Effects of size, shape and material

This approximation scheme allows us to treat the localised surface plasmon as the oscil- lation of a collective coordinate associated with the centre of mass motion, damped by interactions with the relative-coordinate system.

As has been established by multiple photoabsorption experiments [65] the spectra of metallic nanoparticles typically display a ‘giant dipole resonance’ close to the classical Mie values for the dipole resonance frequency of a classical small metal sphere. Thus this resonance should be interpreted as being due to the collective motion of the delo- calised valence electrons, with the external electromagnetic field coupling directly with the electronic centre-of-mass. In this way Hc.m., which is formally equivalent to the

Hamiltonian of a point dipole, is the dominant term in the Hamiltonian for an LSP.

2.2.3 Effects of size, shape and material

Clearly the optical properties of nanoparticles depends on the dielectric environment, as seen explicitly in equation 2.23, as well as the material constituent of the nanoparticle via the plasma frequency. The effect of temperature on the absorption spectra is minimal for commercial electronics operating ranges [66,67], as seen in Figure 2.7. However, it is the geometry of nanoparticles that is of interest, as one can fine tune the plasmonic properties by manipulating the size and shape of nanoparticles [60].

There are a wide range of nanofabrication techniques with varying levels of resolution and throughput [68]. With electron beam lithography it is possible to achieve sub-nanometer resolutions [69,70], whilst few nanometer resolution is readily realisable [71]. There are a host of other methods such as the chemical synthesis of nanoparticles, including the use of DNA as a template to organise few nanometre nanoparticles into single chains with long-range order [72]. Consequently it is possible to precisely engineer the size, shape and arrangement of nanoparticles and thus the optical response.

Chapter 2 - Plasmonics 20

Figure 2.7: Temperature dependence of the absorption spectra for gold nanoparticles with radii of 22nm. The absorption spectra are measured at 18◦C (solid line) and 72◦C (dashed line) using spectrophotometry. Reprinted with permission from [66]. Copyright 1999 American Chemical Society.

Equation 2.23 suggests that the resonance frequency of metallic nanoparticles is size independent, however, spectroscopy measurements show that in reality this is not true. Whilst reports on the governing relationship between the size of nanoparticles and their spectra can be in conflict [73], as this scaling is dependent on other parameters of the particle and environment, in general an increase in particle size corresponds to a red- shift of the resonance peak [74], as seen in Figure 2.8. We can intuitively understand this shift as due to an increase in the separation of oppositely charged surfaces either side of the nanoparticle as its size increases, leading to a reduction in the restoring force and hence a decrease in the resonant frequency.

As well as the resonance peak, the width of the optical spectrum is also size-dependent, with two key mechanisms. Nanoparticles on the scale of ten nanometres have dimensions comparable to or less than the electron mean free path, leading to a modified scattering rate of the dielectric function2.12due to collisions with the nanoparticle surface, leading to a 1/nbroadening of γ as the particle size decreases. On the other hand as the particle

size increases radiation damping, which scales like n3, also causes a significant broad-

ening. Accounting for these two key contributions, one typically models the scattering rate as [75]

Chapter 2 - Plasmonics 21

Figure 2.8: Plot of the peak plasmon resonance wavelength of individual nanoparticles as a function of their size, for three different shapes. Here size refers to: the diameter of spheres, the length between opposite corners of pentagons, and the length of a side of triangles. In each case there is a red-shift with increasing size, as well as a clear dependence on the shape itself. Reprinted from [74], with the permission of AIP Publishing.

Figure 2.9: (a) Optical absorption spectra of spherical gold nanoparticles, normalised to the spectral peak maximum, for various diameters. (b) The bandwidth ∆λ of the spectra (full-width-half-maximum when fitted with a Lorentzian curve). Reprinted with permission from [66]. Copyright 1999 American Chemical Society.

γ = γ0+

AvF

n +

2ω4_{M n}3

3c3 , (2.30)

where γ0 is the Ohmic damping rate, vF is the Fermi velocity of conduction electrons

and A is a geometrical factor [66, 76]. Thus for each nanoparticle and environment there is a dimensional sweet spot where the total broadening effects are minimal. In the example of Figure2.9with gold nanoparticles we can see this occurs at aboutn= 10nm.

Chapter 2 - Plasmonics 22

Figure 2.10: From left to right we see a 30nm thick nanoparticle rod, disk and two triangles, in scanning electron micrographs (top), dark-field images (middle), and dark- field spectra (bottom). The white scale bar is 300nm. Reprinted with permission from [77]. Copyright 2007 John Wiley & Sons.

The shape of a nanoparticle is also a key parameter in determining the optical response. In Figure 2.8 we see that the spectral peaks of different shapes are well separated in frequency. This effect is visually evident in Figure2.10where different shaped nanopar- ticles appear different colours in dark-field microscopy [77]. We also see that there are additional types of resonances in differently shaped nanoparticles. The nanorod has two discernible dipole resonances associated with the major and minor axis, which are red and blue shifted respectively. The triangular nanoparticles also display two resonances, in this case these correspond to a dominant dipole mode and a quadrupole mode at a smaller wavelength [78].

The multipolar characteristics of nanoparticles are pertinent to the reliability of mod- els used throughout this thesis, which consider only the dipole response. Typically for nanospheres and similar geometries on the scale of a few tens of nanometers the electro- magnetic response is overwhelmingly dipolar in nature [79,80]. Even as the quadrupole mode becomes appreciable in nanoparticles on the scale of several tens of nanometres,

Chapter 2 - Plasmonics 23

Figure 2.11: Exact electrodynamic calculation of the extinction spectra of oblate spheroids, all with the same equivalent volume, corresponding to a sphere radius of 80nm. In this figure the extinction is normalized to the area of a circle with radius equal to the semi-major axis. As the asymmetry is increased the dipole peak is red- shifted while the quadrupole mode shrinks in intensity. Reprinted with permission from [60]. Copyright 2003 American Chemical Society.

the dipolar mode is typically still dominant and well separated in frequency. Moreover, it is possible to introduce an asymmetry into the geometry so as to quench the quadrupole mode [60] as shown in Figure2.11.