Nanotechnology-Enabled Sensors

6.6 Nanotechnology Enabled Optical Sensors 355
tween size and oscillation frequency. It only describes that size changes
the intensity of light.
Experimentally, a strong size dependence of the plasmon bandwidth is
observed. 106 As a modification to the Mie theory for small particles, the dielectric
function of the metal nanoparticles itself is assumed to become
size dependent ε ′ + i ε ′
= f ( a,
λ)
.
The size dependence of the dielectric constant is introduced as the diameter
of the particle becomes smaller than the mean free path (MFP) of
the conduction band electrons.
Experimentally, it was first suggested by Kreibig 107 that a 1/abulk dependence
of the plasmon bandwidth exists in measurements. His idea was
in agreement with experimental results for nanomaterials with sizes down
to 2 nm.
A 1/R dependence of the plasmon bandwidth is also predicted by a
quantum mechanical theory by Persson. 108 When the particle radius is
smaller than the mean free path of the bulk metal, conduction electrons are
scattered by the surface and an effective radius aeff can be defined as:
1
a
eff
1 1
= + , (6.78)
a a
where aeff is calculated from the effect of abulk and the real a.
For larger nanoparticles (for instance for gold where 2R > 25 nm) the
extinction cross section also depends on higher-order multipole modes
within the full Mie equation as a result the extinction spectrum is then
dominated by quadrupole and octopole absorption as well as scattering. 109
These higher oscillation modes depend on the particle size. When the size
increases the plasmon absorption maximum is shifted to longer wavelengths
as the bandwidth increases. The excitation of such higher-order
modes can be explained in terms of an inhomogeneous polarization of the
nanoparticles by the electromagnetic field as the particle size becomes
comparable to the wavelength of the incoming radiation. The broadening
of the plasmon band is then usually ascribed to retardation effects. 110 On
the other hand, the increased line width (or the faster loss of coherence of
the plasmon resonance) could also be qualitatively described as a result of
the interactions between the dipole and the quadrupole (and higher-order)
oscillatory motions of the electrons, which deteriorates the phase coherence.
110 The size dependence resonance is a useful phenomenon for sensing
applications. 111 Some examples are presented below.
bulk