= ε1 + iε2) (wavelength dependent) and the dielectric function of the medium (εm) in direct contact with the surface. This can be quantitatively described by the Gustav Mie theory of light scattering for any spherical particles with radius = R, for which a simplified version (when 2R ≪ λ) is shown in equation (2.3)
Cext=24π2R3εm3/2λ·ε2(ε1+2εm)2+ε22.(2.3)
Here Cext is the extinction cross-section and gives the quasi-polarizability of the nanoparticle. SPR is a property of nanomaterials due to the scale of wavelength verses particle size, and the increased surface area offers a more pronounced effect. Furthermore, the wavelength of the resonance peak and thus the colour emitted by the SPR is dependent on the components highlighted above, the primary ones being the particle composition (material), and the radius. However, what equation (2.3) does not describe is the effect of shape/architecture, as the Gustav Mie equation is only for a simple spherical particle. Surface plasmons are highly dependent on the overall surface architecture, and not limited simply to size. Varying morphology affects the SPR frequency and thus emission since varying the shape alters the surface architecture and this alters the electric field distribution on the surfaces, again altering the surface plasmon resonance. This tends to result in an increase in the range of resonance frequencies; this is certainly true for cubes, rods, cages/shells and egg shapes (figure 2.5(ii)). Furthermore, all other volumes have a larger surface area to volume ratio than a sphere, so all other shapes will offer increased surface area. This is particularly true for hollow shapes like cages.
The classic historical example is gold nanoparticles. Spherical gold shows a range of colour dependence on size, from green for particles around 5 nm, up to red for particles of about 50 nm (figure 2.5(ii)). However, silver spherical nanoparticles have higher frequency SPR resonance (higher energy), emitting from green up to ultraviolet compared to gold nanoparticles of the same size and shape (gold emitting red to green), demonstrating the dependence SPR has on material as well as the size. Finally, since this is a surface effect and the surface is interacting with the surrounding media, we cannot forget the SPR frequency dependence on the dielectric constant of the media. Furthermore, SPR is affected by the locality of near materials exhibiting SPR [4]. Both are incredibly useful for applications as colorimetric indicators/sensors as very subtle changes to either the nanoparticle surface, the surrounding media, or the distance nanoparticles are from each other can be optically detected. In addition, SPR is so intense that the scattering of a single nanoparticle can be seen by the eye in an optical scattering microscope. The blue light emitted by a silver nanoparticle has a cross-sectional area a million times greater than the dye molecule fluorescein, making noble metal nanoparticles exhibiting SPR very attractive materials for ultra-sensitive sensors [4]. There have been many excellent tutorial papers and reviews on SPR that the reader is directed to for more in-depth explanation, as well as the many applications, from sensing to in vivo and in vitro biomedical applications [2–5].
2.3.2 Optical: quantum dots fluorescence
Metals conduct electricity because their mobile electrons can move through the material. Electrons are confined to their atomic orbitals in an insulator material. So what is it that allows electrons to move in metals and not in insulators? The answer is there are simply vacant orbitals (holes) to move into at the same (or very similar) energy of the mobile electrons in metals. This is described by band theory, where there is an energy band full of electrons, so there are no holes available for electron movement. This is the valance band. Above this is the conduction band. Metals have electrons in the conduction band, which is not full, so there are plenty of vacant sites to move between. Semiconducting materials have a full valance band and no electrons in the conduction band (similar to an insulator), but they differ from an insulator as they have a very small band gap between the two bands. As such they become conducting when the temperature is increased sufficiently for the electrons to be promoted into the higher energy conducting band, leaving an electron hole in the valence band (with both mobile electron and holes contributing to conduction). Examples of semiconducting materials include CdSe, InAs and GaP. Describing electronic properties with respect to bands is only possible in the bulk phase (as bands are a continuous macroscale phenomenon). As the particle size reduces in size to the nanoscale, the electrons become spacially confined (with fewer vacant orbitals to move into) and the bulk models break down. When the size of the particle becomes comparable to or smaller than the electron/hole pair (exciton) Bohr radius (usually ⩽10 nm), the electrons are confined in all directions and the material is considered to have zero dimensions (hence a dot); these materials are known as quantum dots or ‘artificial atoms’. The spacial confinement results in an increase in the size of the band gap and the banded continuum (valance and conducting bands) breaking into a quantised energy level, perfectly demonstrating how the nanoscale is the intermediate between the bulk and the atomic scale. When energy is supplied to a quantum dot (in the form of electromagnetic radiation), an exciton is created as the electron is promoted into the conduction band, and this then emits fluorescence when the electron relaxes back down into the valance band, recombining with the hole. The wavelength of this fluorescence is dependent on the size of the band gap (ΔE(r)) which is dependent on the band gap of the bulk material (Egap), plus an increase in this size as the radius of the quantum dot (r) decreases (this relationship is shown in equation (2.4) and figure 2.6). The higher energy (shorter wavelength/higher frequency) emission is generated from relaxation across a larger band gap which occurs for smaller quantum dots. Equation (2.4) can be rearranged to equation (2.5) to show the relationship of the quantum dot size (radius r) to the change in band gap energy (h is Planck’s constant, me is the mass of a free electron and mh is the mass of the hole).
ΔEnano(r)=ΔEbulk+h28r21me⁎+1mh⁎(2.4)
r=h28(ΔEnano−ΔEbulk)1me⁎+1mh⁎(2.5)
Quantum dots have exceptional and far superior fluorescent properties in comparison to traditional organic dye molecules. (1) Due to the relationship between size and emission their wavelength can be precisely tuned. (2) They can absorb a broad range of energy but have a very narrow emission band. (3) Their fluorescence is unparalleled with respect to brightness, lifetime and resistance to photobleaching, which makes them excellent optical probes. One downside is many of the best semiconductor materials for high florescence quantum dots with the band gap at the best wavelength for visualisation are often the most toxic (such as CdSe and Cd/S). This is being addressed by coating these materials in less toxic semiconductors, such as ZnS, and also developing new quantum dot materials by doping less toxic semiconductors to tune the band gap. Again, different morphologies can offer further tuning. Further reading on quantum dots and particularly their biomedical uses and the implications of their toxicity can be found in [6].
Figure 2.6. Description of how a band gap in a semiconductor increases as the size of the particle decreases. The left-hand side shows the band gap in the bulk material, while on the right an increased band gap is shown for the nanoparticle (bulk gap is shown as a dashed line on the right for comparison) as the frequency of the light emitted is proportional to this energy (and dependent on the material), the light emitted varies with size. CdSe/ZnS quantum dots of decreasing sizes from left to right are shown in the centre (image from [2] reproduced by permission of the Royal Society of Chemistry).
2.3.3