Perturbation theory for weakly coupled two-dimensional layers
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Paul Cazeaux and Mitchell Luskin School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA
Petr Plechác Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA
Eric Cancès Université Paris Est, Ecole des Ponts and INRIA, 77455 Marne-la-Vallée, France (Received 14 September 2015; accepted 1 March 2016)
A key issue in two-dimensional structures composed of atom-thick sheets of electronic materials is the dependence of the properties of the combined system on the features of its parts. Here, we introduce a simple framework for the study of the electronic structure of layered assemblies based on perturbation theory. Within this framework, we calculate the band structure of commensurate and twisted bilayers of graphene (Gr) and hexagonal boron nitride (h-BN), and of a Gr/h-BN heterostructure, which we compare with reference full-scale density functional theory calculations. This study presents a general methodology for computationally efficient calculations of twodimensional materials and also demonstrates that for relatively large twist in the graphene bilayer, the perturbation of electronic states near the Fermi level is negligible.
I. INTRODUCTION
The physics of two-dimensional systems, composed of one or several layers that have thickness of a single atom or a few atoms, is becoming ever more interesting as the realization of such systems with essentially any desirable combination of materials are within grasp of experiment.1–3 A key issue in such structures is their emergent properties. The environment of an isolated layer is modified by its incorporation into layered assemblies and it is highly desirable to be able to predict the combined system properties from those of isolated parts (layers). There are weak but significant interactions between layers, typically of van der Waals nature, and these need to be taken into account. A big challenge in this direction is that different materials have different inplane lattice constants, leading to incommensurate structures when they are combined. Even for layers of the same material, it is possible that the layers are not in perfect registry as in the ideal three-dimensional solid, or that one layer is twisted by an arbitrary angle relative to its neighbors, either by accident or by design.4–8 These issues make the modeling of two-dimensional layered
Contributing Editor: Lain-Jong Li a) Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2016.99 J. Mater. Res., Vol. 31, No. 7, Apr 14, 2016
structures crucial for the design and fabrication of materials with desirable properties imperative.9,10 Previous theoretical studies have modeled incommensurate twisted bilayers under various approximations such as the tight-binding virtual crystal approximation by Ghader et al.,11 generalized long-wavelength theory by Pal et al.,12 and also by approximating the incommensurate phases to the nearest commensurate ones.13 Notably, work by Bokdam et al.14 implements a many-body firstpr
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