Electronic Structure of Incommensurate Bilayer Heterostructures

University of Minnesota Ph.D. dissertation.August 2018. Major: Mathematics. Advisors: Mitchell Luskin, Peter Polacik. 1 computer file (PDF); v, 67 pages.In this work, we study the electronic structure of incommensurate 2D heterostruc- tures. 2D heterostructures consist of stacked sheets, each consisting of a periodic array of atoms in two directions, while in the third it is only a few atoms thick. These struc- tures are very small and possess a wide assortment of electronic properties, tunable by layering different materials on top of each other with a variety of stackings. While indi- vidual sheets are periodic, and hence can easily be studied using Bloch Theory, bilayer stackings can form aperiodic, or incommensurate, ensembles. Typically, these materials are studied using supercell techniques, where one of the sheets is artificially strained to force a periodic structure on some large cell called a supercell. These techniques introduce unquantified errors, and become numerically intractable for small relative twist angles, which is a widely studied case. In this work, we prove that the electronic density of states (DOS) for 2D incommen- surate layered structures is well-defined as the thermodynamic limit of finite clusters. In addition, we obtain an explicit representation formula for the DOS as an integral over local configurations. Next, based on this representation formula, we propose a novel algorithm for com- puting electronic structure properties in incommensurate heterostructures, which over- comes limitations of the common approach to artificially strain a large supercell and then apply Bloch theory. Further, we present an analogous scheme formulated in momentum space, which we prove has significant computational advantages in specific incommensurate systems of physical interest, e.g., bilayers of a specified class of materials with small rotation angles. We use our theoretical analysis to obtain estimates for improved rates of convergence with respect to total CPU time for our momentum space method that are confirmed in computational experiments