Visualization of a 2D heterostructure. Image courtesy of Paul Cazeaux (University of Minnesota, Twin Cities)
The IMA is pleased to host a hot topics workshop on the Mathematical Modeling of 2D Materials, from May 16-19, 2017. This workshop will be the first to focus on the mathematical modeling and computing aspects of 2D materials, which are a class of nanomaterials that are only one or two atoms thick and behave as two-dimensional structures. Despite being so miniscule, these materials possess the remarkable properties of being exceptionally strong, lightweight, flexible, and excellent conductors of heat and electricity.
There has been an extraordinary level of research activity on the electronic, optical, and mechanical properties of 2D materials in the materials science community, most recently in a new study led by researchers at the University of Minnesota on manipulating 2D materials to make modern day devices faster, smaller, and better.
Interest in the mathematics community has recently emerged with the goal of developing rigorous foundations and efficient and accurate computational methods. Some of the topics to be explored in the workshop include the electronic, optical, and mechanical properties of incommensurate 2D heterostructures, defects, edge states, and surface plasmonics.
The explosion of scientific and technological interest in 2D materials was sparked by the 2004 isolation of freestanding graphene sheets. However, workshop organizer and speaker Efthimios Kaxiras (Harvard University) notes that “graphene, despite its exciting and unique properties, has its limitations, mostly due to the absence of a band gap which is a necessary feature in most optical and electronic applications.”
However, only one decade after the discovery of graphene, a host of other 2D materials with semiconducting or insulating behavior have since been discovered.
“Even more exciting and promising is the possibility of interleaving these materials to create stable structures with any combination of desired electronic, optical, magnetic, and thermal properties with atomic-scale features,” said workshop organizer Mitchell Luskin (University of Minnesota, Twin Cities).
Luskin has many exciting ongoing research projects. He and Kaxiras are leading a Multidisciplinary University Research Initiative (MURI) sponsored by the Army Research Office to develop mathematical and computational models for the 2D heterostructures being studied in the laboratory of Philip Kim (Harvard University), a speaker at the workshop. One of the main issues encountered in the mathematical and computational modeling of 2D heterostructures is that the lattice periodicities of different layers do not match.
Working with workshop organizer and speaker Eric Cances (École des Ponts) and postdoc Paul Cazeaux (University of Minnesota, Twin Cities), Luskin has recently developed a mathematical theory for the transport properties of 2D heterostructures. Instead of the Floquet-Bloch waves used to study periodic structures, the new approach uses the continuously varying local configurations of incommensurate structures.
Rapid advances are also being made in surface plasmonics for 2D materials, offering tremendous possibilities of manipulating light at small scales. A goal for Luskin, workshop organizer and speaker Dionisios Margetis (University of Maryland), and Matthias Maier (University of Minnesota, Twin Cities) is to generate electromagnetic waves that propagate with sufficiently small energy loss and small wavelength at the interface between a conducting material, such as graphene, and an insulating material. A related goal is to design complex geometries of 2D materials that enable the excitation and control of such waves, such as via homogenization of a suitable microstructure.
Numerical simulation of a electro-magnetic wave propagating along the 2D material.
Image courtesy of Matthias Maier (University of Minnesota, Twin Cities)