SUPPORTING INFORMATION
for
Plasmon Coupling in Silver Nanocube Dimers:
Resonance Splitting Induced by Edge Rounding.
Nadia Grillet, Delphine Manchon, Franck Bertorelle, Christophe Bonnet, Michel Broyer,
Emmanuel Cottancin, Jean Lermé, Matthias Hillenkamp and Michel Pellarin*
Université de Lyon, Université Lyon 1, CNRS, Laboratoire de Spectrométrie Ionique et
Moléculaire (U.M.R. 5579) 43, Bld du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
[email protected]
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1. Symmetry Breaking Effects on the LSPR of Silver Nanocube Dimers
Figures S1 and S2 illustrate symmetry breaking effects in the optical response of silver
nanocube dimers. In Figure S1, we have first considered the case of aligned cubes having
different sizes (51nm and 31nm). The LSPR splitting in Figure S1c and Figure S1d is not
related to size asymmetry but rather to the conditions that at least one particle has rounded
edges and that the corresponding curved region is in close proximity to the opposite particle
surface. This explains why no splitting is observed in Figure S1b where the curved edges of
the large particle do not pertain to the interparticle gap region (are not facing the small particle
surface). In Figure S2, the dimer asymmetry is now induced by a misalignment of both
particle edges. When the cubes are perfect, the symmetry breaking is responsible of LSPR
changes that are really noticeable only for the largest misalignment which is not encountered
in our experiments. Even in this case, the appearance of a satellite peak on the right side of the
main LSPR does not reproduce the marked splitting observed in experimental spectra. In fact,
a misalignment between cubes is not necessary to induce the LSPR splitting as it can be seen
in the lowermost spectra in the middle and right columns of Figure S2 (S=0nm). It is
sufficient that at least one cube has rounded edges in the capacitor region (see Figure S1). For
large misalignments, symmetry breaking is only responsible of a fragmentation of the low
wavelength resonance ( λ2R in the manuscript). This is consistent with the hypothesis of a
quadrupolar character for this plasmon mode that is expected to be very sensitive to the exact
surface charge distributions at facing cube edges and corners.
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Figure S1 Computed extinction cross sections (DDA) for dimers of silver nanocubes with
different sizes (L=51 nm and L=31 nm edge lengths). The incident electric field is polarized
along the interparticle axis. The interpaticle distance is d=1 nm. (a) perfect cubes; (b) small
perfect cube and large rounded edge cube (re=10nm); (c) small rounded edge cube (re=10nm)
and large perfect cube; (d) both edge rounded cubes (re=10nm).
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Figure S2 Computed extinction cross sections (DDA) for dimers of silver nanocubes as a
function of the lateral shift S. The dimers have the same cube size (L=51 nm) and interparticle
spacing (d=1 nm). The cubes can have both sharp edges (left) or rounded edges (right). The
case of the heterodimer is given in the middle column. The incident electric field is polarized
along the interparticle axis.
2. LSPR Dependence on the Geometry of Single Nanocubes and Nanocube Dimers:
DDA versus FEM Simulations.
Available techniques for simulating the optical response of nanoparticles (Discrete Dipole
Approximation (DDA), Finite Element Methods (FEM), Finite Difference Time Domain
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Calculations, Generalized Mie Theory for multiple scattering problems…) are based on
different mathematical approaches. Because of numerical limitations and inherent
approximations, they do not provide an exact solution of the Maxwell Equations and their
related physical quantities (local electric field, Joule heating, scattering efficiency…). In
consequence, the quantitative analysis of the experimental results may sometimes depend on
the theoretical approach used, especially if accurate numerical simulations are planned. On the
other hand, there are many ways to smooth cube edges so as to simulate TEM images. One
can wonder for example if, instead of simply rounding cube edges, an additional rounding of
their corners will lead to different conclusions. Figures S3 to S4 are given here to illustrate
both points and confirm that, at least on a qualitative basis, the LSPR splitting is not sensitive
to such refinements.
Figure S3a shows a comparison between DDA and FEM calculations in the case of a cube
of fixed size (51 nm) and subjected to an increasing level of edge rounding, with or without
additional corner rounding. A direct comparison between DDA and FEM for a perfect cube is
not possible because a slight smoothing (re=0.5 nm) is required to avoid a spurious spatial
singularity in FEM calculations. However, both spectra are quite similar. The same holds for
other edge (corner) rounding values and the main features of the cube LSPR (position, width,
asymmetry…) are not drastically different in both calculations (FEM spectra are somewhat
broader). On the other hand, the influence of the cube corner geometry (specifically rounded
(red curves) or not (black curves)) is weak but not negligible. Red and black curves are closer
one to each other in DDA than in FEM calculations. This is understandable since in FEM the
particle surface meshing is very close to the real particle surface and differences in corner
geometries are suitably reproduced. On the contrary, the particle surface roughness induced by
the cubic structure of the dipole array in DDA tends to smooth the differences in corner
shapes (Figure S3b). This effect will be the less important that the mesh size of the dipole
array will be small compared to the particle size. This is illustrated in Figure S3c where
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bottom spectra clearly differ around 400nm. The singular corner geometry obtained by simply
rounding edges (see inset) which is obviously better reproduced by a finer mesh, may lead to
spurious size mesh-dependent spectral features. For the corner rounded cube (top spectrum)
the roughness is minimal and the mesh effects are lowered.
Figure S4 illustrates differences between DDA and FEM calculations and effects of corner
rounding in the case of nanocube dimers. The LSPR splitting pattern is only weakly modified
by an additional rounding of cube corners. One can observe a slight blue shift and a small
change in the relative intensities of both resonances (Figure S4a). On the other hand, the DDA
and FEM calculations for the same geometric structures are more different (bottom and
middle spectra in Figure S4a). It is possible to make them closer just by increasing the
interparticle distance and edge rounding of about 10 % in FEM calculations (top spectra in
Figure S4a). This essentially shows that the effective radius of edge curvature in DDA is
certainly larger than the ideal geometrical value because of truncation effects in the cubic
dipole array (see Figure S3b). Moreover, the relation between the physical particle surface and
the envelope of the outermost dipoles in the DDA description is not straightforward and may
explain a possible shift in the effective interparticle distances.
All these discrepancies are not significant enough to change the main conclusions about the
LSPR splitting due to cube smoothing. Figure S4b shows FEM optical response calculations
of nanocube dimers when both edges and corners are progressively rounded. The birth and
development of the LSPR splitting is very similar to what can be observed in Figure 6a of the
manuscript (DDA calculations for edge rounding only) even if the geometrical parameters are
not strictly identical.
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Figure S3 (a) Comparison between DDA (left) and FEM (right) simulations of the extinction
cross sections for cubes (edge length L=51 nm) with increasing levels of edge rounding re.
The red curves are obtained when corners are rounded with the same curvature (rc=re). (b)
Section of the cubic mesh (dipole array) generated for DDA calculations on a 51 nm cube
with a re=10 nm edge rounding. In the magnified edge region is drawn the ideal circular
section with radius re. (c) DDA calculations for a single cube (L=41 nm) and for two different
mesh sizes of the dipole array (the red curves corresponds to a 5 times larger number of
dipoles as compared to the black ones).
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Figure S4 (a) Comparison between DDA (bottom) and FEM (middle and top) simulations of
the extinction cross sections for nanocube dimers (L=51 nm) and an incident light polarized
along the interparticle axis. Black curves correspond to edge rounding only and red curves to
both edge and corner rounding. The middle FEM curves and the bottom DDA curves are
obtained for the same rounding parameters. The top FEM curves are obtained for another set
of parameters. (b) FEM extinction spectra for nanocube dimers (L=51 nm, d=1nm) for an
increasing edge and corner rounding.
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3. LSPR Splitting in Silver Nanocube Dimers: Effect of the Refractive Index
To account for their interaction with the substrate, particles are supposed to be embedded in
a homogeneous dielectric medium having a refractive index (neff) intermediate between the
one of air and the one of the substrate material. Since this effective index is taken as an
adjustable parameter when experimental spectra are tentatively simulated by numerical
calculations, it is important to understand its influence on the LSPR splitting. This is
illustrated in Figure S5. One can see that increasing neff mainly results in a red shift of both
resonances (Figure S5a) in such a way that their ratio remains almost unchanged (Figure S5b).
The magnitude of the resonance with dipolar character λ1R becomes also larger. These effects
are not exactly the same as those induced by increasing the edge rounding or decreasing the
interparticle distance d (Figure 6 of the manuscript) which justifies that d and neff can be
treated as independent parameters in numerical simulations.
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Figure S5 (a) Calculated extinction cross sections (DDA) for silver nanocube dimers (L=51
nm, d=1nm) as a function of the effective refractive index neff (b) The same spectra drawn on
a reduced wavelength scale (
λ
λ2R
).
4. FEM Simulations of Nanocube Dimer Extinction Spectra
The experimental extinction spectra of nanocube dimers shown in Figure 3 are simulated
with FEM for comparison with DDA calculations shown in Figure 7 for both of them. The
geometric parameters are those listed in Table 1. Three parameters have been adjusted: neff, d
and the rotation angle α of particle 2. d is the minimum distance between particles considering
their faces may be no longer parallel (α≠0). For dimers D1 and D2, these parameters are close
to those used for DDA simulations. The results are shown in Figure S6 together with a
comparison between TEM images and FEM input geometries. In Figure S6c are compared top
views of the input geometry of dimer D2 for two different orientations of the projection plane.
This illustrates the effect of tilting the TEM electron beam relatively to the direction
perpendicular to the substrate where cubes lay down flat. Compared to the ideal case of
homodimers presented in Figure 4 and 6 of the manuscript, the main LSPR peaks are more or
less deformed and let appear substructures when symmetry is clearly broken (size and
alignment).
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Figure S6 (a) Calculated extinction cross sections (FEM) for silver nanocube dimers. The
fitting parameters are (neff=1.3, d=2.6 nm, α=1,5°) for D1, (neff=1.2, d=0.75 nm, α=−1°) for
D2, (neff=1.35, d=1.15 nm, α=0.5°) for D3 and (neff=1.35, d=1.3 nm, α=0°) for D4. (b) TEM
images and corresponding FEM geometries. (c) Illustration of the effect on the apparent
interparticle overlap when the direction normal to the substrate is tilted of about 5° (case of
dimer D2).
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