Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 88

a b

E field E z

z

H y

dielectric e d

x

----- +++ ----- metal e m z +++

Fig. 2.29 ( a ) SPP waves at the dielectric/metal interface and ( b ) field penetration depth in the
dielectric and the metal

in label-free biosensors, since they enhance electromagnetic waves at the metal-
dielectric interface and are very sensitive when the propagating media changes.
There are several methods to generate SPPs, which involve metal-dielectric
interfaces, nanoparticles, or nanowires. A recent review on this subject is found
in ( Dragoman and Dragoman 2008 ). The dispersion relation of SPPs can be derived
considering as example an interface between a metal and a dielectric (see Fig. 2.29 ),
with SPPs propagating along it. Around the interface, as shown in Fig. 2.29 ,thereare
periodical spatial oscillations of regions with positive and negative charges, which
decay rapidly away from the dielectric-metal interface and impose the periodicity
of the electric field in the dielectric and metal.
The dispersion relation is found assuming that SPPs are coherent charge oscilla-
tions that propagate on both sides of the interface situated in the .x; y/ plane and are
confined along the z -direction, which is perpendicular to the interface. SPPs are a
combination of longitudinal and transverse components of the electromagnetic field
assumed propagating along the x direction. The solutions of Maxwell's equations
for the two media (metal and dielectric) can be separated into two categories: s
(TE) and p (TM) polarized waves. The TE solution corresponds to the case when
the electric field is parallel to the interface, whereas in TM waves, the magnetic
field is parallel to the interface. SPP are TM waves because they propagate along
the interface, i.e., along the x axis, and have a nonvanishing E z component of the
electric field perpendicular to the interface.
From Maxwell's equations written for the two media (metal and dielectric) with
permittivities " m and " d and wavevector components along z denoted by k zm and
k zd , we get the SPP dispersion relation in the form

k zm =" m C k zd =" d D 0; (2.62)

which implies that SPP waves exist only if the two media have dielectric permittiv-
ities of opposite signs:

sgn." m / D sgn." d /: (2.63)

So, one of them must have a negative dielectric permittivity, condition that is
satisfied by all metals in the visible optical spectrum. The wave number of SPPs
is determined from