Surface flaws control strain localization in the deformation of Cu$\vert$Au nanolaminates

We carried out matched experiments and molecular dynamics simulations of the compression of nanopillars prepared from Cu$\vert$Au nanolaminates with 25 nm layer thickness. The stress-strain behavior obtained from both techniques are in excellent agreement. Variation of the layer thickness in simulations reveals an increase of the strength with decreasing layer thickness. Pillars fail through the formation of shear bands whose nucleation we trace back to the existence of surface flaws. Our combined approach demonstrates the crucial role of contact geometry in controlling the deformation mode and suggests that modulus-matched nanolaminates should be able to suppress strain localization while maintaining controllable strength.

surface and at the interface of 370 nm and 480 nm, respectively. The actual test volume was composed of a 40 layer stack of 25 nm individual thickness giving a total sample thickness of 1 µm (Fig. 1a). The nanopillars were compressed in situ in a scanning electron microscope (SEM, FEI Nova NanoLab 200 and Nanomechanics InSem nanoindenter) to observe their behavior during deformation.  (Fig. 1a). Deformation then led to the gradual compression of the pillar and eventually to the nucleation of a shear band (indicated by a "1" in Fig. 1b and c). Shear banding localized further deformation and led to the extrusion of a wedge-shaped region near the top of the pillar. Further compression nucleated a second shear band, initiated right where the wedge had slid enough to create a surface step that concentrated stress (position "2" in Fig. 1c and d). Deformation then continued along this secondary shear band and eventually resulted in the failure of the pillar.
The experimental observations pose two important questions: First, it is unclear which process sets the strength of the material and which role the layer thickness plays in that process. We note that there is no evidence for slip along the interface in these pillars. Experiments on pillars with tilted interfaces and simulations on representative volume elements suggest that the interfacial shear strength is ∼ 0.3 GPa (see Supplementary Section S-I) but the Schmid factor for sliding along the interface for the loading geometry shown in Fig. 1 is zero. Second, homogeneous deformation was followed by the traversal of a shear band that led to the failure of the pillar. From the experiments alone, it remains unclear what conditions led to the nucleation of these shear bands. Experimental pillars often have defects from growth and FIB preparation, as for example surface roughness. We here hypothesize a primary reason must be symmetry breaking due to the existence of surface flaws on either pillar or indenter tip.
To test this hypothesis, we carried out molecular dynamics (MD) calculations with varying layer thickness from 5 nm to 25 nm, resulting in systems of up to 380 million atoms with a total pillar height of 300 nm (Fig. 2a). These pillars are smaller than their experimental counterparts but have identical layer thickness and aspect ratio. The interaction between Cu and Au was modeled using a tailor-made embedded atom method potential. 12 The flat, rigid indenter was obtained by freezing the structure of a Cu 50 Zr 50 metallic glass obtained by melting a random solid solution at 2500K and quenching it down to 0K at a rate of 6 K ps −1 . A purely repulsive Lennard-Jones potential with interaction parameters Cu = 0.4093, σ Cu = 2.338, Au = 0.4251, σ Au = 2.485 acts between pillar and indenter. 23 Note that the disordered nature of the indenter introduces finite friction between indenter and pillar. We pressed the indenter onto the pillar by displacing it at a constant applied strain rate ofε app = 0.8 × 10 8 s −1 . The whole pillar was kept at 300K using a using the Nosé-Hoover thermostat 24 with a relaxation time constant of 0.5 ps. A few rows of atoms at the bottom were fixed in space to anchor the pillar to the substrate.  the pillar by cutting atoms above a plane that follows random self-affine scaling 25,26 with Hurst exponent 0.8 and root-mean square (rms) slope of 0.1 (Fig. 2). 3) Bulk defects: As a representative volume defect, we introduced screw dislocations at random positions and orientations.
We quantified experiment by estimating the stress σ inside the pillar before the nucleation of the first shear band (i.e. between the states shown in Figs A key observation in our simulations is that perfectly flat surfaces always lead to homogeneous deformation (Fig. 4a) while rough surfaces show heterogeneous deformation and failure (Fig. 4b,c,e and f).
To clarify the role played by roughness we created pillars with the simplest model for "roughness", a single atomic step on the surface (Fig. 4b). This model "roughness" already led to a deformation mechanism dramatically different from perfectly flat surfaces.   the deformation localized ( Fig. S-1). The deformation was stable and shear in the direction of the interface was not observed. At the maximum strain a shear stress acting along the interface of ≈ 0.2 GPa was observed. In case of the 17 • pillar, more pronounced steps on the pillar side-face were observed (marked by arrow in Fig. S-1d), while the pillar did not fail catastrophically. The maximum shear stress along the interface was ≈ 0.3 GPa.

B. Simulation
We used representative volume elements to compute the interfacial shear strength of the Cu|Au using molecular dynamic calculations. The system represented in Fig Before straining, the systems was relaxed at 300 K for 500 ps using the Nosé-Hoover/Andersen 24 ensemble without any strain. Simple shear strain was applied along [112](111) directions for shear parallel to the nanolaminate interfaces by homogeneously deforming the box. Our notation [abc](hkl) for simple shear reports both the direction of shear [abc] and the plane of shear (hkl). We used a strain rate of 10 8 s −1 in all cases; strain rate dependence of stress is negligible at these rates in FCC metals. 32 For an atomically sharp interface, the nanolaminate responded to this deformation with a shear stress of a few MPa (Fig. S-2b). shown in Fig. S-1a. We computed the exact area A from the convex hull of the cross section at the given height (red line in Fig. S-3a). We also computed the length of the semi-minor and semi-major axes of the pillar (as shown by the dashed lines in Fig. S-3a). With these measurements we determined the lateral strain in the pillar, ε = ln (1 + (d − d 0 ) /d 0 ) where d 0 is the initial diameter. Fig. S-3b shows the results obtained for the different definitions of the cross-sectional area A (smallest and largest cross section, exact convex hull) for an exemplary calculation. We observe for all the cases a yield at σ ≈ 4 GPa and ε ranging from 0.1% to 1% followed by some strain softening. The maximal lateral strain is achieved for the largest cross-section definition with ε ≈ 25% the smallest cross section reaches ε ≈ 22%

S-III. DEFORMATION OF SINGLE-CRYSTALLINE AU PILLARS
We carried out control calculations using single crystal Au pillars of 60 nm height, equal to the total pillar height for the nanolaminate pillars with λ = 5 nm layer thickness. Fig. S-4 show that the pillar deforms homogeneously even in the presence of a surface step. Alongside the atomic position we also show an analysis of the dislocation structure obtained with the dislocation extraction algorithm (DXA, Ref. 33 ). We obtain the same results for selfaffine roughness (not shown here). We observed that after a dislocation nucleates at the surface ( Fig. S-4b) it crosses the full pillar, vanishing at the sidewall and leaving behind a nucleating from the top pillar surface (Fig. S-4e). While some dislocations escape the pillar, others react in the bulk or pile up against the fixed layer at the bottom (Fig. S-4f-i).