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Lecture

Mathematical Modeling and Numerical Analysis for Incommensurate 2D Materials

Description

This lecture covers real space and momentum space methods for relaxed 2D heterostructures, incommensurate lattices, tight-binding model, equidistribution of lattice shifts, density of states approximation, shifted Hamiltonian, thermodynamic limit, exponential convergence of local density of states, mechanical relaxation, energy landscape in bilayer graphene, and error estimates for Hamiltonian domain reduction.

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Related concepts (32)

Bilayer graphene

Bilayer graphene is a material consisting of two layers of graphene. One of the first reports of bilayer graphene was in the seminal 2004 Science paper by Geim and colleagues, in which they described devices "which contained just one, two, or three atomic layers" Bilayer graphene can exist in the AB, or Bernal-stacked form, where half of the atoms lie directly over the center of a hexagon in the lower graphene sheet, and half of the atoms lie over an atom, or, less commonly, in the AA form, in which the layers are exactly aligned.

Mathematical model

A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science).

Graphene

Graphene (ˈgræfiːn) is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure. The name is derived from "graphite" and the suffix -ene, reflecting the fact that the graphite allotrope of carbon contains numerous double bonds. Each atom in a graphene sheet is connected to its three nearest neighbors by σ-bonds and a delocalised π-bond, which contributes to a valence band that extends over the whole sheet.

Twistronics

Twistronics (from twist and electronics) is the study of how the angle (the twist) between layers of two-dimensional materials can change their electrical properties. Materials such as bilayer graphene have been shown to have vastly different electronic behavior, ranging from non-conductive to superconductive, that depends sensitively on the angle between the layers. The term was first introduced by the research group of Efthimios Kaxiras at Harvard University in their theoretical treatment of graphene superlattices.

Mathematical psychology

Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior (in practice often constituted by task performance). The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations.

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