Date of Award

Fall 11-2020

Document Type


Degree Name

Doctor of Philosophy (PhD)




The electromagnetic response of conducting nanostructures in the 1 to 10 nm size range is investigated using the quantum box model (QBM). The core purpose of this dissertation is to understand how quantum effects emerge in both the electric and magnetic response of nanostructures in this size range, and in particular, explore how these effects can improve the design of future technology in the fields of plasmonics, metamaterials, and nano-composite materials. One of the primary objectives is to study how quantum effects enhance magnetism in nanoconductors and determine whether quantum resonances give rise to negative permeability, a highly desirable property in the design of plasmonic and optical devices. This aspect is frequently overlooked since classical theory predicts that most materials are magnetically inert.

With these goals in mind, this dissertation treats the quantum mechanical theory of the electromagnetic response of nanoconductors using perturbation theory. Theoretical expressions for bi-anisotropic susceptibility tensors are derived, and the gauge invariance of the tensors is then assessed. Verifying gauge invariance is of high importance since it is a fundamental property of nature, and it must hold true if the model is to be trusted. By formally including a gauge transformation in the perturbation theory, the susceptibility tensors are proven to be universally gauge-invariant.

After treating the theoretical foundations of the QBM, the theory is then applied to study quantum size effects in the permittivity of metal nanoparticles. Convergence to classical Drude theory is observed for large systems (≳ 10 nm), but finite-size calculations are found to deviate from classical behavior in several ways. First of all, insulator-like, positive real permittivity is found at low frequencies in contrast to the large negative permittivity predicted classically. Secondly, when compared to classical calculations of absorption spectra, quantum calculations predict plasmon peak positions that are either red-shifted or blue-shifted, depending on the embedding material and size distribution of the particles. Finally, discrete quantum resonances emerge in the permittivity of systems sized smaller than 10 nm.

Explicit upper and lower bounds are derived for these resonances, placing limits on the enhancement of the permittivity and relaxation rates due to quantum confinement effects. These bounds are verified numerically, and the size dependence and frequency dependence of the empirical Drude size parameter is extracted from the model. Comparisons with available experimental data suggest that the common practice of empirically modifying the Drude function can lead to inaccurate predictions for highly uniform distributions of nanoparticles with mean radius � < 10 nm.

Next, the QBM methodology is applied to study the magnetic response of conducting nanostructures. Calculations presented in this dissertation find that the magnetic susceptibility is enhanced in metal nanorods, thin slabs, and nanocubes. The calculations find both strong paramagnetism and diamagnetism orders of magnitude larger than semi-classical Landau diamagnetism. The degree of enhancement depends on the structure’s geometrical proportions and its orientation with respect to the magnetic field, but the calculations do not find negative permeability for the systems studied.

Because temperature plays an important role in magnetism, we also present a systematic study of temperature-dependent magnetic properties in nanosized rings and nanocubes using grand canonical statistics. Different domains of temperatures with distinct behavior of susceptibility are identified. In the case of nanocubes, calculations performed with size averaging collapse to the bulk Landau value in the thermodynamic limit, clearly demonstrating the transition from finite three-dimensional systems to semi- classical bulk Landau diamagnetism.

In the case of nanorings, we find that the magnetic susceptibility can exhibit multiple sign flips at intermediate and high temperatures depending on the number of electrons in the ring (�) and whether or not Zeeman splitting effects are included. When the temperature is increased from absolute zero, the susceptibility begins to flip sign above a characteristic temperature that scales inversely with the size of the ring according to �'( or �'(/*, depending on the presence of spin effects and the value of � mod 4. Analytical results are derived for the susceptibility in the low and high temperature limits, explicitly showing how spin affects the ring Curie constant.

The studies of paramagnetic-diamagnetic transitions in thin conducting rings is then extended to the canonical ensemble and compared with the commonly used grand- canonical approximation. Exact calculations of the canonical partition function and magnetic susceptibility are evaluated numerically using a recursive method. Persistent differences between the canonical and grand-canonical calculations are found at both low and intermediate temperatures. Criteria for convergence between the ensembles is provided, establishing the temperature and system size requirements for reaching the thermodynamic limit in quantum rings.