Substrate influence on the plasmonic response of clusters of spherical nanoparticles

The plasmonic response of nanoparticles is exploited in many subfields of science and engineering to enhance optical signals associated with probes of nanoscale and subnanoscale entities. We develop a numerical algorithm based on previous theoretical work that addresses the influence of a substrate on the plasmonic response of collections of nanoparticles of spherical shape. Our method is a real-space approach within the quasistatic limit that can be applied to a wide range of structures. We illustrate the role of the substrate through numerical calculations that explore single nanospheres and nanosphere dimers fabricated from either a Drude model metal or from silver on dielectric substrates and from dielectric spheres on silver substrates.

Figure 1

An illustration of the system we consider in this paper, for the case where we have a nanosphere dimer. The substrate occupies the half space $z<0$, $h$ is the distance between the south poles of the spheres and the substrate, and $d$ is their surface-surface separation. Sphere $j$ of the dimer has dielectric function ${\varepsilon }_j(\omega )$ and radius $a_j$. The medium above the substrate has dielectric function ${\varepsilon }_{+}(\omega )$ while that of the substrate is ${\varepsilon }_{-}(\omega )$. The two black dots schematically represent image multipoles in the substrate seen by an observer in the half space $z>0$.

Figure 2

The dimensionless dipole moment $\mathrm{p̄}(\omega )$, for a Drude metal particle on a substrate of dielectric function (a) ${\varepsilon }_{-}=1$, (b) ${\varepsilon }_{-}=2$, and (c) ${\varepsilon }_{-}=10$. For the vacuum case, i.e., ${\varepsilon }_{-}=1$, we obtain the Mie result at $\hslash \omega =\hslash {\omega }_P/\sqrt{3}\approx 1.73\hspace{0.5em}\mathrm{eV}$. For all plots, we have $h=0.05a$, ${\omega }_P=3\hspace{0.5em}\mathrm{eV}$, $\gamma =0.03\hspace{0.5em}\mathrm{eV}$, and $L=50$.

Figure 3

The dimensionless dipole moment $\mathrm{p̄}(\omega )$, for one of the particles in a Drude metal dimer, on top of a substrate of dielectric function (a) ${\varepsilon }_{-}=1$, (b) ${\varepsilon }_{-}=2$, and (c) ${\varepsilon }_{-}=10$. For all plots, we have $h=0.05a$, $d=0.1a$, ${\omega }_P=3\hspace{0.5em}\mathrm{eV}$, $\gamma =0.03\hspace{0.5em}\mathrm{eV}$, and $L=50$.

Figure 4

The plots show $\mathrm{p̄}(\omega )$ for one of the particles in a Drude metal dimer for (a) $h=2a$, (b) $h=0.3a$, and (c) $h=0.05a$. For all plots, we have ${\varepsilon }_{-}=10$, $d=0.1a$, ${\omega }_P=3\hspace{0.5em}\mathrm{eV}$, $\gamma =0.03\hspace{0.5em}\mathrm{eV}$, and $L=30$.

Figure 5

Position of the lowest-energy resonance as a function of $h$, in the case where ${\mathit{E}}_0\parallel \mathit{ẑ}$. The blue dotted curve shows the result when the calculation is done in the dipole approximation (i.e., $L=1$), while the red solid curve $L=30$ shows the converged results. For all cases, a Drude dimer with $d=0.1a$ was assumed, and the substrate dielectric function was ${\varepsilon }_{-}=10$.

Figure 6

The square of the electric field $\left|\mathit{E}\right|{}^2/\left|{\mathit{E}}_0\right|{}^2$, at point 1 (blue solid curve) and point 2 (green dotted curve) as a function of frequency, for the Drude dimer. The results are for the case where ${\mathit{E}}_0\parallel \mathit{x̂}$, and the substrate dielectric function is ${\varepsilon }_{-}=10$. The other parameters are the same as in Fig. 3.

Figure 7

The field enhancement $\left|\mathit{E}\right|/\left|{\mathit{E}}_0\right|$ in the $\mathit{xz}$ plane. At (a) $\hslash \omega =1.35\hspace{0.5em}\mathrm{eV}$, the highest field enhancement is found between the sphere and the substrate; at (b) $\hslash \omega =1.45\hspace{0.5em}\mathrm{eV}$, the highest field enhancement is found on the line connecting the two spheres. As shown in the figure, ${\mathit{E}}_0\parallel \mathit{x̂}$. The system parameters are ${\varepsilon }_{-}=10$, $h=0.05a$, $d=0.1a$, and $L=50$. Of particular interest is the area between the two lowest-frequency peaks, where the location of maximum field enhancement flips between the two points indicated.

Figure 8

The dimensionless dipole moment for an Ag sphere placed a distance $h=0.05a$ above a dielectric substrate of (a) ${\varepsilon }_{-}=1$, (b) ${\varepsilon }_{-}=2$, and (c) ${\varepsilon }_{-}=10$. The equation system was truncated at $L=50$.

Figure 9

The response for an Ag dimer close to a substrate, as described by the dimensionless dipole moment of one of the spheres. As in previous cases, $d=0.1a$, $h=0.05a$, and $L=50$ were used. (a) ${\varepsilon }_{-}=1$, (b) ${\varepsilon }_{-}=2$, (c) ${\varepsilon }_{-}=10$.

Figure 10

Intensity enhancement $\left|\mathit{E}\right|{}^2/\left|{\mathit{E}}_0\right|{}^2$, as a function of frequency of the applied field at points 1 and 2 for the Ag dimer on a substrate of dielectric function ${\varepsilon }_{-}=10$. The remaining parameters are the same as in Fig. 9.

Figure 11

The dimensionless dipole moment of a member of a dielectric dimer placed in close proximity to an Ag surface. The dielectric functions for the two spheres forming the dimer are both (a) ${\varepsilon }_j=2$ and (b) ${\varepsilon }_j=10$. In both cases, we have $d=0.1a$, $h=0.05a$, and $L=50$.

Figure 12

Intensity enhancements at two selected points for a dielectric dimer placed in the near vicinity of an Ag surface. The dielectric function of the spheres is $\varepsilon =10$, and other parameters are the same as in Fig. 11.

Figure 13

The field enhancement $\left|\mathit{E}\right|/\left|{\mathit{E}}_0\right|$ in the $\mathit{xz}$ plane for a dielectric dimer above an Ag substrate. At (a) $\hslash \omega =3.02\hspace{0.5em}\mathrm{eV}$, the highest field enhancement is found between the spheres; at (b) $\hslash \omega =3.39\hspace{0.5em}\mathrm{eV}$, the highest field enhancement is found on the line connecting the two spheres. As shown in the figure, ${\mathit{E}}_0\parallel \mathit{x̂}$. The parameters are ${\varepsilon }_j=10$, $h=0.05a$, $d=0.1a$, and $L=50$.