Waves, Novel Two-Dimensional Materials, and Applications

At the heart of many realized and envisioned advances in science and technology --from communication networks to quantum computation -- is the ability to robustly transport, channel, and manipulate forms of energy, e.g. electronic and optical, on very small spatial and temporal scales. This is achieved by fabricating novel media in which energy, which propagates as waves, interacts with engineered microfeatures. A current example of a novel, naturally occurring, medium of intense interest is graphene a one-atom thick mono-layer of carbon atoms, arranged in-plane in a honeycomb structure, extending to the macro-scale. Graphene is the most conductive substance known at room temperature (electronically and thermally). This material has been at a center of a revolution in Condensed Matter Physics and Materials Science, known as the Two-Dimensional Material. Many of the novel phenomena observed in graphene relate to general mathematical properties of waves in media with novel symmetry, and hence physicists and engineers have engineered and investigated materials with similar general properties, dubbed 'artificial graphene,' with a view on applications to optics, communications, information storage and computing. PI Weinstein will study a range of problems in fundamental and applied mathematics related to the existence, stability and transport properties of waves in such novel media, with honeycomb structures as a paradigm for 2D materials, as well as questions concerning the dynamics of waves in nonlinear systems. This work is firmly aligned with Quantum Leap, one of the NSF's 10 Big Ideas. This award will support one graduate student to be trained in a multifaceted approach to applied mathematical research using modeling, analysis and computation, with emphasis on applications of wave phenomena in complex media to the scientifically important questions.

PI Weinstein will investigate (A) Energy propagation along line-defects in novel 2D periodic media governed by continuum PDE models, topologically protected states and the relation between bulk properties of periodic media and edge effects, and (B) Dynamics of coherent structures in discrete and continuous nonlinear wave systems, in particular (i) transport in nonlinear lattices and (ii) dynamics of a free boundary problem in compressible fluids.

(A) Many remarkable properties of 2D materials (e.g. extraordinary conductivity) are related to the subtle properties of its band structure (the collection of dispersion surfaces and eigenmodes of Floquet-Bloch theory) which dictate energy transport. The conical singularities (Dirac points) of the dispersion surfaces of graphene are at the core of its remarkable electron mobility, and the topological properties of its band structure properties give information about the types of energy-transport can occur when graphene is interfaced with free-space or another material. Thus, it is through a convergence notion from Analysis, PDE and Topology that one can make process understanding these novel materials and their artificial analogues.

(B) This research is part of the PI's long-standing research program in nonlinear waves, which now emphasizes i) coherent structure propagation in discrete nonlinear dispersive systems and ii) the nonlinear PDE dynamics of coherent structures with infinitely many internal degrees of freedom, as exemplified by the nonlinear oscillations of a bubble in the fluid.

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.