Elsevier

Microelectronic Engineering

Volume 86, Issue 11, November 2009, Pages 2324-2329
Microelectronic Engineering

A mathematical model for slip phenomenon in a cavity-filling process of nanoimprint lithography

https://doi.org/10.1016/j.mee.2009.04.011Get rights and content

Abstract

Squeeze flow theory has been used as an effective tool to clarify how and which process conditions determine cavity-filling behavior in nanoimprint lithography (NIL). Conventional squeeze flow models used in NIL research fields have assumed no-slip conditions at the solid-to-liquid boundaries, that is, at the stamp-to-polymer or polymer-to-substrate boundaries. The no-slip assumptions are often violated, however, in micrometer- to nanometer-scale fluid flow. It is therefore necessary to adopt slip or partial slip boundary conditions. In this paper, an analytical mathematical model for the cavity-filling process of NIL that takes into account slip or partial slip boundary conditions is derived using squeeze flow theory. Velocity profiles, pressure distributions, imprinting forces, and evolutions of residual thickness can be predicted using this analytical model. This paper also aims to elucidate how far the slip phenomenon is able to promote the process rate.

Introduction

Nanoimprint lithography (NIL) is a promising technology for fabrication of integrated micrometer- and nanometer-scale patterns [1], [2]. To imprint such patterns successfully using NIL, processing conditions, such as temperature, pressure, and time, should be appropriately selected. Doing so requires a clear understanding of the behavior of the polymer material during the NIL process. To determine optimal processing conditions, many researchers have investigated the behavior of the polymer material during NIL using numerical simulations. They have assumed that the polymer resist is a nonlinear, elastic solid [3], [4], a viscoelastic solid [5], a viscous fluid [6], [7], [8], or even a viscoplastic material [9], [10]. In contrast, analytical models based on classical squeeze flow theory have been used as effective tools to predict the cavity-filling behavior and to clarify the parametric relations among process conditions [11], [12], [13]. Such models typically adopt no-slip boundary conditions. However, slip phenomena occur regularly in micrometer- to nanometer-scale fluid flow [14], [15]. Thus, the conventional analytical models give inaccurate results for analyses of NIL process under such conditions.

In this paper, using the assumption of an incompressible, pure, viscous fluid, a mathematical analytical model for the cavity-filling process of NIL considering a slip or partially slip boundary condition is described in detail, as is its derivation. Several examples are presented to illustrate how the NIL process can be influenced by slip phenomena at the boundaries using the analytical model derived here. Moreover, it is shown that potential improvements in NIL processing time are possible if boundary slip conditions are promoted.

Section snippets

Need to account for the slip phenomenon in NIL

Since the quasi-steady state solution of a Newtonian fluid was introduced by Stefan [16] and Bird et al. [17], the squeeze flow theory, which analyzes the compression and squeezing phenomena of fluid between two parallel, circular disks, has remained a classical theme in fluid mechanics [18], [19]. So far, analytical models based on squeeze flow theory have been referenced and used by many researchers in NIL research fields because the imprinting process can be compatibly simplified as a

Velocity profiles

For various slip boundary conditions, the lateral velocity profiles were calculated using Eq. (19) (they are depicted in Fig. 3). In the figure, the parallel and vertical axis are the non-dimensional lateral velocity vxS/(-h˙)h at x = S and y/h, respectively. Under the no-slip boundary condition (δ = 0) the velocity profiles have generic parabolic shapes. As the slip ratio δ increases, the velocity profile descends gradually and eventually becomes uniform under the full lubrication condition (δ = 1).

Imprinting load

Conclusions

This paper aimed to elucidate how the cavity-filling process of NIL can be influenced by slip phenomena at boundaries and to what degree those phenomena increase the process rate. To do this, an analytical model of the cavity-filling process of NIL that takes into account slip or partial slip boundary conditions, based on squeeze flow theory, was derived. In the model, the polymer resist was assumed to be an incompressible, pure, viscous fluid. From the model, the velocity profile, stress and

References (21)

  • S.Y. Chou et al.

    Microelectron. Eng.

    (1997)
  • N.W. Kim et al.

    Microelectron. Eng.

    (2008)
  • W.-B. Young

    Microelectron. Eng.

    (2005)
  • H. Schulz et al.

    Microelectron. Eng.

    (2006)
  • M. Gad-el-Hak

    Mec. Ind.

    (2001)
  • H.M. Laun et al.

    J. Non-Newtonian Fluid Mech.

    (1999)
  • L.J. Guo

    J. Phys. D: Appl. Phys.

    (2004)
  • Y. Hirai et al.

    J. Vac. Sci. Technol. B

    (2001)
  • Y. Hirai et al.

    J. Vac. Sci. Technol. B

    (2004)
  • J.-H. Jeong et al.

    Fibers Polym.

    (2002)
There are more references available in the full text version of this article.

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