Curvature-induced and thermal strain in polyhedral gold nanocrystals

Gold nanocrystals are attracting increasing interest within the scientific community due to the inherent differences between their optical and electronic properties and those of their bulk counterparts.¹ Gold nanocrystals are important for a wide range of potential applications in photonics, nanoelectronics, biological imaging, and biosensors^2–5 due to their ability to enhance signal in surface plasmon resonance (SPR) absorption and surface enhanced Raman spectroscopy (SERS) measurements, as well as their chemical and thermal stability. The optical and electronic properties of these nanocrystals are strongly dependent on the morphology, size, and strain.^6,7 Thus, the investigation of nanoscale internal structure, shape, and lattice strain is crucial for the development of suitable synthesis techniques and in understanding growth dynamics on the nanoscale. It has been unclear whether the strain distribution in metallic nanocrystals is intrinsic or if it is a consequence of surface contamination and interfacial stress from the substrate.⁸ Au nanocrystals with a face centered cubic (fcc) lattice are characterized by different surface energies for different crystallographic planes. The low-index facets (e.g., (111)) have the smallest specific surface energies that can be found in fcc crystals.⁹ The Wulff construction theorem,¹⁰ which has been used to predict the equilibrium shape of nanoparticles, offers a geometrical approach to determining the facets which minimize the surface energy of a free floating nanoparticle. In this letter, we map the internal strain of Au nanocrystals and find the strain is influenced both by their interaction with the substrate as well as the details of the facet geometry.

Coherent x-ray diffractive imaging (CXDI) is a technique that probes the overall shape of a nanostructure¹¹ and the displacement of individual atoms from an ideal periodic lattice.^12–16 A mathematical inversion of the coherent x-ray diffraction pattern results in a complex-valued density function. The amplitude of this complex function is proportional to the electron density in the crystal, while the phase can be interpreted as the deformation of the lattice as projected onto the momentum transfer vector, which is determined by the direction of incident and scattered x-ray, of the Bragg peak.¹⁷ The fact that the intensities (I = |E|², where E is the amplitude of the diffracted wave) are measured with the loss of information regarding the phase of the waves gives rise to the so-called phase problem.¹⁸ The phase problem has been a recurring issue in crystallographic reconstructions for many decades. However, it can be mitigated on the condition that the sampling frequency in reciprocal space is at least twice the bandwidth of the real space frequency.^19,20

In this study, Au nanocrystals were grown using thermal chemical vapor deposition (CVD) onto a Si (100) substrate with the native oxide using a catalyst-free thermal approach with AuCl₃ as a precursor. Approximately 1.0 g of powdered AuCl₃ was loaded in a boat and placed within a quartz tube at the center of a Lindberg Blue furnace at 475 $°C$⁠, where the silicon substrate still remains unoxidized because the oxidation of silicon is performed above 800 $°C$⁠.²¹ An Ar carrier gas was flowed through the quartz tubing while the downstream end of the CVD setup remained isolated from the outside atmosphere. The system was left at 475 $°C$ for about an hour, after which it was allowed to cool down to room temperature. The Ar flow through the quartz tube was maintained for the duration of the cooling process. Through this process, nanocrystals form in the absence of any foreign catalyst. The approach we used is similar to one that has been used to form Ni nanocrystals²² and yields single-crystal gold nanocrystals with well-defined facets (see Figs. 1(b) and 1(c) for two examples). In contrast, most synthesis techniques for Au nanocrystals employ surfactants to control shape or size.^23–26 These surfactants remain after the synthesis process is complete and can affect the physical properties of the nanocrystals.

This synthesis method can be used to create many different shapes of gold nanocrystals, ranging from triangular plates and octahedra to multiple twinned particles such as decahedra and icosahedra. These equilibrium nanocrystal shapes can be understood by employing the Wulff construction theorem.¹⁰ We further studied the three-dimensional electron density, lattice displacement and strain distribution in these gold nanocrystals by using synchrotron-based coherent x-ray diffractive imaging.¹² 

In addition to the nanocrystals enclosed by identical polygonal surfaces, plate-like flat crystals are also formed during synthesis. These plate-like structures exhibit a triangular morphology with basal surfaces bounded by two (111) facets and three (100) facets along the sides.⁹ This shape is not explained by conventional thermodynamical arguments due to a relatively large surface area-to-volume ratio. However, it can be explained from the growth of a nanocrystal seed in the presence of a stacking fault.²⁷ If the planar seed exhibits a stacking fault, Au atoms added in close proximity to the fault give rise to crystal growth.^27,28 Regardless of this difference in process of formation, gold nanocrystals with both triangular plate and octahedral shapes retain symmetries inherent to the fcc unit cell.²⁹ 

The coherent x-ray diffraction measurements were performed at Sector 34-ID-C of the Advanced Photon Source (APS) synchrotron facility. Figure 1(a) shows a schematic image of the setup we used for measurements of coherent x-ray diffractive imaging. An upstream monochromator was used to select E = 9.0 keV x-rays, which were focused onto the sample using Kirkpatrick-Baez mirrors that restricted the beam size to a $2 μm × 2 μm$ region. The resulting diffraction patterns were measured using a charge-coupled device (CCD) with 22.5 $μm$ pixels located along the detector arm at a distance of 0.9 m beyond the sample.

The measured 2D diffraction patterns for a triangular nanoplate and an octahedral nanoparticle are shown in Figs. 1(d) and 1(e), respectively. These coherent diffraction patterns were recorded for the rocking curves of the (111) Bragg reflections by rotating the sample through the Bragg peak in increments of 0.005°. 121 frames were collected in this manner for each nanocrystal, covering a total angular range of 0.6°. Isosurfaces of the 3D diffraction patterns, obtained by stacking up the 2D diffraction data collected during rocking scans, are shown in Figs. 1(f) and 1(g). The fringes, seen as spatial modulations in the signal, originate from the interference between waves diffracted from pairs of sharply terminated crystalline facets. Accordingly, the four pairs of fringe modulations in the diffraction pattern imply the presence of eight facets (i.e., an octahedron). The lack of inversion symmetry in the diffraction pattern reflects the strain in the nanocrystal.³⁰ 

The displacement field projected along the {111} Bragg plane direction was measured to quantify the internal nanoscale strain field within the nanocrystals. The missing phase of the diffracted wave was retrieved by using an iterative algorithm. The retrieval algorithm begins by taking a Fourier transformation of the amplitude from the measured diffraction pattern and coupling it with a random guess for the phase. The algorithm iterates back and forth between real and reciprocal space while applying constraints in every cycle.^31,32 The real space constraints ensure that the amplitude, corresponding to the electron density, outside of the support, in which the object is assumed to exist is set to zero. In reciprocal space, the computed modulus is replaced with the square root of the measured intensity. Alternations of error reduction (ER) and hybrid input output (HIO) phase retrieval algorithms are applied.³² In addition, the shrink-wrap algorithm is applied periodically to optimize the support as the algorithm progresses.³³ 

The geometry of gold nanocrystals on a silicon substrate with momentum transfer wave vector $Q→$ for the (111) Bragg reflection is shown in Fig. 2(a). $Q→$ is normal to the substrate for both crystals. Figures 2(b) and 2(c) show the shapes of the reconstructed crystals. The edge length of the octahedral nanocrystal is about 220 nm. The triangular plate has an edge length of about 600 nm and a thickness of about 60 nm. The phase (shown on the color scale) indicates a projection of the deformation onto the reciprocal lattice vector $Q→$⁠. Thus, the phase is the deformation along the normal to the substrate surface in both nanocrystals. In the case of the triangular plate, due to the small thickness of a plate-like nanocrystal, area of contact with the substrate is so large that the physical deformation is closely associated with the direct interaction between a nanocrystal and its substrate.

Two perpendicular cross-sectional planes, both containing the {111} $Q→$ vector, were chosen for detailed analysis of displacement and strain distributions as shown in Fig. 3(b). Figures 3(c) and 3(d) show the phase shifts on the planes depicted in Fig. 3(b). Both images clearly show that the displacements along the vertical direction, which corresponds to the {111} $Q→$ vector, occurred primarily near the surfaces. Moreover, the phase shifts near the top and bottom surfaces are colored blue (negative) and red (positive), respectively. This pattern indicates that the atoms near the top and bottom surfaces moved towards the center of nanocrystal. A distinct contraction of the surface layers of a gold nanocrystal has been confirmed by Huang et al. in models of experimental diffraction patterns and molecular dynamics simulations.³⁴ 

The normal strain is generally defined as the derivative of the displacement, $εxx=∂ux(r)∂x$⁠, where $εxx$ is the normal strain along the x-direction. The 3D strain distribution of the entire volume along the {111} $Q→$ vector direction was obtained by taking the derivative of the displacement field resulting from algorithmic phase reconstruction. Figures 3(e) and 3(f) show the perpendicular slices that represent strain fields on the planes depicted in Fig. 3(b). The concentration of compressive strain in particular regions, as well as the overall inward contraction, is readily apparent. Strong compressive strains are located mainly at the vertices or edges, whereas the weaker strain distribution appears on the flat surfaces. The center of the crystal remains nearly strain-free. This observation is in good agreement with recent studies of strain distribution in metal nanocrystals. These studies found evidence of compressive strain under the locally rounded surface regions of metallic nanocrystals where (111) planes intersect, and exhibit relatively less compressive strain near flat surface regions.^35,36

We observed that the strain near the bottom surface of the octahedral nanocrystal is significantly stronger than that near the top surface. The measured strain is $−4.2×10−4$ and $−5.0×10−4$ near the top and bottom surfaces, respectively, as in Fig. 3(e). The phase-retrieval real space density distribution is well-known to have a two-fold degeneracy because the combination of inversion symmetry and complex conjugate to any complex function solution is also a solution, resulting in identical experimental data.³⁷ Therefore, while Figs. 3(e) and 3(f) show an octahedron with greater compressive strain at the bottom facet based on diffraction data alone, it is not possible to completely rule out an inversion symmetry, equivalent solution where the top facet is unusually strained instead. However, we postulate that it is much more likely that the bottom facet has the greater compressive strain, since such orientation of octahedron (see Figs. 3(e) and 3(f)) would be consistent with the strain induced by the underlying substrate, whereas the top facet solution lacks any clear explanation for such substantial strain modification. This assumption is further supported by greater compressive strain induced by the substrate also observed in triangular nanoparticles, a case in which the correct orientation can be determined from independent SEM measurement.

The strain on the rounded surface of a particle of radius R can be estimated quantitatively by the Young-Laplace equation for a given bulk modulus, K, of the material. Surface stress $σS$ results in a pressure difference between the inside and outside regions of a material that effectively compresses an isotropic particle. For a lattice constant of a

Surface stresses for metals are usually tensile in nature, in the range of 2 N/m.^38,39 The bulk modulus of gold is 180 GPa and the radius of the locally rounded region is 18 nm in the upper right portion of Fig. 3(e). According to Eq. (1), the strain under the rounded surface should be $−4.1×10−4$⁠. This value is in good agreement with the measured strain of $−4.2×10−4$⁠. However, the strain at the bottom surface is approximately 20% higher than that at the top surface. In general, inhomogeneous internal strain in nanocrystals can be caused by irregular surface relaxations.⁴⁰ Surface stress, which is coupled with internal strain, depends on the chemical composition of the surface and its interfaces with neighboring particles, substrates, solvents, surfactants, etc.³⁶ In our experimental environments, the irregular relaxations at the surfaces may be attributed to the dissimilarity in interfacial energies at the substrate. The different surface energies at Au/Air and Au/Si interfaces determine an internal strain accumulation during the growth of a nanocrystal.⁴¹ Furthermore, when the nanocrystals are cooled down to room temperature, the strain at the interface is induced by the difference in the thermal expansion coefficients between the Au nanocrystals and the Si substrate.

Within the temperature range of the synthesis process, the thermal expansion coefficient of Au, $αAu=1.4×10−5K−1$⁠, is greater than that of Si, $αSi=0.3×10−5K−1$⁠. As a result, the Au nanocrystal contracts more than the Si substrate as they are cooled down. Therefore, compressive strain fields along the $Q→$ vector are induced by the tension along the bottom surface, which is parallel to the substrate surface. As shown in Figs. 3(e) and 3(f), these compressive strains are not distributed uniformly across the bottom surface. This distribution can be attributed to a partial relaxation of the strain due to an incomplete contact between the nanoparticle and the substrate surface.

Since strain is calculated with respect to the normal direction of the substrate surface, the strain distribution at the right and left sides of Figs. 3(e) and 3(f) lie nearly in-plane, while the strain distribution in other regions show out-of-plane components. Most metal surfaces exhibit tensile surface stress because they tend to reduce interatomic distances in surface planes.³⁹ Accordingly, we are seeing the signature of tensile surface strain in the octahedral nanocrystals.

Figures 4(a) and 4(b) represent phases and strains, respectively, averaged over the depth of the triangular plate. A primarily negative distribution of phases shows that the triangular plate deformed downwards. The compressive strain along the $Q→$ vector is induced in much the same way as the compressive strain near the bottom of the octahedral nanocrystal, as discussed in the preceding paragraph. The height-to-length ratio of the triangular plate is so small that the displacement of the entire volume is affected predominantly by the difference in thermal expansion between the nanocrystal and the substrate.

In conclusion, we have observed the atomic-level structures and internal strain distributions in Au nanocrystals with the morphology of an octahedron and triangular plate grown by CVD at a temperature of 475 $°C$⁠. Phase reconstruction reveals the lattice distortion of both nanocrystals along the {111} direction. In the octahedral nanocrystal, it is observed that there is a contraction of the lattice near the surfaces, accompanied by a compressive strain beneath these surfaces, especially near locally rounded surface regions. The strain on the surface of these rounded regions is in good accordance with theoretical predictions from the Young-Laplace equation. However, the strain near the bottom surface shows a discrepancy. We conclude that this inhomogeneous internal strain distribution is caused not only by the dissimilar surface energies of the interfaces during the growth process, but also by differing thermal expansions between gold nanocrystals and the silicon substrate when they are cooled down to room temperature. In the case of a thin nanoplate crystal, the difference in thermal expansion is the primary contributor to the strain fields of the entire particle. This is the study of both curvature-induced strain under a locally rounded surface and thermal strain at the nanocrystal-substrate interface in a single Au nanocrystal.

J. W. Kim thanks Professor Ian Robinson for helpful discussion. The coherent x-ray imaging work at UCSD was supported by US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-SC0001805. Crystal growth was supported by NSF Award DMR-0906957 and DMR-1411335. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. D.O.E. under Contract No. DE-AC02-06CH11357.