Answer

Surface Plasmons

Plasmons are quantized collective oscillations of excited electrons under the influence of electromagnetic radiation. Put simply, they are quantum of plasma oscillations that exist mainly in metallic bodies, where electronics are loosely attached to the atoms, and thus are free to roam. They are formed when photons are excited by electromagnetic radiation. Plasmons exist in three major forms: surface plasmons, bulk or volume plasmons, and localized surface plasmons.

Surface Plasmons (SPs) are coherent electron excitations or collection electron oscillations that exist at the mental-dielectric or metal-vacuum interface. As the name suggests, they are surface electromagnetic (EM) waves that flow (propagate) over the surface of a metal in a direction parallel to the interface between the metal and dielectric or vacuum medium. These plasmons are not trapped, meaning they are not strictly standing EM waves; they bounce or hop back and forth between voids, creating a standing wave. In this sense, surface plasmons feel the symmetry of thin film surfaces and, depending on the orientation, absorb different amounts of light at different energies.

It can be inferred from the above description that surface plasmons are “collective oscillations in the electron density at the surface of a metal” (Pitarke et al., p. 412); that is, the quanta of surface-charge-density oscillations. The destination of surface plasmons are polaritons explained, in part, by surface charge oscillations that are coupled naturally to EM waves. Figure 1 below depicts an electron density wage moving along a metal-dielectric interface.

Figure 1: Illustration of an electron density wage moving along a metal-                            dielectric interface.

Coupling between electromagnetic waves and charge density oscillations (free electron propagations) on the metal surface causes the occurrence of surface plasmon polariton waves. These waves exist for TM polarization (see figure 2) and can be efficiently excited with light in the visible range of the EM spectrum.

Figure 2: Surface Plasmon Polaritons occur when Electromagnetic Wave   couples with free electron propagation on a metal surface.

The behavior and characteristics of surface plasmon polariton can be represented using the Drude Model as shown in the following expressions.

Where;

e = electronic charge

m_e = the effective of free electrons

Г = damping coefficient

E = amplitude of the incident electric field

ω = frequency of incident field

Polarization =

Where N = free electron concentration

Surface plasmon polariton can be computed for two system types: semi-infinite system (energy dispersion and skin depth) and thin films.

For a Drude thin slab in a vacuum;

The above equation shows that surface waves decouple at short wavelengths; that is, at , each surface or interface maintains independent oscillations at a low frequency of , which characterizes a semi-infinite electron gas having a single plane boundary.

However, at long wavelengths (), normal oscillations occur at  and tangential 2-dimensional oscillations at  

Excitation of SPs requires excitation requires fulfillment of both momentum and energy conservation. This can be visualized by analyzing the dispersion relation between energy and momentum in terms of angular frequency ω and wave vector in the direction of propagation, respectively. The direction of propagation is given by the following expressions;

At low frequencies, surface plasmon polariton is closer to Sommerfled-Zenneck wave, which is a point where the relation between wavevector and frequency, namely the dispersion relation is the same as in free space. As the frequency increases, however, the dispersion relation bends significantly, reaching an asymptotic limit known as plasma frequency (Maradudin, Sambles, & Barnes, 2014). This behavior of SPP at low and high frequencies is illustrated in figure 2 below.

            Figure 2: Dispersion Curve for SPs illustrates how the SP curve (shown in red) moves                              towards the photon curve (shown in blue) at low frequency (k_x). 

Localized Surface Plasmons (LSPs)

LSPs, unlike SPs, occur in metal nanoparticles, especially silver and gold nanoparticles. LSPs are non-propagative excitation of plasmons, meaning their EM field is remains localized or confined to a nanoscale region around the interface of a metal nanoparticle and a dialectic medium. The fact that translational invariance is nonexistent implies that LSPs cannot be effectively described in terms of wavevector like in the case of SPPs. Moreover, EM modes of these plasmons are discrete as opposed to continuous, and the plasmons can be excited through using incident waves. The resonances of LSPs depend largely on the shape of the nanoparticle, with spherical particles being easier to study analytically.