1 Nanoplasmonics: From Present into Future
5
scale of the energy exchange between the electric and magnetic components of an
electromagnetic wave, does not define the limit of the spatial localization of energy.
Because the size of the system R is smaller than any electromagnetic length scale,
of which smallest is l s , it is R that defines the spatial scale of the optical energy
localization. Thus the optical fields are confined on the nanoscale, and their spatial
distribution scales with the system’s size. This physical picture is at the heart of the
nanoplasmonics.
Consider as an example a gold nanosphere of radius R < l s , e.g., R ∼ 10 nm,
subjected to a plane electromagnetic wave, as shown in Fig. 1.1b. The field penetrates
the metal and causes displacement of electrons with respect to the lattice resulting in
the opposite charges appearing at the opposing surfaces, as illustrated in Fig. 1.1c.
The attraction of these charges causes a restoring force that along with the (effective)
mass of the electrons defines an electromechanical oscillator called a SP. When the
frequency ω sp of this SP is close to the frequency of the excitation light wave, a
resonance occurs leading to the enhanced local field at the surface, as illustrated in
Fig. 1.1b.
This resonant enhancement has also an adverse side: loss of energy always associated with a resonance. The rate of this loss is proportional to Im ε m [30]. This leads
to a finite lifetime of SPs. The decay rate of the plasmonic field γ is ∝ (Im ε m )
−1 . In
fact, it is given below in this chapter as Eq. (1.49) in Sect. 1.3.4. This expression has
originally been obtained in Ref. [31] and is also reproduced below for convenience,
γ =
Im s(ω)
∂Re s(ω)
∂ω
≈
Im ε m (ω)
∂Reε m (ω)
∂ω
,
(1.3)
where
s(ω) =
ε d
ε d − ε m (ω)
(1.4)
is Bergman’s spectral parameter [29]. Note that γ does not explicitly depend on the
system geometry but only on the optical frequency ω and the permittivities. However,
the system’s geometry determines the SP frequency ω and, thus, implicitly enters
these equations. The approximate equality in Eq. (1.3) is valid for relatively small
relaxation rates, γ ∪ ω. Apart from γ, an important parameter is the so-called quality
factor
Q =
ω
2γ
≈
ω
∂Reε m (ω)
∂ω
2Im ε m (ω)
(1.5)
The quality factor determines how many optical periods free SP oscillations occur
before field decays. It also shows how many times the local optical field at the surface
of a plasmonic nanoparticle exceeds the external field.
Note that another definition of the quality factor, which is often used, is
Q =
−Re ε m (ω)
Im ε m (ω)
.
(1.6)
5
scale of the energy exchange between the electric and magnetic components of an
electromagnetic wave, does not define the limit of the spatial localization of energy.
Because the size of the system R is smaller than any electromagnetic length scale,
of which smallest is l s , it is R that defines the spatial scale of the optical energy
localization. Thus the optical fields are confined on the nanoscale, and their spatial
distribution scales with the system’s size. This physical picture is at the heart of the
nanoplasmonics.
Consider as an example a gold nanosphere of radius R < l s , e.g., R ∼ 10 nm,
subjected to a plane electromagnetic wave, as shown in Fig. 1.1b. The field penetrates
the metal and causes displacement of electrons with respect to the lattice resulting in
the opposite charges appearing at the opposing surfaces, as illustrated in Fig. 1.1c.
The attraction of these charges causes a restoring force that along with the (effective)
mass of the electrons defines an electromechanical oscillator called a SP. When the
frequency ω sp of this SP is close to the frequency of the excitation light wave, a
resonance occurs leading to the enhanced local field at the surface, as illustrated in
Fig. 1.1b.
This resonant enhancement has also an adverse side: loss of energy always associated with a resonance. The rate of this loss is proportional to Im ε m [30]. This leads
to a finite lifetime of SPs. The decay rate of the plasmonic field γ is ∝ (Im ε m )
−1 . In
fact, it is given below in this chapter as Eq. (1.49) in Sect. 1.3.4. This expression has
originally been obtained in Ref. [31] and is also reproduced below for convenience,
γ =
Im s(ω)
∂Re s(ω)
∂ω
≈
Im ε m (ω)
∂Reε m (ω)
∂ω
,
(1.3)
where
s(ω) =
ε d
ε d − ε m (ω)
(1.4)
is Bergman’s spectral parameter [29]. Note that γ does not explicitly depend on the
system geometry but only on the optical frequency ω and the permittivities. However,
the system’s geometry determines the SP frequency ω and, thus, implicitly enters
these equations. The approximate equality in Eq. (1.3) is valid for relatively small
relaxation rates, γ ∪ ω. Apart from γ, an important parameter is the so-called quality
factor
Q =
ω
2γ
≈
ω
∂Reε m (ω)
∂ω
2Im ε m (ω)
(1.5)
The quality factor determines how many optical periods free SP oscillations occur
before field decays. It also shows how many times the local optical field at the surface
of a plasmonic nanoparticle exceeds the external field.
Note that another definition of the quality factor, which is often used, is
Q =
−Re ε m (ω)
Im ε m (ω)
.
(1.6)