Collective electronic effects in scanning probe microscopy

The surface plasmon dispersion relations are calculated for a metal coated dielectric probe above a dielectric half space with and without metal coating. Employing prolate spheroidal coordinate system this configuration was modeled as confocal single-sheeted hyperboloids of revolution superimposed on planar domains. The involved media are characterized by frequency dependent, spatially local dielectric functions. Due to subwavelength dimensions of the region of interest, nonretarded electrodynamics is utilized to derive exact analytical expressions describing the resonant surface modes. The dispersion relations are studied as functions of the parameter that defines the hyperboloidal boundaries of the tip and the corresponding coating, and as functions of the involved coating thicknesses. Both parallel and perpendicular polarizations are considered. The results are simulated numerically and limiting cases are discussed with comparison to the Cartesian thin foil case.; Using this new type of probe-substrate configuration, the surface plasmon coupling mechanism is investigated experimentally utilizing a scanning probe microscope, and the signal strength acquired by the probe is measured as a function of the distance between the probe and the sample. This is repeated at three different wavelengths of the incident p-polarized photons used to stimulate surface plasmons in the thin metal foil. The results are compared with the theory.; Utilizing the prolate spheroidal coordinate system, the related and relevant problem of the Coulomb interaction of a dielectric probe tip with a uniform field existing above a semiinfinite, homogeneous dielectric substrate was studied. This is of interest in atomic force microscopy when the sample surface is electrically charged. The induced polarization surface charge density and the field distribution at the bounding surface of the dielectric medium with the geometry of a single-sheeted hyperboloid of revolution located above the dielectric half space interfaced with a uniform surface charge density is calculated. The force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The potential and the field distributions are calculated in the neighborhood of the apex of the tip. The calculation is based on solving Laplace's equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The integral expansion is analytically proved and numerically verified. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions similar to those arising in the Mehler-Fock integral transform.