Discussion

As the mean size and shape of Ag particles were kept approximately constant when increasing the particle density, the blue shift of the plasmon energy for dome-like preparations is mainly attributed to the electric dipole-dipole interactions, i.e., the coupling between the plasmon dipoles in the Ag particles.

Increasing the particle density is equivalent to a decrease of the inter-particle distance and hence a strengthening of the dipole-dipole coupling. The blue shift of the Mie resonance observed for dome-like particles results from a destructive dipole-dipole coupling that enhances when the dipoles approach each other. This behavior is typical for the interaction between the (1,0) Mie modes in neighboring particles (Chap. 1, sect. III.2.3), thus providing a clear indication that the measured photon emission in the spectra is due to the radiative decay of the (1,0) Mie mode. The latter conclusion is supported by the following observations as well: (i) The disk-like particles are characterized by a smaller aspect ratio than the dome-like ones. The energy of the (1,0) mode for flatter particles is expected to be higher than for round ones (Chap. 1, sect. III.2.3). This is in agreement with the observation in this experiment (figure 3.8). (ii) The field-emitted electrons from the tip impinge perpendicular to the substrate plane and thus preferentially excite the (1,0) mode. The in-plane oscillations are primarily excited by secondary processes, e.g., hot electron thermalization, and the excitation cross-section is much lower. (iii) The photon detection system in the experimental setup is more sensitive to light emitted from the (1,0) mode, as the corresponding radiative decay results in a photon emission parallel to the substrate plane. The decay of the (1,1) mode, on the other hand, results in a photon emission normal to the substrate surface and is therefore mainly blocked by the STM head.

The energy increase of the (1,0) mode for dome-like particles is caused by the additional energy due to the dipole-dipole interactions between the (1,0) plasmons as the interparticle distance decreases. In the general case, the interaction energy between two identical dipoles at a distance d from each other is proportional to p².d^-3, where p is the electric dipole moment.

However, when assuming a square particle network, the fit of the measured increase of the plasmon energy reveals a dependence according to d^-1.8 for dome-like particles (figure 3.8, dotted line). Thus, in a 2D network of Ag particles, the dipole-dipole interaction energy increases slower than for two isolated dipoles as a function of distance d.

For disk-like particles, a blue shift of the (1,0) mode with increasing density is expected as well. However, no shift could be resolved most likely due to the weaker coupling. As the average height of disk-like particles is ~30% smaller than that of the dome-like ones, the

dipole-dipole interaction is almost two times smaller than for the round particles of comparable densities. In addition, the recorded Mie resonance is situated near 3.76eV, which corresponds to the surface plasmon resonance in a continuous Ag film, i.e., as d approaches zero. This energy marks therefore the upper limit of plasmon excitations in silver surfaces [12,116]. Besides, the absence of a shift could be related to the slightly increasing aspect ratio measured for increasing densities (figure 3.6), which lowers the Mie (1,0) energy and thus compensate the shift due to the dipole-dipole interaction.

The blue shift of the (1,0) and the red shift of (1,1) plasmon modes with increasing particle density have been observed before for lithographically fabricated particle ensembles [117,118,119]. Whereas the large particle sizes and the interparticle distances in those experiments needed an interpretation of the optical data within the framework of retardation and multipole effects [43,118], a treatment within the dipole approximation was sufficient in ref.[119]. In our experiment, the particles have smaller sizes compared to the emitted photon wavelengths. Therefore, the system is viewed in the quasi-static regime and only the dipolar effects have to be considered.

Our interpretation of the experimental results is supported by model calculations for the optical properties of dome-like and disk-like particle ensembles, as performed with the GranFilm program [115]. The code permits the determination of the polarizability of supported particles. The basic idea of the calculation method is to consider truncated spheres, oblate or prolate spheroids as geometries of the supported particles, and solve the Laplace equation for the electrostatic potential in the quasi-static regime. The mathematical form of the potential is described as a series expansion in a multipolar basis in either spherical or spheroidal coordinates to account for different particle shapes [115,120].

We computed the plasmon energies of Ag particle ensembles on a bulk Al2O3 substrate by calculating the absorption coefficient for varying densities using experimentally established bulk dielectric functions [121], and neglecting the NiAl substrate. As no specific lateral arrangement of the prepared particles was observed on the substrate, the mean field approach was chosen to describe their inhomogeneous lateral distribution. The particles were modeled as truncated spheroids determined by a perpendicular (r⊥) and a parallel (r⎪⎢) radius corresponding to the experimental particle height and radius, respectively. Hence, the dome-like particles are characterized by r⊥ > r⎪⎢ and the disk-like particles by r⊥ < r⎪⎢.

Several combinations of radii were tested in order to reproduce the experimental results.

However, optimum agreement was achieved only for a narrow range of radii. In figure 3.8, two curves are shown representing the calculated evolution of the plasmon energy with particle density for the dome-like (dashed line) and the disk-like (dash-dotted line) particles.

The theoretical curves describe the measured data rather well, and thus, illustrating the

dominant role of the dipole-dipole interaction in the experiment. The experimental data point at a density of 1.31×10¹¹cm^-2 deviates to higher energies from the calculated evolution for round particles. This emphasizes the increased particle-particle coupling due to the high local density, as the Ag particles preferentially nucleate along the defect lines of the alumina surface (figures 3.2 and 3.5).

The optimal input values of particle radii were r⊥ = 46Å, r⎪⎢ = 32Å, and r⊥ = 27Å, r⎪⎢ = 40Å for round and flat particle shapes, respectively. The calculated values of r⊥ overestimate the measured heights by almost a factor of two. This means larger dipolar moments are needed to reproduce the experimental energy positions of the (1,0) plasmon. This effect is attributed to the neglect of the NiAl support below the Al2O3 film in the calculations.

Therefore, the induced image dipoles in the NiAl were not taken into account. As the image dipole couples constructively with the driving (1,0) dipole in the particle, it enhances the total polarizability of the system (Chap. 1, sect. III.2.3). To judge the importance of this effect, we simulated the case of Ag particles deposited directly on a NiAl support. Fitting this model to the experimental dependence of the plasmon energy on the density leads to a much better agreement between calculated und measured shapes. It results in an r⊥ = 33Å for the round and r⊥ = 20Å for the flat particles, which are comparable with the heights deduced from the measurements. However, computed line widths and energy positions of the Ag Mie plasmon deviate more strongly from the experimental results for a NiAl support compared to Al2O3

surface. This underlines the combined influence of the oxide film and the metal substrate on the optical properties of the supported Ag particles.

III. Effect of long-range order