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Atomic force microscope-(AFM-) based indentation tests were performed to examine mechanical properties of parallel single-crystal silicon nanolines (SiNLs) of sub-100-nm line width, fabricated by a process combining electron-beam lithography and anisotropic wet etching. The SiNLs have straight and nearly atomically flat sidewalls, and the cross section is almost perfectly rectangular with uniform width and height along the longitudinal direction. The measured load-displacement curves from the indentation tests show an instability with large displacement bursts at a critical load ranging from
480

Silicon (Si)-based nanomaterials [

In this paper, we carry out nanoindentation
experiments to characterize the mechanical properties of single-crystal Si nanolines
(SiNLs). These SiNLs are fabricated by a process combining electron beam
lithography (EBL) and anisotropic wet etching [

Figure

SEM images of parallel silicon nanolines, with 74 nm line
width and 510 nm height; the pitch distance is 180 nm. (a) A plan view, (b) a
perspective view with

An atomic force
microscope (AFM)-based nanoindentation system (Triboscope by Hysitron,
Inc.) was used to characterize the mechanical properties of the SiNLs. A conically shaped diamond indenter
with the tip radius around 5

(a) Schematic of the nanoindentation test on parallel SiNLs. (b) A set of three load-displacement curves from the nanoindentation tests on the 74 nm SiNLs, with no residual deformation after unloading of the indenter.

Additional
indentation tests with the SiNLs were carried out with various indentation loads,
as shown in Figure

(a) A set of five load-displacement curves from the nanoindentation tests on the 74 nm SiNLs, with irrecoverable displacements due to fracture of the SiNLs. (b) A SEM image of the fractured SiNLs, with debris of isosceles triangular shape.

In this section, we calibrate a finite element model
to simulate the nanoindentation test on SiNLs. The model system consists of a
spherical indenter and parallel SiNLs standing on a substrate, as illustrated
in Figure

Illustration
of the parallel SiNLs under spherical indentation (not to the scale), showing
the cross sections of two lines and the contact forces. The tip alignment is
indicated by an offset

Calibration of the finite element model using the experimental data. The elastic indenter model and the rigid indenter model give nearly identical load-displacement curves, both in good agreement with the experimental data.

Three-dimensional finite element (FE) models are
constructed for the SiNLs using ABAQUS [

A finite
element model of parallel SiNLs under indentation, showing three half-lines
with symmetric boundary conditions at the planes

To model the indentation load, contact surfaces are
defined between the indenter and the lines. The contact property, either
frictionless or frictional, is specified. As discussed in the next section, the
friction between the indenter tip and the SiNLs is very important in
determining the critical buckling load. On the other hand, contact and friction among
SiNLs are ignored because of the relatively large spacing between the lines (

We begin with a simplest model, assuming frictionless
contact between the indenter and the lines. The tip of the indenter is aligned
with a trench center between two lines (i.e.,

Indentation load-displacement curves from finite element models with different numbers of lines. The indenter tip is aligned with the trench center between two center lines. The initiation of contact with the first four pairs of the lines is indicated by the arrows with numbers 1–4.

All the force-displacement curves in Figure

Buckling of
SiNLs and mode transition simulated by the finite element model. The contours show
the distributions of normal stress in the 2-direction. (a)–(c) correspond to
the points A, B, C on the load-displacement curve in Figure

While the Riks method used in the numerical
simulation gives a single equilibrium path for each model, experimental curves
depend on the control of the load or displacement. If the experiments were under
a displacement control, similar curves as shown in Figure

The indentation tests of the present study were not
equipped with sufficient lateral resolution for accurate positioning of the
indenter with respect to individual SiNLs. The possible location of the indenter tip
therefore could vary from a trench center between two lines to the center of
one line (another case of symmetric loading). As illustrated in Figure

The critical load of buckling transition versus the indenter tip location, obtained from the finite element models. The critical loads for the left and right center lines are obtained from a single-line model.

As a special case, when the tip of the indenter is aligned
exactly with the line center (i.e.,

As illustrated
in Figure

The critical load of buckling transition as a function of the friction coefficient at the contact, obtained from the finite element models with tip locations at the trench center, line center, and 3 nm off the line center.

When the indenter
tip is aligned with the line center, higher critical loads are obtained except
for the case with the large friction coefficient

Simulated
buckling modes of SiNLs under indentation with the indenter tip aligned 3 nm
off the line center. (a) With frictionless contact between the indenter and the
lines, (b) with coefficient of friction,

The critical
loads from the experiments, ranging from 480

In the FE simulations
discussed above, the SiNLs have
been assumed to be supported by a rigid foundation. Consideration of an elastic
substrate requires a significant increase in the number of FE elements and the computational
time. To illustrate the effect of the elastic substrate, a FE model is
constructed with six lines under a symmetric loading at the trench center. The dimensions
of the substrate are 490 nm (height) by 900 nm (width) by 3000 nm (length). The
material properties for the substrate are the same as those for the SiNLs, with
Young’s modulus

A comparison of indentation load-displacement curves from the finite element models of parallel SiNLs on a rigid substrate and on an elastic substrate, respectively.

The stress distribution after the buckling transition,
obtained from the FE model with the elastic substrate, is shown in Figure

Stress distributions from a finite element model with SiNLs on an elastic substrate under a symmetric trench center loading. (a) and (b) show contours of the normal stresses in the 2- and 3-directions, respectively.

A buckling instability is observed in the nanoindentation tests of the parallel silicon nanolines. A systematic analysis of the buckling modes is presented based on finite element modeling. In particular, the effects of indenter tip location and contact friction on the critical load are discussed as possible causes for the scattering of the experimental data. The result from the present study demonstrates a potential methodology to study buckling, friction, and fracture at the nanoscale via a combination of experiments and modeling.

The authors are grateful for the financial support by National Science Foundation through Grant no. CMMI-0654105. The work was performed in part at the Microelectronics Research Center of the University of Texas at Austin, a member of the National Nanofabrication Infrastructure Network supported by National Science Foundation under Award no. 0335765.