^{1,a)}

### Abstract

The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of siliconnanowires. We accomplish this by using the surface Cauchy–Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite elementeigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free siliconnanowires with unreconstructed surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

H.S.P. gratefully acknowledges support of the NSF through Grant No. CMMI-0750395.

I. INTRODUCTION

II. THEORY

A. Bulk Cauchy–Born model for silicon

B. Surface Cauchy–Born model for silicon

C. Finite elementeigenvalue problem for nanowire resonant frequencies

III. NUMERICAL EXAMPLES

A. Constant cross-sectional area

B. Constant length

C. Constant surface area to volume ratio

IV. DISCUSSION AND ANALYSIS

A. Comparison to experiment

V. CONCLUSIONS

### Key Topics

- Nanowires
- 117.0
- Silicon
- 53.0
- Boundary value problems
- 26.0
- Elasticity
- 19.0
- Finite element methods
- 16.0

## Figures

Illustration of the diamond cubic lattice structure of silicon. Black atoms represent standard fcc unit cell atoms, while green atoms represent the interpenetrating fcc lattice. The drawn bonds connect atoms in fcc lattice B to atoms in fcc lattice A.

Illustration of the diamond cubic lattice structure of silicon. Black atoms represent standard fcc unit cell atoms, while green atoms represent the interpenetrating fcc lattice. The drawn bonds connect atoms in fcc lattice B to atoms in fcc lattice A.

Illustration of the nine atom surface unit cell for the surface with a normal of a diamond cubic crystal. Black atoms represent fcc lattice A, while green atoms represent the interpenetrating fcc lattice B. The drawn bonds connect atoms in fcc lattice B to atoms in fcc lattice A.

Illustration of the nine atom surface unit cell for the surface with a normal of a diamond cubic crystal. Black atoms represent fcc lattice A, while green atoms represent the interpenetrating fcc lattice B. The drawn bonds connect atoms in fcc lattice B to atoms in fcc lattice A.

Nanowire geometry considered for numerical examples.

Nanowire geometry considered for numerical examples.

Minimum energy configuration of a fixed/free silicon nanowire due to surface stresses as predicted by (top) MS calculation, (bottom) SCB calculation.

Minimum energy configuration of a fixed/free silicon nanowire due to surface stresses as predicted by (top) MS calculation, (bottom) SCB calculation.

Normalized resonant frequencies for constant CSA silicon nanowires.

Normalized resonant frequencies for constant CSA silicon nanowires.

Normalized resonant frequencies for constant length silicon nanowires.

Normalized resonant frequencies for constant length silicon nanowires.

Normalized resonant frequencies for constant SAV silicon nanowires.

Normalized resonant frequencies for constant SAV silicon nanowires.

Normalized Young’s modulus for both fixed/fixed and fixed/free boundary conditions for constant CSA nanowires.

Normalized Young’s modulus for both fixed/fixed and fixed/free boundary conditions for constant CSA nanowires.

Normalized resonant frequencies for fixed/fixed constant CSA, SAV, and length nanowires plotted against the nanowire aspect ratio.

Normalized resonant frequencies for fixed/fixed constant CSA, SAV, and length nanowires plotted against the nanowire aspect ratio.

Normalized resonant frequencies for fixed/free constant CSA, SAV, and length nanowires plotted against the nanowire aspect ratio.

Normalized resonant frequencies for fixed/free constant CSA, SAV, and length nanowires plotted against the nanowire aspect ratio.

## Tables

Summary of geometries considered: constant SAV ratio, constant length, and constant CSA. All dimensions are in nanometers (nm).

Summary of geometries considered: constant SAV ratio, constant length, and constant CSA. All dimensions are in nanometers (nm).

Summary of constant CSA nanowire fundamental resonant frequencies for fixed/free boundary conditions as computed from: (1) The analytic solution given by Eq. (24), (2) BCB, and (3) SCB calculations. All frequencies are in megahertz (MHz), the nanowire dimensions are in nm.

Summary of constant CSA nanowire fundamental resonant frequencies for fixed/free boundary conditions as computed from: (1) The analytic solution given by Eq. (24), (2) BCB, and (3) SCB calculations. All frequencies are in megahertz (MHz), the nanowire dimensions are in nm.

Summary of constant CSA nanowire fundamental resonant frequencies for fixed/fixed boundary conditions as computed from: (1) The analytic solution given by Eq. (25), (2) BCB, and (3) SCB calculations. All frequencies are in MHz, the nanowire geometry is in nm.

Summary of constant CSA nanowire fundamental resonant frequencies for fixed/fixed boundary conditions as computed from: (1) The analytic solution given by Eq. (25), (2) BCB, and (3) SCB calculations. All frequencies are in MHz, the nanowire geometry is in nm.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content