A brittleness transition in silicon due to scale

Abstract

To understand the brittleness transition in low-toughness materials, the nucleation and kinetics of dislocations must be measured and modeled. One aspect overlooked is that the apparent activation energy for plasticity is modified at very high stresses. Coupled with state of stress and length scale effects on plasticity, the lowering of the brittle-to-ductile transition (BDT) in such materials can be partially understood. Experimental evidence in silicon single crystals in the length scale regime of 40 nm to 1 mm is presented. It is shown that high stress affects both length scale and temperature-dependent properties of activation volume and activation energy for dislocation nucleation and/or mobility. Nanoparticles and nanopillars of single-crystal silicon demonstrate unexpectedly high fracture toughness at low temperatures under compression. A thermal activation approach can model the three decades of size associated with the factor of three absolute temperature shift in the BDT.

Appendices

Appendix A

Previous work on dislocation shielding at a crack tip for the emitted slip being blocked at a grain boundary, with grain size, d, is given elsewhere.21,23 While this was originally conducted for a blocked slip band in a polycrystalline array of grain size, d, here we assume a slip band in a sphere of diameter, d. The basis is the original analysis by Li47 using the function

$$F(\xi) \,= \, \alpha I(\xi) - \phi \frac{1} {\sqrt{1 - {\xi ^2}}} - S\quad,$$

((A1))

where α = 2τ_ys/πA; A = μ/ 2π(l-v); d = blocked slip band length. The physical parameters are τ_ys, shear yield strength, μ, elastic shear modulus, and v, Poisson’s ratio. The spatial parameters coupled to a shielded crack giving rise to mode II fracture toughness, K_IIC, are

$$\beta \, = \, \frac{K_{\text{IIC}}} {A\sqrt {2 \pi b}} ; \xi ^2 = \frac{d - c} {d} ; \, S \, = \, f \, \left\{{E(\xi),\,H(\xi)} \right\}$$

((A2))

with b the Burgers vector, c, the nearest dislocation in the band to the crack tip, and S a function of complete elliptical integrals. With Eqs. (A1) and (A2) and relating N shielding dislocations to the mode II fracture toughness, a further conversion to mode I fracture toughness21 gave

$$K_{\text{IC}}\, = \,\frac{2} {\sqrt 3} \, \tau_{\text{ys}} \sqrt {\frac{d}{\pi}} \, + \, \frac {\sqrt \pi \mu Nb} {2 \sqrt 3 (1 - {\text{v}}) \sqrt d}\quad.$$

((A3))

This is the same result reported previously except the fracture toughness was reported as K_IC rather than K_IIC. Using the appropriate relationship between the two gives the correct relationship at Eq. (A3). Simplifying with a blocked slip band from Eshelby et al.,48 one proposes to first order that

$$\tau_\text{ys} = \frac{2\mu Nb}{\pi (1 - \text{v})} \quad,$$

((A4))

which can be rationalized for the very initial stages of crack nucleation wherein the surface site represents an extremely small crack length. In effect, yielding and crack nucleation are considered as concomitant activities. Interestingly, if one then used Eq. (A4) to eliminate d from Eq. (A3), one finds with v = 1/3 that the two terms in Eq. (A3) are nearly identical, i.e.,

$$K_{\text{IC}} = 0.450 \left[ \mu Nb \sigma_{\text{ys}} \right]^{1/2} + 0.554 \left[ \mu Nb \sigma_{\text{ys}} \right]^{1/2}.$$

((A5a))

The assumption here is that τ_ys = σ_ys/2 since the orientation of a given nanosphere as deposited is random. As the sum of the coefficients is almost exactly unity, we use

$$K_{\text{IC}} = \left[ \mu Nb \sigma_{\text{ys}} \right]^{1/2},$$

((A5b))

which is Eq. (2) in the main text. It is understood that the boundary conditions for nucleating a crack in a sphere are different than those for a crack nucleating at a planar surface of a semi-infinite medium. As such, it would be appropriate to repeat such a derivation using discretized dislocation theory.

Appendix B: Fracture Toughness as Related to Plastic Strain and Activation Volume

In particular for the compression of spheres, the Tabor or geometric estimate of contact strains may be used. This can be given by either

$${{\varepsilon }_{\text{p}}} = 0.2\left( {\frac{a}{R}} \right); = \delta /R\,,$$

((A6))

with the second formulation biased towards the strains at the top and bottom contacts rather than an average value of δ/2R. For strains between 5 and 15 percent, these agree within about 25 percent. For initially dislocation-free spheres, δ ~ Nb and with Eq. (A6) gives

$$Nb \approx R{\varepsilon _{\text{p}}}.$$

((A7))

Using Eq. (A7) to eliminate N in Eq. (2), one finds

$$K_{\text{IC}} = \left[ \mu \sigma_{\text{ys}} R \varepsilon _{\text{p}} \right]^{1/2}.$$

((A8))

With a desire to relate fracture behavior to the physical parameters controlling dislocation nucleation and mobility, an activation volume, V^*, is invoked. Previously, V^* had been determined45 for different size silicon spheres and a resulting empirical fit gives

$$V*/b^3 = \alpha\left(\frac{d - d_0} {d_0} \right)^{0.9},$$

((A9))

with a cutoff diameter of d₀ = 25 nm, α = 5 and β = 0.9. Such a fit resulted due to the inverse stress dependencies for both d and V*. To simplify and relate V*directly to the nanosphere radius, the data in Table I demonstrate for radii between 40 and 500 nm that

$$V* = \eta b^2 R.$$

((A10))

First, the data fit well with η = 1/8. Combining Eqs. (A7), (A8), and (A10) gives

$$K_{\text{IC}} = \left[ \mu \sigma_{\text{ys}} \varepsilon_{\text{p}} \frac{8V*}{b^2} \right]^{1/2}.$$

((A11))

Finally, since the mode I fracture toughness is related to the mode I strain ener_/gy release rate, we use standard relationships $K_{{\text{IC}}}^2 = E{G_{{\text{IC}}}}{\text{/}}\left( {1 - {{\text{v}}^2}} \right)$ and μ = E/2(1+v) to find

$${G_{{\text{IC}}}} = \left[ {V*\frac{{{\eta _0}{{\sigma }_{{\text{ys}}}}{{\varepsilon }_{\text{p}}}*}}{{{b^2}}}} \right],$$

((A12))

where η₀ lies between 2.7 and 3 for Poisson’s ratio between 1/4 and 1/3. For this study, η₀ = 3 as used in Eq. (3) of the main text.

Appendix C

To check the plausibility of using a simpler technique for determining K_IC of a sample nanopillar previously published, another three-dimensional (3D) finite element analysis was invoked. The previously published 3D finite element, finite difference technique had obtained a value of 2.9 MPa$\sqrt {\text{m}}$. Using Shivakumar and Newman’s49 elastic–plastic Finite Element Method (FEM) analysis for shallow cracks, an eta factor is required which is similar to the compliance derivative. With values of η_pr for plastic deformation of shallow cracks, the J-integral toughness is given by

$$J_{\text{p}} = \frac{\eta_{\text{pr}}}{b_{\text{p}}B}\int_{0}^{\delta_{l}} Pd\delta ,$$

((A13))

where δ_l is the load-line displacement under load, P and bB are ligament and thickness dimensions associated with a rectangular solid. Given the dimensions of the right cylinder, this can be directly used but may be inappropriate given the change in compliance. Nevertheless, the value of bd, where d is the specimen diameter of 400 nm and b is the ligament of 1200 nm, gave a J_p equal to 87.5 J/m² using the 0.6 mN load. Here, η_pr ≈ 1 for the 180-nm-long crack. Converting this to K_IC gave 3.83 MPa$\sqrt {\text{m}}$, which is somewhat larger than the linear-elastic fracture mechanics (LEFM) result of 2.9 MPa$\sqrt {\text{m}}$ as expected. Based on this similarity, Eq. (A13) was used for nanospheres assuming that similar wedge-opening cracks resulted as was found for the nanopillars. The surface cracks for the nanospheres were much smaller using surface roughness and oxide film thickness as a guide. This resulted in η_pr ≈ 0.3, which gave realistic estimates of the strain energy release rate. A caveat is that one would not expect η_pr to be the same for different spheres, which probably accounts for much of the scatter in Fig. 4(b).