Surface Plasmon Dispersion Relation Research Articles

Plasmons in multicomponent superlattices

The dispersion relation for surface and bulk plasmons in a two-component optical semiconductor superlattice (2COSSL) arranged in periodic fashion is derived in the non-retardation limit using the one-dimensional transfer matrix method. Detailed calculations are presented for a [(GaAS/Al0.3Ga0.7AS)/glass] geometry, treating one of the alternating layers as frequency dependent and the other as frequency independent, and for a [(GaSb/InAS)/SiO2] geometry, treating both the layers as frequency dependent. An extension of this theory has been carried out in a three-component optical semiconductor superlattice (3COSSL). Spectral properties of bulk plasmons are demonstrated for the 3COSSL in symmetrical (air/GaSb/air) and unsymmetrical (air/GaSb/SiO2) geometries. It is indicated that a three-component periodic multilayered system is a better option for designing optical devices such as optical filters, filter-polarizers, monochromators and etlons.

Surface plasmons in Drude metals

We present here a detailed derivation of the dispersion relations for surface plasmons in classical Drude metals, taking into account the fact that the relative permittivity at high frequency may be different from 1, as in the case of dielectrics. We also retain the imaginary part of the dielectric response in the derivation of the dispersion relations for the surface plasmon, which exists up to a maximum frequency, ω s. This treatment has not previously been published. We compare the model with experiment, and show the importance of the relative permittivity in understanding the value of ω s in Ag. We show that the observed ratios of surface to volume plasmon peak energies in electron energy loss spectra can be understood in terms of the simple model, but that considerable care is needed in interpreting the physical circumstances of such experiments.

Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation

We calculate the surface plasmon dispersion relations for a periodic chain of spherical metallic nanoparticles in an isotropic host, including all multipole modes in a generalized tight-binding approach. For sufficiently small particles ($kd \ll 1$, where $k$ is the wave vector and $d$ is the interparticle separation), the calculation is exact. The lowest bands differ only slightly from previous point-dipole calculations provided the particle radius $a \lesssim d/3$, but differ substantially at smaller separation. We also calculate the dispersion relations for many higher bands, and estimate the group velocity $v_g$ and the exponential decay length $\xi_D$ for energy propagation for the lowest two bands due to single-grain damping. For $a/d=0.33$, the result for $\xi_D$ is in qualitative agreement with experiments on gold nanoparticle chains, while for larger $a/d$, such as $a/d=0.45$, $v_g$ and $\xi_D$ are expected to be strongly $k$-dependent because of the multipole corrections. When $a/d \sim 1/2$, we predict novel percolation effects in the spectrum, and find surprising symmetry in the plasmon band structure. Finally, we reformulate the band structure equations for a Drude metal in the time domain, and suggest how to include localized driving electric fields in the equations of motion.

Dispersion relation of surface plasmons near photonic band gaps: Influence of the interaction with light

Abstract We present an experimental and theoretical study of the photonic band gap in the propagation of surface plasmons (SPs) on periodically corrugated surfaces. Our main purpose is to investigate the case where the band gap width is larger than the energy distance between the SP dispersion curve for a flat surface and the light line. We introduce a physical model of the interaction of light waves with SPs and derive an analytical expression for the SP wave vector near band gaps based on the coupled-mode approach involving three interacting modes (two of them are SP modes and one is a light mode). By using the interferometric measurement we have studied, for the first time, the SP propagation parameters in the vicinity of the photonic band gap (10 μm wavelength region). The predictions of our theory are in good agreement with the experimental data.

Dispersion relation of surface plasmons near photonic band gaps: Influence of the interaction with light

We present an experimental and theoretical study of the photonic band gap in the propagation of surface plasmons (SPs) on periodically corrugated surfaces. Our main purpose is to investigate the case where the band gap width is larger than the energy distance between the SP dispersion curve for a flat surface and the light line. We introduce a physical model of the interaction of light waves with SPs and derive an analytical expression for the SP wave vector near band gaps based on the coupled-mode approach involving three interacting modes (two of them are SP modes and one is a light mode). By using the interferometric measurement we have studied, for the first time, the SP propagation parameters in the vicinity of the photonic band gap (10 μm wavelength region). The predictions of our theory are in good agreement with the experimental data.

Influence ofdelectrons on the dispersion relation of Ag surface plasmons for different single-crystal faces

The surface plasmon dispersion relation of Ag is calculated for different single-crystal orientations. To describe the dynamical response properties of both delocalized $5s$ electrons and more tightly bound $4d$ electrons, the jellium model is combined with the so-called dipolium model, in which the occupied Ag d bands are represented in terms of polarizable spheres located at the sites of a semi-infinite fcc lattice. The nonlocal susceptibility characterizing the s electron response in the surface region is derived using density functional theory. The screening of the Coulomb interaction between conduction electrons via the lattice of dipoles, and of the dipole interaction via the surrounding sea of conduction electrons, is treated self-consistently. Electron energy loss spectra are calculated for all three low-index faces. The surface plasmon energy is found to increase with parallel wave vector for all cases. The magnitude of the positive slope depends on the crystal orientation and, for Ag(110), on the propagation direction. These results are in qualitative agreement with electron energy loss measurements.

Hydrodynamic model calculation of surface plasmons on periodically corrugated metal surfaces

Surface plasmon (SP) modes on a periodically corrugated metal surface have been investigated with the hydrodynamic model. Miniband gaps in the SP dispersion relation are predicted, due to the coherent scattering of the SP waves by the periodically corrugated surface. The scattering between the bulk plasmons (BPs) and SPs caused by the corrugated surface causes the SPs with high frequencies to decay into BP modes. Even in the dispersionless case, the corrugated surface is expected to lift the degeneracy of the SP modes and introduce an effective dispersion in the SP dispersion relation.

Surface plasmon dispersion at silver single-crystal surfaces

We develop a model for the calculation of screening at the surface of single Ag crystals, including the response of both conduction and bound electrons. We obtain the electron-energy-loss spectra on the low index faces, and from it, the corresponding surface plasmon (SP) dispersion relations. Our results depend strongly on the face orientation and, for Ag(110), and also on the propagation direction. With an appropriate choice of surface density profile for the conduction electrons, our model yields large positive SP dispersions, in qualitative agreement with experiment.

The reflectivity of a film with a grating in contact with a semi-infinite bimetallic superlattice

We study the influence of a semi-infinite bimetallic superlattice on the `excitation' of surface polaritons appearing in a film with a grating surface. The transfer-matrix formalism and the Rayleigh-Fano approach are applied to study the optical response for p-polarized light. We analyse the reflectivity and the dispersion relation of the surface plasmon-polaritons, considering Al and Mg for the semi-infinite superlattice and Mg for the film. The `optical' dispersion relation of surface plasmons is constructed using the minima of the reflectivity. This relation shows a dependence on the corrugation strength such that, as this corrugation grows, the corresponding curve shifts to lower frequencies.

Electrostatic surface shape resonances of a finite number of ridges

The resonance properties of localized electrostatic surface modes associated with a finite number of ridges on an otherwise planar surface are investigated. Numerical solutions of the homogeneous integral equations that describe the electromagnetic fields in the vicinity of the ridges are used to obtain the dispersion relation of surface plasmons. The frequencies of the electrostatic surface shape resonances are calculated for ridges with Gaussian, Lorentzian, sinusoidal, exponential, and triangular profiles. We show the existence of splittings of the plasmon frequencies, which depends on the surface profile function and on the distance between the ridges. Considering the ridge with a sinusoidal profile, we obtain the limit on the number of ridges which generates a frequency splitting of the electrostatic surface shape resonances, whose frequency values converge to those of the dispersion relation of surface plasmons on one-dimensional sinusoidal grating.

Surface dielectric response: Exact solution in the semiclassical infinite-barrier model with diffuse scattering

The dielectric response of a semi-infinite medium with a sharp electron profile at the surface is reexamined within the framework of the diffuse scattering model. The problem is shown to have an analytical solution for an arbitrary wave vector and frequency dependent bulk dielectric function $\ensuremath{\epsilon}(q,\ensuremath{\omega})$. The explicit surface energy-loss function and the surface plasmon dispersion relation are derived, which extend the conventional ones $({L}_{s}(\ensuremath{\omega})=\ensuremath{-}\mathrm{Im}1/[\ensuremath{\epsilon}(\ensuremath{\omega})+1]$ and $\ensuremath{\epsilon}(\ensuremath{\omega})=\ensuremath{-}1$, respectively$)$ to the spatially dispersive case. These solutions are applied to the models with hydrodynamic, Lindhard, and Lindhard-Mermin dielectric functions. The results obtained give the deeper insight into the analytical structure of the surface dielectric response, in particular, showing that the coupling of the bulk plasmon with a charge outside a solid corresponds not to a pole, as is the case for a charge inside a solid, but to the branching point singularity in the energy-loss function. Comparison with Ritchie-Marusak theory of specular reflection shows that while retaining all the advantages of an analytical solution, the diffuse scattering model yields the more realistic description of the dynamical response of metal surfaces. The results are illustrated by the application to the medium with aluminum parameters and discussed in conjunction with charged particles energy losses.

Exact surface-plasmon dispersion relation for spatially dispersive solid with an abrupt surface

The exact surface-plasmon dispersion relation for a semi-infinite medium with an abrupt surface and an arbitrary wave-vector and frequency dependent dielectric function ϵ(q,ω) is obtained. The result is applied to the models with hydrodynamic and Lindhard dielectric functions and the comparison with other theories and the experiment is fulfilled.

SURFACE PLASMONS ON UNIAXIAL CRYSTALS WITH GRATING SURFACES

In this paper we calculate the dispersion relation of surface plasmons propagating on uniaxial crystals, upon which a periodic grating has been built. The Rayleigh–Fano method is used, resulting in an infinite number of branches. We consider different orientations of the optical axis of the crystal, with respect to the direction of propagation. It is observed that the splitting of the surface plasmons depends not only on the ratio between the amplitude of the grating and its period (ζo/a), but also on the factor of anisotropy [Formula: see text], which changes for each optical orientation. Numerical results for the dispersion relation are obtained for α- SiO 2, and show the influence of the factor γ on the convergence of matrices.

Collective surface modes in small spherical metallic systems within the Bloch-Jensen hydrodynamical model

An exact generalization of the classical Bloch-Jensen hydrodynamical model for the homogeneous free-electron gas is obtained for a two-layer spherical metal system. The dispersion relation of surface plasmons together with the analytical expression of the self-consistent electric field are calculated exactly. It is found that small and large systems exhibit notably different behaviours. In particular, the surface-plasmon frequencies strongly depend on the layer thickness. As is the case for the metal-vacuum interface, the electric field is shown to present a sharp maximum at the interface between the two metals. A simple physical interpretation of this effect is provided. The present two-layer system also gives interesting indications about the part played in collective surface modes by the ion-density profile near the surface of small metal clusters, which may be helpful in interpreting recent puzzling experimental data.

Thin-foil surface-plasmon modification in scanning-probe microscopy.

The modification of the surface-plasmon dispersion relation in a thin foil by the near-zone effects of a scanned probe offers optical data that yields interesting forms of microscopy and spectroscopy. An exact solution for this effect is obtained here by taking advantage of the applicability of nonretarded electrodynamics in prolate spheroidal coordinates. Using the known optical properties of the materials, the modifying parameters and dispersion relations are discussed. Both planar and curved foils are examined.

Splitting of the dispersion relation of surface plasmons on a rough self-affine fractal surface.

The surface-plasmon dispersion relation for a statistically rough surface can display a splitting under specific conditions. Calculations of the dispersion-relation splitting are performed in terms of a correlation model for self-affine fractal rough surfaces with analytic form of the associated roughness spectrum, G(K)\ensuremath{\propto}(1+${\mathit{aK}}^{2}$${\ensuremath{\xi}}^{2}$${)}^{\mathrm{\ensuremath{-}}1\mathrm{\ensuremath{-}}\mathit{H}}$. The dependence of the splitting on the roughness exponent H is investigated.

Excitation of surface plasmons on titanium nitride films: determination of the dielectric function

For the first time it is possible to observe the excitation of surface plasmons (SP) on thin titanium nitride (TiN) films. From angle- and wavelength-dependent reflection measurements the dielectric function of TiN, and thus the optical constants, were calculated. Therefore, the main aim of this work was the determination of the real and imaginary parts of the dielectric function ε(ω) and thus the dispersion relation of SP of TiN. This method is an alternative to the standard technique of ellipsometry for determination of the complex dielectric function.

Joan Bausells

Abstract The response of a plane surface to a q- and ω-dependent excitation within a model valid for general non-local dielectric functions, representing the electronic surface diffuseness by a double step density has been obtained. The surface plasmon dispersion relation obtained in this model is compared with a sum-rule calculation which assumes a jellium positive background for the ionic part of the metal and uses an analytical approach to a Lang-Kohn calculation for the electronic density profile at the surface. The role played by the different physical assumptions made for each model is stressed. Finally, a comparison of the results with experimental data and some previous theoretical calculations is made.

Response of simple-metal plane surfaces

The response of a plane surface to a q- and ω-dependent excitation within a model valid for general non-local dielectric functions, representing the electronic surface diffuseness by a double step density has been obtained. The surface plasmon dispersion relation obtained in this model is compared with a sum-rule calculation which assumes a jellium positive background for the ionic part of the metal and uses an analytical approach to a Lang-Kohn calculation for the electronic density profile at the surface. The role played by the different physical assumptions made for each model is stressed. Finally, a comparison of the results with experimental data and some previous theoretical calculations is made.

Surface plasmon dispersion relation for normal and tangential modes

A theoretical investigation has been made for the surface plasmon dispersion relations for the case of normal and tangential modes to the surface. The hydrodynamical model has been used. We have obtained the dispersion relation by using the continuity of φ and dφ dx , where φ is the scalar potential. The dispersion relations obtained in the present work have been successfully compared with the theoretical and experimental work of several authors.