In situ deformation of silicon nanospheres

Abstract

As a natural response to the ongoing trend of device miniaturization, many effects of scaling on the properties of materials have become well documented. However, the mechanical properties of individual nanoparticles are not well understood and the direct observation of nanoparticle deformation has only recently been achieved. This work investigates the mechanical behavior of silicon nanospheres in the transmission electron microscope (TEM) using an in situ indentation sample holder. In situ TEM studies provide information which is not accessible by more traditional means, including particle orientation prior to deformation and the type and location of any preexisting defects. In this study, isolated nanoparticles were located and compressed between a diamond tip and a sapphire substrate. Here, the deformation behavior of individual particles is investigated and analogous strain fields between small particles are discussed.

Appendix A

For analyzing the surface energy between two spheres as shown in Fig. 6, it was first considered that the result represented by Johnson’s Equation (5.51) in Ref. [35] could be utilized directly given by

$$ \gamma _{{\text{eff}}} \;\; = \;\;2\gamma _{\text{s}} \; - \;\gamma _{{\text{gb}}} \;\; = \;\;\frac{{4E^* a^3 }} {{9\pi R^2 }} $$

(3)

where γ _gb is a grain boundary energy. In using the value of R equal to 56.4 nm from Eq. (2) this gave too large a result since the contact radius used, a _c, was that at zero load while Eq. (3) is appropriate to a _adh at the maximum negative tensile load necessary to separate the particles (represented by Pt. B in Fig. 5.8, Ref. [34]). Using the JKR result [33, 35],

$$ a^3 \;\; = \;\;\frac{{PR}} {K}\,\left[ {1\;\; + \;\;\frac{{3\pi \gamma _{{\text{eff}}} R}} {P}\;\; + \;\;\sqrt {\frac{{6\pi \gamma _{{\text{eff}}} R}} {P}\;\; + \;\;\frac{{\left( {3\pi \gamma _{{\text{eff}}} R} \right)^2 }} {{P^2 }}} \;} \right] $$

(4)

and setting P = 0 appropriate to Fig. 6, one obtains

$$ a_{\text{c}}^3 \;\; = \;\;\frac{{6\pi \gamma _{{\text{eff}}} R^2 }} {K}. $$

(5)

With the definition of $K\;\; = \;\left( {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \right)\left( {{E \mathord{\left/ {\vphantom {E {\left( {1 - \nu ^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \nu ^2 } \right)}}} \right)\;$, this becomes

$$ \gamma _{{\text{eff}}} \;\; = \;\;\frac{{Ea_{\text{c}}^3 }} {{9\pi \left( {1\; - \;\nu ^2 } \right)R^2 }} $$

(6)

which is Eq. (1) as utilized in the main text. Equations (3) and (6) are consistent with Johnson’s Fig. 5.8, Ref. [35].