Nanoparticles for in-vitro and in-vivo biosensing and imaging

20 Metal nanoparticles
In eq. 1.11, ω p is the so-called plasma frequency and ɛ ∞ includes the contribution
of the bound electrons to the polarizability. The plasma frequency is given by
ω p = √ ne 2 /ɛ 0 m ∗ with n and m ∗ being the density and effective mass of the conduction
electrons, respectively.
However, the complex electronic structure of metals is not described very accurately by
this model, so most calculations involving metals use a dielectric function known from
measurements. The measurements by Johnson and Christy [41] are generally considered
to be most reliable and were obtained on metal films under high vacuum conditions
Figure 1.6 shows a comparison between the Drude model and the values measured
by Johnson and Christy [41]: whereas the low energy values are well described by the
Drude-Sommerfeld model, additional contributions are present at higher energies (E>2
eV). The reason for this discrepancy is related to the excitation of electrons from deeper
bands into the conduction band, the so-called interband excitations. An additional
reason for the derivation from the Drude-Sommerfeld behaviour is that the conduction
band is increasingly non-parabolic for higher energies.
Figure 1.6: (a) Real part of the dielectric function of gold; (b) imaginary part. Experimental data
(JC) from Johnson and Christy (1972) [41] compared to calculated values using the quasi-freeelectron or
Drude model with the parameters indicated in the graph. The shaded area around the experimental data
indicates the measurement uncertainty. The interband contribution (broken line) is calculated from the
difference of the experimental data to the Drude model.
The effect of the dielectric constants of the metal and the surrounding medium on
the polarizability is illustrated in the simulations in figure 1.7; the solid line shows the
calculated extinction spectrum of a spherical Au particle with a diameter of 50 nm in
water. The peak in the extinction is found at a wavelength of about 520 nm, i.e., green
light is absorbed, giving the particles their red color. The spectrum of a Ag sphere of the
same size in the same medium is also shown in Figure 1.7 (dashed line). The extinction
peak is found at a smaller wavelength (at 420 nm), a direct consequence of the different
dielectric constants for Ag than for Au.
Changing the embedding medium to glass instead of water (i.e., ɛ m from 1.45 to 1.33)
causes a peak red shift for Ag by about 25 nm (see dash-dotted line). The glass effectively