ABSTRACT

This project investigates the physical properties of multi-layered materials for the purpose of guiding the efficient and novel design of opto-electronic devices, such as lasers, antennas, and filters. A popular example of such a material is stacked sheets of hexagonal carbon graphene. The aim is to develop flexibility in design from the multitude of ways in which the layers can be placed relative to each other to achieve a desired interaction of optical and electromagnetic waves, ultimately delivering practical options for harnessing light. The layering process naturally intertwines the geometrical structure of the layers with the mathematical equations for the underlying physics. This project focuses on the interaction of these mathematical objects with the intent of building a foundational theory of layered media. Because the investigation targets the underlying physical phenomena, its results will be applicable across the diverse aspects of society that rely on electromagnetic devices. Students at the graduate and undergraduate levels will be integrated into a dedicated group of researchers from diverse backgrounds and will be given the opportunity to present their work at scientific conferences to help them broaden their scientific horizons and forge professional connections.

There is a physical and mathematical object that lies at the center of the investigations in this project, called the Fermi surface, which together with the associated Bloch surface contains core information about electromagnetic waves in crystalline or periodic materials, by encoding a relation between two fundamental physical quantities, energy (or frequency) and momentum (or wavenumber). These objects are analytic or algebraic varieties, depending on the details of the periodic medium, and their structure encodes information about the layering method. The layered materials can be either two-dimensional, such as multi-layer graphene sheets, or three-dimensional, where the layers are internal to the bulk and provide a sort of hidden degree of freedom. Specific objectives are to establish connections between the layering method and the Fermi and Bloch varieties, particularly their reducibility and singularities; to analyze the resonant consequences of reducibility of these varieties; to classify internal symmetries that result in reducibility and the persistence of reducibility under symmetry breaking; to create novel bulk media by homogenization of layered graph models; and to apply these investigations to multi-layer graphene and other physically viable layered materials. The study will invoke the mathematical areas of complex analysis, partial differential equations, algebraic geometry, and spectral theory of differential operators.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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