Surface plasmons

Colliex, C.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 403-404

Section 4.3.4.3.4. Surface plasmons

C. Colliex^a

4.3.4.3.4. Surface plasmons

| top | pdf |

Volume plasmons are longitudinal waves of charge density propagating through the bulk of the solid. Similarly, three exist longitudinal waves of charge density travelling along the surface between two media A and B (one may be a vacuum): these are the surface plasmons (Kliewer & Fuchs, 1974). Boundary conditions imply that $[\varepsilon_A(\omega)+\varepsilon_B(\omega)=0. \eqno (4.3.4.30)]$ The corresponding charge-density fluctuation is contained within the (x) boundary plane, z being normal to the surface: $[\rho({\bf x}, z)\simeq\cos({\bf q}\cdot{\bf x}-\omega t)\delta(z), \eqno (4.3.4.31)]$ and the associated electrostatic potential oscillates in space and time as $[\varphi({\bf x},z)\,\alpha\cos({\bf q}\cdot{\bf x}-\omega t)\exp(-q|z|). \eqno (4.3.4.32)]$ The characteristic energy $[\omega_s]$ of this surface mode is estimated in the free electron case as:

In the planar interface case: $[\left.\matrix{\omega_s = \displaystyle{\omega_p\over \sqrt 2}\hfill \cr \quad\raise2ex\hbox{(interface metal--vacuum)\semi}\hfill \cr \omega_s = \displaystyle{\omega_p \over (1+\varepsilon_d)^{1/2}}\hfill \cr \quad\hbox{(interface metal--dielectric of constant}\,\, \varepsilon_d); \hfill\cr \omega_s =\displaystyle\left({\omega^2_{p_A}\,+\,\omega^2_{p_B} \over2}\right)^{1/2}\hfill \cr \quad\raise2ex\hbox{(interface metal {\it A}--metal {\it B}).}\hfill} \right\} \eqno (4.3.4.33)]$

In the spherical interface case: $[(\omega_s)_l= {\omega_p\over[(2l+1)/l]^{1/2}} \eqno(4.3.4.34a)]$ (metal sphere in vacuum – the modes are now quantified following the l quantum number in spherical geometry); $[(\omega_s)_l={\omega_p\over [(2l+1)/(l+1)]^{1/2}} \eqno (4.3.4.34b)]$ (spherical void within metal).

Thin-film geometry: $[(\omega_s)^\pm={\omega_p\left[{1\pm\exp(-qt) \over 1+\varepsilon_d}\right]^{1/2}} \eqno (4.3.4.35)]$ (metal layer of thickness t embedded in dielectric films of constant $[\varepsilon_d]$ ). The two solutions result from the coupling of the oscillations on the two surfaces, the electric field being symmetric for the $[(\omega_s)^-]$ mode and antisymmetric for the $[(\omega_s)^+]$ .

In a real solid, the surface plasmon modes are determined by the roots of the equation $[\varepsilon(\omega_s)=-1]$ for vacuum coating [or $[\varepsilon(\omega_s)=-\varepsilon_d]$ for dielectric coating].

The probability of surface-loss excitation [P_s] is mostly governed by the $[{\rm Im}\{-1/[1+\varepsilon(\omega)]\}]$ energy-loss function, which is analogous for surface modes to the bulk $[{\rm Im}\{-1/[\varepsilon(\omega)]\}]$ energy-loss function. In normal incidence, the differential scattering cross section $[{\rm d}P_s/\!{\rm d}\Omega]$ is zero in the forward direction, reaches a maximum for $[\theta=\pm\theta_E/3^{1/2}]$ , and decreases as $[\theta^{-3}]$ at large angles. In non-normal incidence, the angular distribution is asymmetrical, goes through a zero value for momentum transfer $[\hbar{\bf q}]$ in a direction perpendicular to the interface, and the total probability increases as $[P_s(\varphi)=\,{P_s(O)\over\cos\varphi}, \eqno (4.3.4.36)]$ where $[\varphi]$ is the incidence angle between the primary beam and the normal to the surface. As a consequence, the probability of producing one (and several) surface losses increases rapidly for grazing incidences.

References

Kliewer, K. & Fuchs, R. (1974). Theory of dynamical properties of dielectric surfaces. Adv. Chem. Phys. 27, 355–541.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 403-404