Moiré Patterns as a Probe of Interplanar Interactions for Graphene on h-BN

Abstract

By atomistic modeling of moiré patterns of graphene on a substrate with a small lattice mismatch, we find qualitatively different strain distributions for small and large misorientation angles, corresponding to the commensurate-incommensurate transition recently observed in graphene on hexagonal BN. We find that the ratio of C-N and C-B interactions is the main parameter determining the different bond lengths in the center and edges of the moiré pattern. Agreement with experimental data is obtained only by assuming that the C-B interactions are at least twice weaker than the C-N interactions. The correspondence between the strain distribution in the nanoscale moiré pattern and the potential energy surface at the atomic scale found in our calculations, makes the moiré pattern a tool to study details of dispersive forces in van der Waals heterostructures. The study of these new hybrid materials is emerging as a strong research area.The superposition of periodic layered structures, with either slightly different lattice constants or different orientations, creates moiré patterns [3][4][5][6][7]. These patterns can yield a wealth of information about the lattice constant mismatch, strain and imperfections of the surface [8][9][10][11][12][13]. The moiré patterns imply a change of the interatomic distances that can affect properties that are important both for applications and for fundamental physics such as the quantum mechanics of electrons in quasi-periodic potentials [3][4][5][6].In recent years hexagonal boron nitride (h-BN) has become a standard substrate for graphene growth due to its flat surface without dangling bonds, the hexagonal lattice with a lattice constant only 1.8 % larger than that of graphene and the fact that h-BN is an insulator [14]. These properties have led to the realization of the first field effect transistor [15]. The difference in lattice constant leads to the appearances of moiré patterns, which can be observed experimentally [16][17][18].Usually, moiré structures are considered from a purely geometrical point of view for the superposition of two rigid lattices where the length L of the moiré patterns is found to depend on the angle θ and the lattice mismatch between the two layers aswhere p is the ratio between lattice constants and a the lattice constant of the substrate [19]. Strain due to the lattice mismatch and/or rotations have been considered in a continuum approach to study the modification of the electronic structures in tight binding calculations [20][21][22][23][24] and the pseudo-magnetic fields resulting from out-of-plane displacements [25,26]. Full atomic relaxation to minimal energy configurations is however necessary to make a detailed comparison to experimental structural information as obtained by scanning probe microscopy [7]. At the same time, we will show that this procedure allows to get quantitative information on the interplanar interactions. It is well known that dispersive forces are beyond the standard local density functional and generalized gradient correct...

Quantum elasticity of graphene: Thermal expansion coefficient and specific heat

“…[72,73]. In this setting, large thermal expansions of 2D materials at low temperatures would result in a strong temperature dependence of lattice mismatch which can be seen via reconstruction of Moire patterns [74][75][76] or via characteristics of graphene bubbles on a substrate [77].…”

Adsorption and STM imaging of polycyclic aromatic hydrocarbons on graphene

“…Moreover, as we will show, there is an excellent agreement with the experimental STM images of PAHs on graphite. Indeed, a graphene layer may adopt a particular structure once adsorbed, especially on a rough SiO 2 substrate, on SiC [47,48], on metallic surfaces [46,49] such as Ru(0001), Rh(111), Pt(111), Ir(111), or Ni(111), or on h-BN [50]. Even if a moiré pattern was observed on Cu(111) with STM [51], as well as preferential sticking of molecules on some parts of the moiré [52], one anticipates that the reconstruction of the graphene layer is small enough to consider an unperturbed substrate for molecular adsorption.…”

Effect of Structural Relaxation on the Electronic Structure of Graphene on Hexagonal Boron Nitride

“…The computational challenge lies in the fact that at such angles these superlattices have unit cells consisting of tens of thousands of atoms, making it impossible to study them using methods such as density functional theory (DFT). A classical atomistic approach instead is able to describe the experimentally observed structures [10], and the tight-binding propagation method (TBPM) described in Ref. [11] can be used to study the electronic properties of systems with hundreds of millions of atoms, circumventing this problem.…”