Graduation Year

2019

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Sherwin Kouchekian, Ph.D.

Committee Member

Brian Curtin, Ph.D.

Committee Member

Ali Passian, Ph.D.

Committee Member

Ivan Rothstein, Ph.D.

Committee Member

Boris Shekhtman, Ph.D.

Keywords

Decay rate, Interaction Hamiltonian, Matrix element, Second quantization, Surface plasmons

Abstract

We investigate field quantization in high-curvature geometries. The models and calculations can help with understanding the elastic and inelastic scattering of photons and electrons in nanostructures and probe-like metallic domains. The results find important applications in high-resolution photonic and electronic modalities of scanning probe microscopy, nano-optics, plasmonics, and quantum sensing.

Quasistatic formulation, leading to nonretarded quantities, is employed and justified on the basis of the nanoscale, here subwavelength, dimensions of the considered domains of interest.

Within the quasistatic framework, we represent the nanostructure material domains with frequency-dependent dielectric functions. Quantities associated with the normal modes of the electronic systems, the nonretarded plasmon dispersion relations, eigenmodes, and fields are then calculated for several geometric entities of use in nanoscience and nanotechnology.

From the classical energy of the charge density oscillations in the modeled nanoparticle, we then derive the Hamiltonian of the system, which is used for quantization.

The quantized plasmon field is obtained and, employing an interaction Hamiltonian derived from the first-order perturbation theory within the hydrodynamic model of an electron gas, we obtain an analytical expression for the radiative decay rate of the plasmons.

The established treatment is applied to multiple geometries to investigate the quantized charge density oscillations on their bounding surfaces. Specifically, using one sheet of a two-sheeted hyperboloid of revolution, paraboloid of revolution, and cylindrical domains, all with one infinite dimension, and the finite spheroidal and toroidal domains are treated.

In addition to a comparison of the paraboloidal and hyperboloidal results, interesting similarities are observed for the paraboloidal domains with respect to the surface modes and radiation patterns of a prolate spheroid, a finite geometric domain highly suitable for modeling of nanoparticles such as quantum dots. The prolate and oblate spheroidal calculations are validated by comparison to the spherical case, which is obtained as a special case of a spheroid.

In addition to calculating the potential and field distributions, and dispersion relations, we study the angular intensity and the relation between the emission angle with the rate of radiative decay.

The various morphologies are compared for their plasmon dispersion properties, field distributions, and radiative decay rates, which are shown to be consistent.

For the specific case of a nanoring, modeled in the toroidal geometry, significant complexity arises due to an inherent coupling among the various modes. Within reasonable approximations to decouple the modes, we study the radiative decay channel for a vacuum bounded single solid nanoring by quantizing the fields associated with charge density oscillations on the nanoring surface. Further suggestions are made for future studies. The obtained results are relevant to other material domains that model a nanostructure such as a probe tip, quantum dot, or nanoantenna.

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