th the burgeouning field of plasmonics.
Both bulk and surface plasmons have associated electromagnetic waves, and can consequently be described by Maxwell's equations. The coherent oscillations of electron motion can be encapsulated in the dielectric constant of the material. A simple model for the motion of an electron in a conductor is a free electron driven by the field of the electromagnetic wave.
This electron motion can then be converted into a dielectric constant by using the definitions of the dipole moment, electric susceptibility, and dielectric constant:
|Fig. A-1: Dispersion relation for bulk plasmons and free-space electromagnetic waves.|
&omegap is known as the plasma frequency. This dielectric constant can then be inserted into the standard electromagnetic wave equation
The wave equation allows for the standard propagating wave solution, given by
This is a valid solution provided that
This is the dispersion relation of bulk plasmon polaritons propagating in a metallic film. The dispersion relation gives the correspondence between the time-dependence of the electromagnetic wave (&omega), and the spatial variation (k);
the wavelength of the wave is given by &lambda=2&pi/k. Figure A-1 shows a plot of the bulk plasmon dispersion relation (solid line), along with the free space dispersion relation (&omega = ck). For frequencies above the plasma frequency &omegap, the metal supports propagating modes. For frequencies below &omegap, k becomes imaginary and the fields exponentially decay inside the metal. For typical metals, h&omegap/2&pi &asymp 10eV (in the ultra-violet); for semiconductors, h&omegap/2&pi < 0.5eV (in the terahertz regime). Although bulk plasmons, which can exist even deep inside a metal, do not exist for &omega < &omegap, there is an additional solution to Maxwell's equations for these frequencies: surface plasmons.
Surface plasmons are propagating waves bound to plasmon oscillations of electrons at a metal-dielectric interface. The field intensity of the wave is at a maximum at the interface and exponentially decays away from the interface. The mathematical form of surface plasmons can be obtained by solving Maxwell's equations at a metal-dielectric interface:
Assuming the metal-dielectric interface lies in the xy plane at z = 0, it can be shown that x-propagating surface waves of the following form are solutions to Maxwell's equations:
with Ex,m = Ex,d (E|| continuous), &epsilonmEz,m = &epsilondEz,d (D&perp continuous), and Hy,m = Hy,d (B|| continuous); all other field components are zero. The subscript j is used to denote fields in the metal or dielectric. The dispersion relation for surface plasmons can be obtained by inserting the equations for E and H into Maxwell's equations and enforcing the boundary conditions:
The dispersion relation, assuming &epsilonm = 1 - (&omegap/&omega)2, is shown in Fig. 1, along with the free-space and bulk plasmon dispersion relations. The surface plasmon dispersion relation always lies to the right of the free-space dispersion relations; as a result, free-space radiation does not have enough momentum to launch surface plasmons.
Additionally, the magnitude of kx for the surface plasmon asymptotically approaches &infin as &omega approaches &omegasp, where &epsilonm = -&epsilond. For &epsilonm = 1 - (&omegap/&omega)2, &omegasp = &omegap / &radic(1 + &epsilond). As kx increases, the surface plasmon wavelength decreases and &kappaz increases, resulting in a rapidly varying surface wave that is more tightly confined to the surface. For smaller values of &omega, the surface plasmon wave vector approaches the light line, the surface plasmon and free-space wavelengths become more similar, and the surface wave becomes less and less strongly confined to the surface.
For the simple free-electron model considered thus far, there is no loss in the system and the surface wave will have an infinite propagation length. In real materials, however, there is loss and the wave will only propagate a finite length. The propagation length will decrease as the surface wave becomes mor