Chapter 3 Dynamic Effects
3.5 Numerical results
simple case of a spherical particle, we have
2 polarization as given in (3.50) and (3.51) to illustrate how good the original MLWA can be improved to, in comparison with the exact electrodynamic results.
3.5 Numerical results
In order to demonstrate the dynamical effects from our MLWA and IMLWA models formulated above for the optical response of a spheroidal nanoshell, we have carried out some numerical studies on the coupled plasmon frequencies and the extinction cross sections of the silver nanoshell systems studied in Ref. [8] within the context of the hybridization model. The dielectric function for silver as given in Ref. [8] has the following form:
S B2 /
i
with S 5.0, B 9.5eV, and 0.15eV. To access the accuracy of our various long wavelength approximations, we first compared them with the exact electrodynamic results (Mie theory) for a spherical shell system which are easily available [10].280 320 360 400 440
300 350 400 450 500 550 600 0
ext kIm1
for a hollow silver spherical shell according to all the three approximations(static, MLWA, and IMLWA with 1) in comparison with the exact electrodynamic result from the Mie theory [10]. As is clear from the figure, while the exact result shows both the split-dipole and the quadrupole modes, all the three approximations can only show the split-dipole modes as expected. Furthermore, while all of them give quite close results for
3.5 Numerical results 33
the frequency of the antibonding mode, significant errors occur in with the static model yielding blue-shifted resonances, and the long-wavelength approximations leading to red-shifted resonances for the bonding modes.
Note that it is a rather general result that for a hollow shell, the high frequency is relatively insensitive to the change in the interaction between the two individual plasmons, and thus retardation effects are quite insignificant for this mode. This is analogous to the in-phase oscillation of two coupled mechanical oscillators with characteristic frequencies quite insensitive to the strength of the coupling. Note also that the blue-shifted peaks from static calculation also occur in a very general way, since retardation effect will lead to an overall weaker interaction between the cavity and the surface plasmons. Most importantly, one sees the significant failure of the static approximation for such a size of the shell
1 60 nm
r and r2 70 nm (predicting in this case a blue shift of ~ 20% for and a
peak ~ 5 times larger in value), and how the MLWA (and especially the IMLWA) can yield
rather accurate results in comparison with the exact Mie theory. This thus confirms the usefulness of the MLWA’s in the calculation of dipole extinction for nanoshells of these
dimensions which are often encountered in experimental studies, and are not too small
compared to the optical wavelengths used in the experiments.
Having established the accuracy of the MLWA’s for spherical nanoshells, we next
apply them to the study of speroidal shells. Figure 3.2 shows the calculation of the
extinction cross section for the prolate silver spheroidal shell studied in [8] with a dielectric
core (9.5) and foci a =40 nm according to the three models: static, MLWA, and IMLWA. The complete geometry of the nanoshell is specified in the figure caption and the incident plane wave is polarized along the axis of rotational symmetry (note that our above formalism for the IMLWA only applies to this type of polarization). We notice that in this
case the high frequency modes are still relatively insensitive to the different models.
While the ―static results‖ compare closely to those given by the hybridization model (see
Fig. 7 in Ref. [8] ), the significance of the corrections from both the long wavelength models is clearly revealed and is qualitatively similar to that found in the case for spherical shells. Again, we expect the result from IMLWA to be closer to the exact one from electrodynamics.
3.5 Numerical results 35
300 400 500 600 700
0 2 4 6 8
Extinction cross section (104 nm2 )
Wavelength (nm)
Static MLWA IMLWA
Figure 3.2 Comparison of the extinction cross sections obtained in the static limit, from MLWA, and from IMLWA. The results are shown for a confocal prolate metallic shell with 40 nm foci and aspect ratios 1/2 (core) and 2/3 (outer surface). The nanoshell has a hematite core (9.5) with the silver shell embedded in vacuum. The electric field is along the rotational symmetry axis of the spheroidal nanoshell.
Having demonstrated the significance of the MLWA’s, we next apply it to study the
effect of different aspect ratios on the extinction of the nanoshell. Figure 3.3 shows the MLWA results for the spheroidal nanoshell in Figure 3.2 except that now the inner aspect ratio is varied. The polarization of the incident field is along [m0, Figure 3.3 (a)] and perpendicular [m1, Figure 3.3 (b)] to the rotational axis, respectively. For the m0 case, one sees similar qualitative features for the resonance frequencies as observed in the static HM calculation [8] such as the red-shifted bonding mode, and the almost-unchanged
antibonding mode with the increase of the inner aspect ratio with fixed outer ratio.
However, the values for the cross section obtained in our MLWA are quite different from those obtained in the HM as expected. For the m1 case, Figure 3.3 (b) shows a stronger extinction for the antibonding mode, and the greater dependence of this resonance frequency on the inner aspect ratio, with a blue-shifted peak as the ratio increases. For clarity, we have not shown the IMLWA results in Figure 3.3 (a) but we expect these results will give less red-shifted resonances, and slightly lower peak cross sections compared with those obtained from the MLWA as shown.
3.5 Numerical results 37
Extinction Cross Section (104 nm2 )
Wavelength (nm)
Extinction Cross Section (104 nm2 )
Wavelength (nm) (a)
m = 0
Figure 3.3 The extinction cross section of the confocal prolate metallic nanoshells (foci is fixed at 40 nm) with a fixed aspect ratio (2/3) for the outer surface and four different aspect ratios (0.2, 0.5, 0.55, and 0.6) for the core. The nanoshell has a hematite core (9.5) with the silver shell embedded in the vacuum. The electric fields are oriented parallel (a) and perpendicular (b) to the rotational symmetry axis of the spheroidal nanoshell. The calculation is based on the MLWA model.
Finally, we also study the split eigen-frequencies using our model. Figure 3.4 shows the resonance frequencies of the coupled bonding and antibonding modes according to the three different models for the same spheroidal shell studied in Fig. 5 of Ref. [8]. We show both the resonance wavelengths [Figure 3.4 (a)] and frequencies in eV [Figure 3.4 (b)] as a function of the inner aspect ratio of the shell. First we point out that our results according to the static model reproduce identical results as obtained from the HM (compare the solid
curves in Figure 3.4 (b) to those in Fig. 5 (c) in Ref. [8]). Those obtained from the MLWA’s, however, will give red-shifted resonance frequencies in general. These red
shifts are particular significant for the bonding modes in the present m0 case; and we have also found (not shown) that they actually become more pronounced for the
antibonding modes in the m1 case for this filled shell (core with9.5). In addition, we note that the dynamic modifications are more significant for lower inner aspect ratio of the shell as expected, since the effective scattering volume is greater in this case, which leads to a manifestation of the corrections from the finiteness of the wavelengths within the MLWA approach.
3.5 Numerical results 39 ratio of the core obtained from the static, MLWA, and IMLWA models, respectively. The aspect ratio of the outer surface and foci are fixed at 2/3 and 40 nm, respectively. The nanoshell has a hematite core (9.5) with the silver shell embedded in the vacuum. The electric field is along the rotational symmetry axis of the spheroidal nanoshell. Note that (a) is in nanometers and (b) is eV for direct comparison with the results in Ref. [8].