the substrate, the capillary force pulls the resist up to the features fabricated on the template. The fluidic flow of the resist can be described as a Newtonian fluidic. For simplification, the mold is considered to be stationary, and the whole system is assumed to be in the standard atmospheric pressure. The elastic deformation of beam structures before collapse during UV-NIL is neglected because the high brittleness of typical mold material.  In the solid mechanics model, the mold is considered as an elastic material, and the pressure difference obtained in the previous step is applied to the walls of beams.
Since the thickness of the resist layer is much smaller than the lateral dimensions of the substrate and the Reynolds number of the resist flow is low, the kinematics of the resist flow are satisfied with the simplified Navier-Stokes equation with lubrication approximation and continuity equation: 15 Where pr is the pressure in the resist, ux and uz are the velocities in and perpendicular to the substrate, μ is the viscosity of the resist.
With the boundary conditions (BCs), Solving Equation (1) and (2) using the above BCs, where γ is the surface tension of the resist, and r1 and r2 are the principal radii of curvature.
As the deformation of mold structures before collapse is considered to be neglected, the capillary rise process of resist liquid can be derived from a Poiseille flow by the Lucas-Washburn equation Consequently, the pressure difference drives the resist liquid flow can be derived from the Darcy- The Darcy friction factor is defined as where Re is the Reynolds number of the resist flow, For a flow between two plates, D=2d, the Equation (11) reduces to, where, Beams of the mold are considered as elastic material in evaluation of the capillary force-induced deformation, and the yield strength is used as the collapse criterion. Capillary force induced stress is distributed inside the beams corresponding to the pressure difference across the beam.
The model structure for calculating the capillary force induced stress distribution is schematically illustrated in Figure 1c. The second axial momentum of area is 3 1 12 From the elastic beam bending model, the maximum stress across the beam could be estimated For the non-wetting mold, the surface tension prevents resist liquid from filling the cavities.
Thus, external pressure is required to overcome the surface tension and push the resist liquid into channels of the mold structures. The model for the non-wetting imprint process is schematically illustrated in Figure 1b. The pressure difference across the liquid-air interface, according to the Young-Laplace equations for the forward contact angle θ, which is the critical pressure maintaining the hydrostatic force balance, is given by When the external pressure exceeds the critical pressure, the resist liquid will fill the channels due to the pressure difference.
Similar to the wetting-wall model, the maximum stress across the beam could be estimated by,

Results and discussion
There are several independent variables that must be specified to solve in our model. These  Table 1.

Time-Dependent Stress Distribution in the Mold
The von Mises distribution in the 50-nm-width beams due to the resist flow in a 50 nm width channel at t = 0.05 μs is plotted in Figure 3a. The maximum stress occurs at the corners, which is in correspondence with previous discussions (Figure 1b). The stress also increases with the filling ratio, due to the increase of the interaction area between the resist flow and mold beams. The relationship between the filling time and the maximum von Mises stress across the beams with different gap widths is summarized in Figure 3b. Apparently, the maximum von Mises stress increases with the filling ratio and reaches the maximum value at ~ 100 % filling ratio, which is also consistent with the theoretical analysis. The relationship of the maximum von Mises stress versus the gap width at ~ 100 % filling is then summarized in Figure 3c. We fitted the FEM simulated data using an inverse proportion model, and the results clearly show that the maximum stress is inversely proportional to the gap width, as predicted in Equation (15).  Max von Mises stress inside the beam for different gap widths near 100% filling ratio. The heights of the beam were held at 100 nm. The data is fitted using an inverse proportion model.

Comparison between FEM Simulated and Theoretically Calculated Results
As discussed in the previous section, the interaction area of the Laplace pressure increases with

Effects of Physical Parameters of Resist and Mold Materials
In principle, a narrower beam can lead to a smaller momentum of inertia, and a higher beam results in a larger moment, which both cause a higher level of stress distribution. The thickness of the resist, which is considered to be constant in this model, is a minor factor that affects the stress distribution. When advancing the dimension of the patterns on the mold to tens of nanometers scale, the Laplace pressure (several MPa) becomes much larger than the pressure in the thin resist film (several kPa), as long as the thickness of the resist is still greater than the height of the beams on the mold. Therefore, the maximum stress would decrease according to Equation (  Physical parameters of resist liquid also play an essential role in the imprint process. As discussed in the previous paragraphs, the viscosity of the resist flow affects only the filling process but not the stress distribution in the mold structures. However, the surface tension and contact angle of resist liquid change the maximum stress inside the beams significantly. The relationships of surface tension and contact angle with the maximum stress inside two 100 nm width, 100 nm gap width beams are summerized in Figure 5d and 5e, respectively. The results indicate that the maximum stress increases linearly with the surface tension, due to the increased capillary force and Laplace pressure, and decreases with the contact angle, because lower contact angle leads to better wettability and increases the capillary force accordingly.

Critical Aspect Ratio of the Mold
In solid mechanics, the structures collapse when the maximum stress exceeds the yield strength of materials. As illustrated in Figure 1b, the maximum stress occurs at the corner of a beam of the imprint mold during the imprinting process. The stress distributed at the corner near the resist liquid flow is compressive; while the stress at the opposite corner is tensile. Therefore, the mold structure collapses through the unbalanced loading. The collapsing process could be predicted by  Figure 6a shows the critical aspect ratio as a function of gap width of a two-parallel-beams imprint mold for different yield strength ranging from 60 MPa to 100 MPa, the result indicates that the highest aspect ratio allowed for transferring the 10 nm period beams to the resist through UV-NIL is only 1.8, even for a mold with yield strength of σY = 100 MPa. Therefore, to improve the critical aspect ratio, materials with higher yield strength is demanded. According to Equation (9), a critical imprint pressure exists in the non-wetting UV-NIL system. Figure 6b shows the critical imprint pressure as a function of the gap width of the beams on the non-wetting imprint mold in UV-NIL, and the width of the beams is the same as the gap width. In the figure, the critical pressure is calculated for a contact angle of 110°, and the result indicates that critical imprint pressure increases when the gap width of the beams shrinks. For example, for gratings with a sub-10 nm period, the critical imprint pressure exceeds 4 MPa, almost 40 atm.
The relationship of the critical aspect ratio for non-wetting molds versus the gap width of the beams is plotted in Figure 6c. The results indicate that for gating patterns with a given period, the critical aspect ratio could be improved through hydrophobic treatment of the mold.

Conclusion
In conclusion, we have established a model to describe the dynamic mechanical behavior in UV-NIL towards sub-10 nm resolution. We have theoretically and numerically investigated the capillary force induced fluidic flow of resist and pattern deformation in UV-NIL. The maximum stress and critical aspect ratio in both wetting and non-wetting molds have been calculated for pattern structures ranging from 5 nm to 200 nm. This study has reported here the key factors affecting the durability of imprint molds and provided guidelines for reducing the pattern deformation on the imprint mold. Based on our study, the critical aspect ratio for grating patterns with a given period is limited by both the mechanical properties of the resist and the mold. Highfidelity replication of nanostructures at sub-10 nm resolution demands proper selection of resist and aspect ratio of the nanostructures on the mold.