A mathematical model for slip phenomenon in a cavity-filling process of nanoimprint lithography
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 at x = S and , 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
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