Fracture modes in micropillar compression of brittle crystals

Abstract

This article describes cracking during microcompression of Si, InAs, MgO, and MgAl₂O₄ crystals and compares this with previous observations on Si and GaAs micropillars. The most common mode of cracking was through-thickness axial splitting, the crack growing downward from intersecting slip bands in pillars above a critical size. The splitting behavior observed in all of these materials was quantitatively consistent with a previous analysis, despite the differences in properties and slip geometry between the different materials. Cracking above the slip bands also occurred either in the side or in the top surface of some pillars. The driving forces for these modes of cracking are described and compared with observations. However, only through-thickness axial splitting was observed to give complete failure of the pillar; it is, therefore, considered to be the most important in determining the brittle-to-ductile transitions that have been observed.

Appendix A

Kendall describes, in terms of global energy changes, the formation of splitting cracks in columns compressed using a punch with a width smaller than that of the column itself.19 Each half-column is then loaded asymmetrically and bends, allowing the compressive force to move downward and do work. In the absence of frictional effects, this driving force goes to zero as the punch width approaches the pillar width. In micropillar compression, where the punch is typically much larger than the pillar top, one might expect no effect. However, an effect arises because the pillar tapers, so that the centroid of each half pillar is closer to the axis at the top of the pillar than it is at the bottom. Kendall’s calculation has been adapted to estimate the magnitude of this effect on the driving force due to through-thickness axial splitting.

Consider the pillar shown in Fig. A1, the centroid of each semicylindrical slice is at a distance y′ from the center of the pillar, where

$$y' = {2r \over 3 \pi},$$

((A1))

FIG. A1.

Schematic of a tapered pillar containing an axial split, with definition of dimensions used in this appendix.

r is the radius of the pillar at some depth in the pillar and depends on the vertical position within the pillar and on the taper angle, α

$$r = r_\text{top} + z \tan \alpha,$$

((A2))

where r_top is the radius of the top of the pillar and z is the distance from the top of the pillar. The second moment of area with respect to the centroid axis is

$$I_{x, \text{C}} = \left( {\pi \over 8} - {8 \over 9 \pi} \right) r^4 = \left( {\pi \over 8} - {8 \over 9 \pi} \right) \left( r_\text{top} + z \tan \alpha \right)^4 .$$

((A3))

The bending moment in each slice through the pillar is produced by the difference in centroid position

$$M = {F \over 2} \,\cdot\,{4 \over 3 \pi} \left( r - r_\text{top} \right) = {2 \sigma r_\text{top}^2 z \tan \alpha \over 3},$$

((A4))

where

$$F = \sigma \pi r_\text{top}^2.$$

((A5))

The bending strain energy in the two beams taken together is given by

((A6))

so that the contribution to the driving force from the taper is

((A7))

where d is taken to be equal to 2r_top. Expressing the stress intensity factor of the crack derived for uniaxial splitting, Equation (2), in terms of a crack driving force gives

$${\text{d} U_\text{E} \over \text{d} c} = Gd = {K_\text{I}^2 \over E} d = {\beta^2 k_s^2 \sigma^2 d^2 \over E}$$

((A8))

The relative value of the two driving forces is therefore given by

((A9))

Taking typical values for silicon micropillars of β = 1.1, S = 0.47 and φ = 54.7° and taking c = z ≈ 0.7 d at the onset of splitting gives${{G_{{\rm{taper}}} } \over {G_{{\rm{split}}} }} = 0.06 - 0.22$ for α between 2 and 4°. The additional driving force due to the taper is therefore considered to be a secondary effect, at least where crack nucleation is concerned.