Universality of periodicity as revealed from interlayer-mediated cracks

A crack and its propagation is a challenging multiscale materials phenomenon of broad interest, from nanoscience to exogeology. Particularly in fracture mechanics, periodicities are of high scientific interest. However, a full understanding of this phenomenon across various physical scales is lacking. Here, we demonstrate periodic interlayer-mediated thin film crack propagation and discuss the governing conditions resulting in their periodicity as being universal. We show strong confinement of thin film cracks and arbitrary steering of their propagation by inserting a predefined thin interlayer, composed of either a polymer, metal, or even atomically thin graphene, between the substrate and the brittle thin film. The thin interlayer-mediated controllability arises from local modification of the effective mechanical properties of the crack medium. Numerical calculations incorporating basic fracture mechanics principles well model our experimental results. We believe that previous studies of periodic cracks in SiN films, self-de-bonding sol-gel films, and even drying colloidal films, along with this study, share the same physical origins but with differing physical boundary conditions. This finding provides a simple analogy for various periodic crack systems that exist in nature, not only for thin film cracks but also for cracks ranging in scale.


Section 1. Supplementary figures
. Period dependency on interlayer track width. Black arrow on top indicates crack growth direction. The bottom left graph is the Fourier coefficient of the digitized crack patterns from optical images. Single dominant peaks were observed for each case, with positive correlation with track width, as indicated in the bottom right graph. Note that saw-tooth-like features in the crack pattern are more pronounced in the wider track width sample. Additionally, the crack running direction and saw-tooth-like features have a strong correlation; no reversed case was observed.

2-1. Heating level estimation
According to diffusion theory, the time needed for heat to diffuse through 2-µm-thick SiO 2 would be t = (2 µm) 2 / α ≈ 4.8 µs, where the thermal diffusivity of oxide is α = 0.84 µm 2 / µs (1). Silicon heat diffusion time is negligible compared to that of oxide because the diffusivity of silicon is 2 orders of magnitude higher. In the experiment, the scanning time was a few seconds; thus, we could conclude that the thermal state of the sample surface is a steady state. If we assume that all the energy was focused on the probe point without scanning and the electron beam encountered the bulk SiO 2 specimen instead of the thin film system, then we could write the (overestimated) expected temperature rise as ΔT = 3V acc I / (2πκR) (2) , where probing current is I = 1 nA, thermal conductivity of SiO 2 is κ = 1.38 W / m ·K, and electron range R of SiO 2 is ~3 µm at V acc = 20 kV, which is much larger than the probe diameter (on the order of ~100 nm), ΔT = 3V acc I / (2πκR) = (3 × 20 kV × 1 nA) / (2π × 1.38 Wm -1 K -1 × 3 µm) = 2.31 K .
This result should be lower if we take into account the scanning of the probe beam and thermal conductivity of the silicon substrate. This result is also consistent with earlier reports of 70 K on the SiO 2 film on the Si substrate, with much higher current (600 nA) level and much smaller exposed area (2 x 2 µm 2 ) (1).

2-2. Estimated thermal stress
When the thickness of the film (2 µm) is much thinner than that of the substrate (500 µm) and this system is exposed to a temperature change ΔT, then the interior of the film away from the edge is subjected to the equi-biaxial in-plane stress change, Δσ = EΔαΔT / (1 − ν) where E: Young's modulus; Δα: difference of thermal expansion between that of film and substrate; ν: Poisson ratio of film (3). Because the thermal expansion of Si is much higher than that of SiO 2 , the film is subjected to tensile stress. For the PMMA interlayer sample, we could observe crack generation from ~ 300 °C. If we take into account the material properties of SiO 2 and Si, then this enhanced tensile stress at 300 °C is

2-3. Electrostatic charging
If electrostatic charging-induced stress is at the 50 MPa level, then we could say that the electrostatic force could drive crack growth. When the surface potential of the ebeam-exposed SiO 2 is U s , then the initial FIB-milled initiation notch could be interpreted as a parallel capacitor plate with a 200-nm gap. Then, the electrostatic force per unit area is Assuming that U s is x volts, then the electrostatically induced stress is σ e = (8.85 × 10 -12 C 2 N -1 m -2 ) ( Thus, if U s is at the ~ 650 V level, then the charging-induced stress is sufficient for crack growth. Movie S1. Real-time recording of e-beam exposure-induced crack growth. Crack is stopped by discontinuity (dot-like) of the PMMA pattern at the end of the movie.
Movie S2. Real-time dark-field recording of a metal crack sample (spiral) with a droplet of DI water. The color change of the crack gap indicates that water moves S6 within the crack pattern via the combined effect of capillary force and the evaporation of water.