# A modified squeeze equation for predicting the filling ratio of nanoimprint lithography

- JiHyeong Ryu†
^{1}, - Ho Lee†
^{1}, - Sang-Ho Lee
^{2}, - HyungJun Lim
^{3, 4}and - JaeJong Lee
^{3, 4}Email author

**4**:14

**DOI: **10.1186/s40580-017-0108-z

© Korea Nano Technology Research Society 2017

**Received: **15 February 2017

**Accepted: **26 May 2017

**Published: **13 June 2017

## Abstract

A numerical method using the modified squeeze model is proposed in this paper in order to overcome the limitation of the established squeeze equation and obtain filling ratios for nanoimprint lithography (NIL). Because the imprinting velocity is overestimated when the ratio of indenter width to polymer thickness is close to unity, the modified equation is critical. For verification, the numerical results are compared with the experimental data according to the various stamp geometries and pressure variation rates, for which a maximum difference of 10% is indicated. Based on these results, additional studies are conducted using the modified squeeze equation in order to obtain filling ratios according to the polymer thickness and temperature. The filling rates are enhanced through the increases in the temperature and the polymer thickness. The results demonstrate that the modified squeeze equation can be used to obtain and predict the filling ratio of sub-nanoscale NIL fabrication. It is expected that this study will assist in optimizing the experimental conditions and approaches for roll-to-roll NIL and step-and-flash NIL.

### Keywords

Modified squeeze model Filling ratio Thermal nanoimprint lithography Pressure variation rate## 1 Background

Nanoimprint lithography (NIL), which was proposed by Chou [1], is widely considered to achieve complicated structures [2, 3] for electronic devices. This method, however, has several technical issues to resolve before becoming more adept than conventional lithography methods. These issues include bubble defects, incomplete filling, stamp deformation, and residual layers. In order to solve these problems, numerical methods have been used to understand the polymer filling behaviors, which are crucial for achieving stable patterns and designing the imprinting conditions. Heyderman et al. investigated the filling characteristics of stamps during thermal NIL using poly(methyl methacrylate) (PMMA, M_{w} = 75 k). They observed two filling mechanisms for microcavities where a viscous flow moved into the cavity center from the edges and mounds were formed by capillary flow [4]. Scheer et al. analyzed the polymer filling behaviors according to the stamp geometry (indenter width, polymer thickness) and polymer properties (surface energy, viscosity, and molecular weight) based on hydrodynamic considerations in thermal NIL. They demonstrated that the squeeze theory could be used to compare different imprint situations [5]. King et al. simulated the polymer deformation according to the polymer thickness, cavity size, and hybrid asymmetric neighbor cavities [6]. They presented that the flow characteristics were defined according to the cavity width to polymer thickness ratio and polymer supply ratio: pipe, squeeze, and Stoke’s flow. Lee et al. investigated various polymer filling behaviors including the numerical methods according to the slip boundary condition, dynamic contact angle, pressure and temperature. It was shown that the polymer filling shape could be varied according to the pressure, temperature, and stamp geometry [7, 8]. Bonning et al. proposed a new simulation technique with a contact mechanical-based approach for thermal NIL. This method has the advantage of requiring 30–100 s for the NIL simulation. They demonstrated that the numerical results were in good agreement with the experimental data [9, 10].

In the present study, the modified squeeze model was developed in order to overcome the limitation of the established squeeze equation for NIL and used to predict the polymer filling behaviors and ratios. The numerical results were compared with the experimental data according to the various stamp geometries and pressure variation rates. Additional studies were conducted to obtain the filling ratio with various polymer thicknesses and temperatures using the modified squeeze equation. Experimental images were captured by scanning electron microscope (SEM) to obtain the filling shapes and filling ratios. It was found that simulation results using the modified equation were well in agreement with experimental data.

## 2 Numerical method

### 2.1 Overview

_{w}= 75 kg/mol) was considered as a incompressible fluid under isothermal conditions, two governing equations [11], i.e. continuity and momentum equations, were used in the simulation. In order to consider the effect of the contact angles according to temperature, the surface energy of PMMA was referred from Fig. 1 in Ref. [12]. The contact angles between a substrate and a fluid were calculated using the followed equation:

Dimensions of the silicon stamp for the NIL experiment

Parameters | Case 1 | Case 2 | Case 3 |
---|---|---|---|

Value (nm) | Value (nm) | Value (nm) | |

W (width) | 600 | 1200 | 1800 |

S (indenter) | 2200 | 2200 | 3400 |

C (depth) | 250 | 250 | 250 |

H | 200, 300, 400 | 200, 300, 400 | 200, 300, 400 |

_{s}and γ

_{l}are the surface energy of the substrate and fluid, respectively. The surface energy of silicon with anti-sticking layer was obtained with 17.06 mN/m

^{2}. Table 2 showed the calculated contact angles using the variation of the surface energy with temperature. The simulations were carried out with same situation of the NIL experiments, where the pressure increased until 10 or 12.5 bar with pressure variation rate of 5.5, 10, 20, 50 bar/s and then, it was maintained for 1 s as shown in Fig. 2.

Surface tension and contact angle according to the temperature

Temperature (K) | Surface tension (mN/m) | Contact angle (°) |
---|---|---|

428 | 33.70 | 65.0 |

433 | 33.35 | 64.5 |

438 | 33.00 | 64.0 |

### 2.2 Polymer model

_{g}) is known as a shear thinning fluid which can be well described by the cross-WLF model presented by Walters et al. [13] as follows:

*η*

_{0}(

*T*) and

*η*

_{0}(

*T*

_{0}) indicate the zero shear viscosity at the NIL temperature (

*T*) and an arbitrary reference temperature (

*T*

_{0}).

*C*

_{1}and

*C*

_{2}are the polymer specific constants, which can be determined from the fitted cross model using the results of the viscosity experiment. The parameters of the cross-WLF model for PMMA with 120 kg/mol, which indicates the shear thinning behavior, was presented in Table 3 [8]. Since the critical stress level is independent of molecular weight and depends only slightly on polymer chemical nature [14], all parameters can be used for the viscosity of PMMA.

Parameters of cross-WLF model for PMMA experiment

Parameters | Value |
---|---|

C | 31.081 |

C | 51.6 |

T | 377.15 |

n | 0.3973 |

τ (Pa) | 35,607 |

The zero shear viscosity is proportional to the molecular weight (M_{w}) below a critical molecular weight (M_{c}). However, zero shear viscosity is dependent to the power of 3.4 of the molecular weight above the critical molecular weight. The critical molecular weight and is approximately 3 kg/mol in case of PMMA as presented by Torres [15]. Because the molecular weight of PMMA that we used is 75 kg/mol, the zero shear viscosity proportional to about the 3.4th power of the molecular weight can be used for this study. The zero shear viscosity was only extrapolated according to the molecular weight of PMMA. After the zero shear viscosity with the molecular weight was calculated, the viscosity values were followed by Eqs. (2) and (3).

## 3 Experiment

^{2}in the stamp size of 25 × 25 mm

^{2}. A fluorinated silane, tridecafluoro-1,1,2,2-tetrahydro-octyl-trichlorosilane (TFS, C8H4Cl3-F13Si) were treated in order to obtain anti-sticking layer for an easy demolding process before the NIL experiment. PMMA (Mr-I PMMA-75 k-300) with a thickness of 300 nm was coated onto a silicon wafer. The NIL process was conducted using laboratory-made equipment named as ANT4 in the Korea Institute of Machinery and Materials (KIMM) [16]. Figure 3 showed the procedure of NIL with imprinting pressure and temperature. A range of pressure variation rates of 5.5, 10, 20, and 50 bar/s, referred to as an ‘increasing pressure step’, were applied until 10 bar was reached at a temperature of 438 K using nanoimprint lithography to verify and compare with various numerical methods. In order to estimate the filling ratio with cavity size, we used the ratio of the cavity width to the tool width (dimensionless cavity ratio) and conducted NIL with the pressure variation rate of 10 bar/s under 12.5 bar at 438 K. After the pressures of 10 and 12.5 bar were reached, they were maintained for 1 s and given the name, ‘constant pressure step,’ in all NIL experiments.

Then, the filled cavity was rapidly cooled to 363 K in order to prevent creeping flow by thermal gradient following the pressing step. The experimental results were imaged using SEM to investigate the polymer filling ratios and behaviors for comparison with the numerical results. The filling ratio was approximated with SEM images and SolidWorks software, using the ratio of the filled polymer area to the cavity area. Experimental results were averaged in order to obtain the mean value, which is indicated by 86, 74, 67 and 64% with the variance of ±4% as the pressure variation rate increased from 5.5 to 50 bar/s. In case of the results of the dimensionless cavity size, the mean value was presented by 69, 81, and 90% with the variance of ±5%. The greatest difference between the means and error bars in the experimental results was approximately 8%.

## 4 Results and discussions

### 4.1 Modified imprinting velocity

*P*(

*t*),

*H*(

*t*), and

*η*(

*t*) are the pressure, polymer thickness, and viscosity at each time step, respectively. The Eq. (5) was created with consideration of the area of the compressed air in the cavity to compare another equations. The established equations, i.e. Eq. (4), can be employed for the NIL condition for which the ratio of indenter width to the polymer thickness (polymer supply ratio) is more than 1. An imprinting velocity from classical squeeze model could be overestimated in the range of the ratio of indenter width of the stamp to polymer thickness that is close to 1 [6]. Thus, a compensated function is necessary to overcome this limitation:

*H*(

*0*) +

*C*was used as an inversely proportional function to prevent the overestimated effect of the initial polymer thickness and consider the flow resistance; this is described in Fig. 3 of Ref. [18]. We calculated the imprinting velocity under the quasi steady state every time step, where pressure, the viscosity, and polymer thickness were changed every time step. Because the pressure as the function of time developed the shear rate which was a parameter for the viscosity. The shear rate and viscosity can be calculated and updated every time step using the changed values of the imprinting velocity every iteration as below:

### 4.2 Verification

### 4.3 Filling ratio with polymer thickness and temperature

Zero shear viscosity of PMMA at each temperature

Temperature ratio | Temperature (K) | Zero shear viscosity (×10 |
---|---|---|

1.13 | 428 | 2.4 |

1.14 | 433 | 1.2 |

1.15 | 438 | 0.6 |

1.17 | 443 | 0.3 |

## 5 Conclusions

The numerical method using the modified squeeze model was proposed to predict filling ratios and surmount limitations of the conventional squeeze equation in this paper. Both the results of the modified method and the experiment had a maximum difference of 10%. Concave shapes were indicated in the results, which were well in agreement. The filling ratio increased with the increases of dimensionless cavity size, polymer thickness, and temperature. It was confirmed that the filling ratio was proportional to cavity width due to air resistance. The filling ratio increases by 2.4, 4.2, and 6.5 times were caused by the augmentation of the polymer thickness of 1.5, 2, and 2.5 times from 200 nm, respectively. In the case of the result of the temperature, the filling ratio augmented 1.9, 3.4, and 5.8 times as the zero shear viscosity reduced by 2, 4, and 8 times from the value found at 428 K. The results demonstrated that the modified squeeze equation can be used for sub-nanoscale NIL simulation and the filling ratio with various polymer thicknesses and temperatures can be predicted to be the square of the increased values from the base polymer thickness and the decreased values to the power of 0.86 from baseline zero shear viscosity, respectively. It is expected that this modified equation can be expanded to simulation at the nano-scale through adjusting the exponential index (n) and these studies will be helpful for creating adequate operating conditions and predicting filling ratios and times.

## Notes

## Declarations

### Authors’ contributions

JR and HL wrote the manuscript. JL guided the manuscript procedure. All authors read and approved the final manuscript.

### Acknowledgements

This research was supported by the R&D program for Industrial Core Technology through the Korea Evaluation Institute of Industrial Technology, supported by the Ministry of Knowledge Economy in Korea (Grant No. 10040225).

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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