First-principles study of the Young’s modulus of Si ⟨001⟩ nanowires

We report the results of first-principles density functional theory calculations of the Young’s modulus and other mechanical properties of hydrogen-passivated Si ⟨001⟩ anowires. The nanowires are taken to have predominantly {100} surfaces, with small {110} facets. The Young’s modulus, the equilibrium length, and the residual stress of a series of prismatic wires are found to have a size dependence that scales like the surface area to volume ratio for all but the smallest wires. We analyze the physical origin of the size dependence and compare the results to two existing models.

Figure 1

(Color online) Cross sections of fully relaxed hydrogen-passivated wires, with each Si atom colored as shown in the legend corresponding to its transverse relaxation in $\mathrm{\AA }$. The widths of wires are (a) 0.61 (b) 0.92, (c) 1.00, (d) 1.39, (e) 1.49, (f) 2.05, (g) 2.80, and (h) $3.92\mathrm{nm}$, respectively. The width is defined as the square root of the cross-sectional area.

Figure 2

Silicon nanowire axial stress and equilibrium elongation strain calculated in DFT as a function of wire size. The solid curve is a fit to $C∕w$ of the elongation strain to 4 data points from $1.49\mathrm{nm}$ and larger wires, with $C=1.9\%\mathrm{nm}$. The predictions of Eq. (1) are also plotted using the stresses from DFT calculations of hydrogenated 14-layer {100} and 15-layer {110} slabs. The asterisklike symbols are from overlaps.

Figure 3

Silicon nanowire Young’s modulus calculated in DFT as a function of wire size. For comparison values of continuum formula (3) are also plotted, using the {100} and {110} surface elastic constants obtained in DFT from hydrogenated 14-layer {100} and 15-layer {110} slabs, respectively. The solid curve $E=E_{\mathit{\text{bulk}}}^{\mathrm{DFT}}-C∕w$, with $C=66.11\mathrm{GPa}∕\mathrm{nm}$, is the best fit to a pure surface area to volume size dependence.