Interference in edge-scattering from monocrystalline gold flakes

We observe strongly dissimilar scattering from two types of edges in hexagonal quasi-monocrystalline gold flakes with thicknesses around 1 micron. We identify as the origin the interference between a direct, quasi-specular scattering and an indirect scattering process involving an intermediate surface-plasmon state. The dissimilarity between the two types of edges is a direct consequence of the three-fold symmetry around the [111]-axis and the intrinsic chirality of a face-centered cubic lattice. We propose that this effect can be used to estimate flake thickness, crystal morphology, and surface contamination.


Introduction
Historically, the field of plasmonics 1 has explored the interaction of light with the free electron gas, with a predominant attention to amorphous and polycrystalline noble metal nanostructures and thin films, 2 while less attention has been devoted to plasmons supported by monocrystalline materials. More recently, chemically synthesized monocrystalline gold flakes have been receiving increasing attention within the plasmonic community. In many aspects, such colloidal gold nanoparticles show superior plasmonic properties, as compared to evaporated polycrystalline films. [3][4][5][6] Atomic flatness and well-defined crystal structure offer larger plasmon propagation lengths and sharper resonances due to lower Ohmic losses and reduced surface scattering. 7 These favorable properties have been utilized in the design and fabrication of various plasmonic devices, such as nano-circuits, 3,8 nano-antennas, [9][10][11] tapers, 12 and plasmon billiards. 13,14 However, flat metal crystals are rarely true single crystals, but rather twins joined at pairs of stacking faults. 15 This is no coincidence, because the strong lateral growth involving [100]-facets requires the presence of at least 2 stacking faults within the seed. 6 These defects play an important role in the crystal growth 5 and might exhibit interesting plasmonic phenomena of electronic 2D states. This along with the well defined material properties of single crystals render them excellent candidates for the observation of quantum effects 16,17 or of anisotropic nonlinear or nonlocal response e.g. due to the deviation of the Fermi surface from a perfect sphere. This is especially true for submicron particles as quantum corrections to the classical electrodynamics manifest increasingly when approaching the subwavelength scale and reaching out for atomic dimensions. 18,19 Differences between the material properties of mono-and polycrystalline metals are both important for applications as well as interesting in their own right. Moreover, the highly ordered atomic structure of single crystals is also reflected in their geometry with very well defined angles, atomically flat surfaces and sharp edges. The quality of these features is well beyond what is currently achievable with state-of-the-art nano patterning of polycrystalline films, 20  Several examples for such studies have been conducted for particles that are small compared to the wavelength of light. [23][24][25][26] Crystal-related morphologic features are of course by no means restricted to subwavelengthsized particles, but also appear in fairly large objects such as the gold flakes studied in this paper with lateral dimensions greater than 10 µm and thicknesses around 1 µm. One such non-trivial feature is the fact that the edges of our nearly hexagonal flakes are asymmetrically tapered and that two different types of such edge terminations alternate around the flake. As a result, each edge is dissimilar to both adjacent edges and the opposing one. This lack of symmetry with respect to mirroring and 180 • rotation can be seen in high-resolution scanning-electron microscope (SEM) images (Figure 1a-c). Yet quite often, this detail is ignored and the edges are simply approximated as rectangular truncations 21,22 . However, it reflects the fact that the face-centered cubic (FCC) gold lattice is symmetric with respect to neither a 180 • -rotation nor a mirror operation through the [111]-axis. It is therefore a large-scale manifestation of the atomic order and can lead to a significant difference in the optical far-field properties as we show here.
In this work we report on a distinct difference in the scattering of visible light from the two types of edge terminations of colloidally grown gold flakes with thicknesses around one micron.
Even though in an optical bright-field (BF) our flake looks perfectly hexagonal ( Figure 1d) and seemingly exhibits six fold symmetry, difference in scattering appears as differently colored edges when observed in an optical dark-field (DF) microscope under low numerical aperture (NA) collection conditions (see Figure 1e-h). We conclude that this is the far-field manifestation of the fact that opposing flake edges meet the substrate at different angles, which in turn is a macroscopic consequence of the inherent chirality along the [111]-axis of the FCC lattice. The strong imbalance in the size of the top and bottom {111}-planes over all others is due to stacking faults (more precisely multiple twin planes), which form in the early stage of the crystal growth. Such defects commonly appear in the metals with FCC crystal structure, especially in gold, as they have some of the lowest defect energies 29 . They lead to quite different growth rates along different crystal axes and thus cause high aspect ratio of the crystals. Therefore, such thin flakes are not strictly speaking monocrystalline, as commonly referred to, but twins.
It is noteworthy that the (111)-plane of the FCC-lattice has only three-fold symmetry even though such a crystal facet might be perfectly hexagonal, as one shown in Figure 1d. This is the result of the chirality of the FCC-lattice along the [111]-axis, which is due to the existence of two different stacking patterns. Macroscopically, it manifests in the aforementioned two types of flake edges, which appear with threefold symmetry. In the following, we refer to edges where the side facet touching the substrate is of (111)-type or (100)-type as type-A or type-B edges, respectively. As it turns out, this three-fold symmetry and therefore the lattice chirality can be directly observed with an optical microscope.
We noticed that flakes which look almost perfectly hexagonal in the optical bright-field and high-NA dark-field (Figure 1d,e) exhibit very different scattering spectra from the two types of edges. This is visible to the naked eye under dark-field conditions and becomes more prominent with decreasing collection NA as illustrated in Figure 1f-h. Images were acquired using the same objective lens and the same illumination conditions (light from DF condenser impinging on the sample at angle θ inc ≈ 12 • , as shown in Figure 2), and filtering the collected light in the Fourier plane, as described in details in the Methods section.
We find behavior that is qualitatively similar to that depicted here, yet with different shades of red, yellow and green commonly for flakes with thicknesses around 1 micron, so it appears to be a geometry-related effect. In order to further understand the underlying process, we first performed 2D finite-element calculations in p-polarization. This is sufficient for qualitative results, because we observed experimentally that with polarized illumination only the edges perpendicular to the incident polarization appear in dark-field and that the scattered light is p-polarized itself.
We then post-processed the data with a far-field filtering procedure that mimics the effect of a low-NA objective (see Methods section for further details). Figure 3 shows experimental spectra acquired with NA≈0.4, which provides good contrast while keeping sufficiently strong signal, for two adjacent edges of the flake shown in Figure 1. Alongside is shown a numerical spectrum and we expect that the scattering spectra of either edge type depend periodically on the parameter h l with a periodicity of for the type-A edge and with α instead of β for the type-B edge. Using the permittivity ε Au = −16 + 1.1i for monocrystalline gold 30 at a vacuum wavelength of λ 0 = 700 nm as an example, and assuming θ inc = 12 • , we find: ≈ 420 nm and ∆h This means that we expect the low-NA spectra observed at type-A or type-B edges to repeat whenever the thickness of the lower part of the flake is increased by 420 nm or 320 nm, respectively.
These values for the periodicity are to be compared to the numerically simulated spectra in the Figure 5, where the upwards scattered power for both types of edges at a fixed wavelength (700 nm) is plotted as a function of the lower flake thickness h l with all other parameters (e.g. θ inc , h u ) kept as in Figure 3. We observe distinct oscillations with a periodicity of ≈ 340 nm for type-B edges, which is in good agreement with the simple interference model. The periodicity of the type-A spectra is slightly less consistent as the spectrum is not a pure sinusoid (distances between minima, maxima and turning points give different "periodicities"). Anyhow, the periodicity is greater than for type-B edges and we extract a value of ≈ 380 nm. This is still in qualitative agreement with the interference model. We attribute the disparity to the existence of a second, weaker resonant effect, potentially a Fabry-Pérot-like standing wave on the lower facet. Yet, the main effect is clearly visible, especially since no alternative explanation predicts oscillations in the 300-400 nm range.
So far, we have not discussed the agreement between experimental and numerical spectra ( Figure 3). Both panels show qualitatively similar behavior. The most striking differences are an overall red-shift by some 100 nm of all features in the experimental spectrum and a significantly reduced amplitude towards the blue spectral range. This is partially due to simplifications and  uncertainties in the numerical model. Firstly, in our simulations we chose a 2D finite-element model with plane-wave illumination impinging normally on the edge. In contrast, the illumination in our dark field experiment was from a range of azimuthal angles covering a sector of 60 • . We expect that non-normal incidence would lead to considerable red-shift of the interference effect.
Secondly, also the elevation angle θ inc in the experiment is not well defined. Incident light arrives at the edge from an indefinite range of angles around approx. 12 • . In the numerical model, we assume a plane wave from θ inc = 12 • . These two angular distributions lead to a smearing and potentially to a partial destruction of the interference pattern, especially at higher orders, i.e. shorter wavelength. Thirdly, we have conducted simulations with different common models for the permittivity of gold and find that variations in the plasmon dispersion relation can easily account for 50 nm shift in the spectral features, too. Although the flakes appear clean under the electron microscope, we suspect that the flake is covered by residue from the fabrication process, which again would lead to spectral red-shift and potentially to increased loss at shorter wavelengths. Within the limits of these uncertainties, we are confident, that we have identified the main origin for the thickness-dependent dissimilar scattering spectra from type-A and type-B edges in our monocrystalline gold flakes. Finally, the strong sensitivity with respect to the plasmon dispersion relation also offers the possibility to independently verify ellipsometrically determined material parameters of monocrystalline metal particles provided they are known to be clean and using plane-wave like illumination e.g. in a goniometer.

Conclusions
To summarize, in this work we have exemplified that the differences in the scattering spectra of the adjacent edges of the gold monocrystalline flakes, which we have first observed experimentally in the DF microscope, are far-field manifestation of the subwavelength-scale morphological features. We have developed a numerical model and filtering method which allows to simulate the experimental conditions fairly accurately. Through a careful analysis of numerical simulations, we found that the height of the lower trapezoid in the cross-section of the flake (h l ) is the main parameter for determination of the scattering spectrum. Guided by analysis of numerical results, we developed an analytic model where the physical mechanism, which gives the main contribution to the observed scattering spectrum, is the interference between a surface plasmon in the lower facet of the flake's edge and free space waves. The difference in lengths of the facets of adjacent edges explains the difference in scattering spectra of those.
We speculate that DF spectroscopy can potentially be used for estimation of the flake thickness or the dielectric function of the monocrystalline gold. Given that the flakes have well-determined geometries, and with possibilities for experimental measurement of length scales at atomic-scale resolution, the later appears to be a more realistic task. While we might not know the exact dielectric function (tensor), we accurately know dimensions of the box containing the plasmons, which is a rare condition considering the situation where geometries are fabricated with top-down approaches such as focused ion beam (FIB) or electron beam lithography (EBL). In other words, the accurate information about geometry may in turn be used to determine the (bulk) optical properties of the metal, by perfectly matching accurate experiments with simulations, with the dielectric function (tensor) being the unknown in the simulations.

Methods
In the following subsections experimental and numerical methods used in this work are described.

Sample preparation
Gold monocrystalline flakes were prepared using the modified Brust-Schiffrin 27 method for colloidal gold synthesis in a two-phase liquid-liquid system via thermolysis 28

Spectroscopy
DF spectroscopy measurements were performed using the Zeiss Observer microscope (Epiplan-Neofluar HD objective 50x, NA=0.80) and Andor Kymera 193i spectrograph equipped with Andor Newton CCD camera. Additionally, two lenses (achromatic doublets with focal lengths 15 and 20 cm) and an iris diaphragm were used to create the so-called 4f correlator system for spatial filtering (i.e. NA selection). For the measurements described in this work we have calibrated diaphragm opening to correspond to NA ≈ 0.4. This value of the collection NA was chosen because it gives good contrast between two types of edges, while keeping sufficiently strong signal and high spatial resolution. Lenses with different focal lengths were chosen intentionally to obtain appropriate image magnification on the camera screen.
A standard tungsten-halogen lamp was used as an illumination unit in this setup. In order to achieve "one-sided" illumination, the DF mirror cube was modified to restrict the range of azimuthal angles of incidence, i.e. DF illumination ring was partially covered with an opaque sheet and only a sector of ≈ 60 • was left open. Reported spectra were normalized to a reference spectrum, obtained by illuminating a white scatterer in same conditions.
Additionally, a linear polarizer and analyzer were used to select appropriate (i.e. perpendicular to the specified edge of the flake) polarization of the incident and scattered light.
In order to avoid any systematic error due to the experimental setup, we measured different edges by rotating the sample while keeping fixed all other settings.

Numerical simulations
Numerical simulations were performed using a commercially available finite-element method (FEM) solver (Comsol Multiphysics 5.3). The geometry of the model is two-dimensional, implying homogeneity along z-axis direction (axis orthogonal to the plane of the flake's cross-section).
We consider this simplification to be appropriate as lateral dimensions of the flake are much larger than the thickness (i.e. w fl h u + h l ). The model assumes a plane wave with wavelength λ 0 incident at angle θ inc . In the first step, the model solves for the electric field distribution in the vicinity of the air/silicon interface. For the refractive index of Si, we used interpolated experimental data by Aspens 31 . In the second step, a gold particle is placed on the substrate with the shape shown in Figure 2 and fields calculated in the first step are used as a background source to obtain the scattered fields. The refractive index of monocrystalline gold is described by interpolated experimental data from Olmon 30 . The model uses triangular meshing (5 nm maximum element size in the metal domain, 8 nm in silicon) and fourth order polynomial basis functions. We have performed mesh refinement study, from which we assess second order convergence and estimate relative error in the reported numerical data to be less then one percent.
In the subsequent step, the simulated fields were post processed using a dedicated filtering method, which mimics operation of the microscope objective, i.e. selects only traveling waves which propagate within a given NA.