Sample Manufacture

Gold crystals were prepared by dewetting a 20 nm thick gold layer, thermally evaporated onto a silicon substrate with a 2 nm titanium adhesion layer. The resulting crystals range from ≈100 nm to a few μm in size and show facets corresponding to {111} and {100} crystal planes (Fig. 1(A)). No FIB-milling was carried out in the vicinity of the unimplanted reference crystal. FIB-milling of crystals A, B and C was carried out at normal incidence, using a 30 keV, 50 pA gallium ion beam and fluences of 4.2 × 104 ions/μm2, 1.3 × 107 ions/μm2 and 1.5 × 108 ions/μm2 respectively. Crystal D was exposed to a fluence of 4.2 × 104 ions/μm2 and a central, nominally 200 nm diameter, region to a fluence of 2.5 × 109 ions/μm2. To allow reliable measurement of multiple reflections from crystals A, B, C and D, FIB was used to remove any other gold crystals within a 20 μm radius. Scanning electron micrographs of crystals A, B, C and D are shown in the SI Fig. S1. X-ray diffraction measurements were carried out 16 to 20 days after sample manufacture.

Ion Implantation Calculations

Ion implantation calculations used the “monolayer collision - surface sputtering” model in the Stopping Range of Ions in Matter (SRIM) code25. For the gold target a displacement energy of 44 eV, binding energy of 3 eV and surface energy of 3.8 eV were used46. Gallium ions were implanted at normal incidence with an energy of 30 keV, gathering statistics over 105 ions. Each ion was estimated to cause on average ~430 target displacements, of which ~30 were replacement collisions. The calculated sputtering rate was ~15.5 gold atoms per gallium ion. For crystal A the amount of material removed by sputtering was negligible. For crystals B and C, an estimated layer of thickness ~3 nm and ~40 nm respectively was removed. Custom MATLAB scripts were used to capture the receding surface effect due to sputtering. The calculated displacement damage and gallium concentration profiles for crystals A, B, C and D, plotted as a function of depth, are shown in the SI Fig. S3.

Experimental Measurements

Synchrotron X-ray micro-beam Laue diffraction at beamline 34-ID-E at the Advanced Photon Source (APS), Argonne National Lab, USA was used to determine the lattice orientations of gold crystals. This served to pre-align crystals for coherent X-ray diffraction measurements at beamline 34-ID-C at the APS. Measurements on the unimplanted reference crystal used an X-ray energy of 9.25 keV, while diffraction patterns from crystals A, B, C and D were collected at 10.2 keV. The X-ray beam was focussed to a size of 1.4 × 2.1 μm2 (h × v) using KB mirrors. Placing the sample in the KB back-focal plane, within the central maximum of the focus, provides the planar wave front required for BCDI. Diffraction patterns were recorded on a Medipix2 area detector with a 256 × 256 pixel matrix and a pixel size of 55 × 55 μm2. For crystals A, B, C and D the detector was positioned 1.85 m from the sample and 3D coherent X-ray diffraction patterns (CXDP) were recorded by rotating the crystal through an angular range of 0.4° and recording an image every 0.0025° with 1 s exposure time. For the unimplanted reference crystal a sample-to-detector distance of 0.635 m was used and CXDPs were recorded by rotating through an angular range of 1.5° in 0.01° steps with 0.5 s exposure time. The sample to detector distances were chosen by starting at the distance required for the measurement of an oversampled diffraction pattern and then moving the detector further back until the diffraction pattern filled the detector matrix. To optimize the signal to noise of the CXDPs, multiple repeated scans of each reflection were performed. Repeated scans were then aligned to maximize their cross-correlation coefficient, and scans with a cross-correlation coefficient greater than 0.99 were summed to produce the CXDP for a specific reflection. For each crystal CXDPs from the following reflections were collected (the number of repeat scans that were averaged is noted in [] brackets): unimplanted reference: {111} [30]; crystal A: (1-11) [18], (11-1) [24], (200) [23], (020) [26], (002) [27]; crystal B: (-111) [16], (1-11) [9], (11-1) [14], (200) [11], (020) [16], (002) [17]; crystal C: (-111) [28], (1-11) [14], (11-1) [27], (200) [26], (020) [22], (002) [18]; and crystal D: (-111) [12], (1-11) [16], (11-1) [17], (200) [9], (020) [12], (002) [14]. Unfortunately the (-111) reflection of crystal A was physically inaccessible. Examples of the coherent diffraction patterns recorded from {111} reflections of all crystals are shown in SI Fig. S7.

Phase Retrieval

The phase retrieval algorithm used to recover the real-space complex electron density is adapted from published work30. Each 3D CXDP pattern was treated independently, using a guided phase retrieval approach with 20 random starts and 5 generations. For each generation 330 phase retrieval iterations were performed using Error Reduction and Hybrid-Input-Output algorithms. Trials using larger numbers of iterations showed no significant further evolution of the solution. Partial coherence effects were accounted for19, and the normalised mutual coherence functions, recovered for all reflections, are consistent with an almost fully coherent illumination. After the fifth generation a sharpness metric was used to select the three best estimates, which were then averaged to return the reconstructed complex electron density. Finally all reconstructions were transformed into an orthogonal laboratory reference frame with isotropic real-space pixel spacing. Agreement between the reconstructed crystal morphologies and scanning electron micrographs is excellent (SI Fig. S1). The normalised cross correlation coefficients, found when comparing the sample shape recovered from BCDI with SEM images, are 0.97, 0.98, 0.97 and 0.97 respectively for crystals A, B, C and D, when considering sample shape projected onto the plane of the substrate. Spatial resolution of the reconstructions was determined by taking the derivative of line profiles of the crystal-air-interface and fitting these with a Gaussian. For each reconstruction six profiles (2 in each spatial direction) were measured and the mean resolution value recorded.

3D Reconstruction of Lattice Displacements, Strains and Stresses

To recover the 3D lattice displacement field, u(r), of a given crystal, any phase wraps in the complex electron densities reconstructed from multiple crystal reflections were unwrapped using the algorithm developed by Cusack et al.47. Next all reconstructions were rotated into the same sample coordinate frame. The phase of the electron density reconstructed from a particular hkl peak, ψ hkl (r), is linked to the scattering vector q hkl and lattice displacement u(r) by ψ hkl (r) = q hkl .u(r). Thus each reconstruction provides a projection of u(r) along the corresponding q hkl . If 3 reflections with linearly independent q hkl are measured, u(r) can be reconstructed. Here 5 (crystal A) or 6 (crystals B, C and D) reflections with non-collinear q vectors were measured from each crystal. Thus the system of equations is over determined, and a least squares fit was used to calculate u(r). The symmetric Cauchy strain tensor, ε(r), is found by differentiating u(r). The strain uncertainty of our measurements, estimated from line profiles of ε(r) extracted from crystal A (Fig. 4), is ~10−4. Stresses were computed from ε(r) using anisotropic elastic constants for gold24.

Finite Element Calculations

Finite element simulations were performed in Abaqus 6.14, using the experimentally determined crystal morphology as a template for generation of the finite element mesh. Custom Matlab and Python scripts developed for this purpose are available upon request. A global seed size of 10 nm was used, based on mesh dependency studies that showed negligible improvements for finer mesh sizes. The resulting model for crystal A is shown in SI Fig. S2. Material properties were captured using anisotropic linear elastic constants for gold24. A uniform volumetric lattice strain, ε v , was imposed within a 20 nm thick layer at the top face of crystal A to represent the effect of ion-implantation damage. ε v = −3.15 × 10−3 provides a good match to the experimentally measured lattice displacement fields in crystal A. Displacements on the bottom surface of the crystal were fixed to capture the substrate effect.

Density Functional Theory Calculations

Ab initio density functional theory (DFT) calculations were performed of a mono-vacancy and of four different self-interstitial defect configurations in fcc gold (100 dumbbell, octahedral site, 110 crowdion and 110 dumbbell). Calculations were carried out in the Vienna ab initio simulation package (VASP)48,49,50,51 using the revised-TPSS exchange functional52,53, and included spin-orbit coupling. Spin-orbit coupling accounts for the band splitting and shape modifications of the 5d bands54,55,56. A plane wave energy cutoff of 450 eV was used and the outermost s- and d-electrons were treated as valence electrons. All samples were relaxed to a stress free condition with residual forces smaller than 0.01 eV/Å. Formation energies and relaxation volumes were calculated by comparing the energies and volumes of a supercell containing each defect type with those of a perfect crystal supercell of similar size and using the same k-point mesh. Visualizations of the supercells used for these calculations are shown in SI Fig. S4.

The lattice constant and elastic constants were calculated using a 4 atom cubic unit cell. The equilibrium lattice constant is 4.075 Å in good agreement with experiments57. The elastic constants were calculated using a finite differences scheme. We obtained c 11 = 210.55 GPa, c 12 = 168.11 GPa and c 44 = 49.96 GPa, which compare well to experimental values at 0 K58. The elastic constants were used to correct both formation energy and relaxation volume of isolated defects according to the method suggested by Varvenne et al.59, which considers the elastic interactions of defects, and image forces due to the finite supercell size and periodic boundary conditions. The formation energies and relaxation volumes are listed in SI Table S1. The relaxation volume of a substitution gallium atom in gold was also calculated.

Data Availability

Diffraction data and selected computer codes used for data analysis and simulations can be obtained from the authors by contacting felix.hofmann@eng.ox.ac.uk.