Collective Buckling of Periodic Soft Nanostructures on Surfaces and Promotion for Nanolithography

Abstract

Mechanical instabilities of periodic soft nanostructures and their interaction with the deformation of a base could lead to collective buckling. Our theoretical calculations suggest multiple modes of buckling associated with distinctive potential energies. This theoretical discovery allowed us to selectively promote certain buckling modes through experiments as an unusual approach for nanolithography. One typical example includes reducing the feature size in a selectively buckled grating from a pitch of 200 to 130 nm.

Effects of surface deformation on the collective buckling of an array of rigid beams on an elastic substrate

“…A major progress in the buckling analysis of periodic soft nanostructures was made in our recent work [Lin et al 2007]. Due to the inclusion of the most basic and important mechanism into our theoretical model, i.e., neighboring beams interact through the deformation of the substrate, we were able to describe the most basic collective buckling behavior of the soft structures and obtain results qualitatively matching experimental findings.…”

“…The weight W = ρg2ah acts at the center of the beam. The beams are assumed to be rigid, as in [Lin et al 2007;Chen et al 2008;. In contrast with these works (and also with [Feng and Li 2009] and [Lin et al 2010]), here we allow the bottoms of the beams to move in both the horizontal and vertical directions, in addition to rotation.…”

“…Due to the inclusion of the most basic and important mechanism into our theoretical model, i.e., neighboring beams interact through the deformation of the substrate, we were able to describe the most basic collective buckling behavior of the soft structures and obtain results qualitatively matching experimental findings. The model in [Lin et al 2007] was later generalized and applied to the case of a two-dimensional array of rigid beams on an elastic foundation [Chen et al 2008], a nonuniform onedimensional array of rigid beams , and a one-dimensional array of elastic beams [Feng and Li 2009;.…”

“…In [Lin et al 2007;Chen et al 2008;Feng and Li 2009;, the beam bottoms in the substrate were allowed to rotate which is the major deformation, but their vertical and horizontal displacements were neglected. While this could describe the collective buckling of beam arrays, the restriction on beam bottom displacements renders the system too stiff.…”

“…In this paper we generalize the analysis in [Lin et al 2007] by removing the restrictions on the beam bottom displacements. The bottoms of the beams can move in both horizontal and vertical directions.…”