Graphene on hexagonal lattice substrate: Stress and pseudo-magnetic field

Abstract

Moiré patterns in the pseudo-magnetic field and in the strain profile of graphene (GE) when put on top of a hexagonal lattice substrate are predicted from elasticity theory. The van der Waals (vdW) interaction between GE and the substrate induces out-of-plane deformations in graphene which results in a strain field, and consequently in a pseudo-magnetic field. When the misorientation angle is about 0.5 o a three-fold symmetric strain field is realized that results in a pseudo-magnetic field very similar to the one proposed by F. Guinea, M. I. Katsnelson, and A. K. Geim [Nat. Phys. 6, 30 (2010)]. Our results show that the periodicity and length of the pseudo-magnetic field can be tuned in GE by changing the misorientation angle and substrate adhesion parameters and a considerable energy gap (23 meV) can be obtained due to out-of-plane deformation of graphene which is in the range of recent experimental measurements (20-30 meV).Stacking different two dimensional materials with slightly different lattice structures on top of each other results in a new superlattice structure which is called Moiré pattern. The van der Waals (vdW) interaction between different 2D-crystals such as graphene (GE), hexagonal boron nitride (h-BN), and molybdenum disulfide (MOS 2 ) results in a multilayer heterostructure [1]. The resulting hexagonal Moiré pattern in graphene on top of other hexagonal lattice substrates affects the electromechanical properties of graphene. For example, hexagonal boron nitride has turned out to be an ideal dielectric substrate which is atomically flat and improves graphene's mobility by more than two orders of magnitude [2,3]. The B-N bond length is close to that of C-C with only a very small (1.6%-2%) lattice mismatch [4,5] which results in the appearance of a Moiré pattern (MP) when GE is put on top of BN. It was found that GE flakes can align with the underlying h-BN lattice within an error of less than 0.05 o [4,6]. Ab-initio and semi-empirical van der Waals studies showed that the interaction between GE flakes and the h-BN substrate is similar to that of a GE-GE stacked structure [7]. On the other hand the different electronegativity of B, N and C atoms leads to a nonuniform attractive force distribution over GE.Non-uniform strain in GE results in a pseudo-magnetic field and consequently results in the opening of an energy gap [8,9]. Earlier density functional theory calculations assumed lattice matching between GE and h-BN which induces in-plane strain and opens a gap in GE's spectrum of 50-60 meV [7]. But recent experiments found a gap in the range of 20-30 meV [10,11]. In this letter, we first develop a general theory for GE over a hexagonal lattice substrate and show that the induced strain has triangular symmetry resulting in interesting pseudo-magnetic field patterns which vary with the misorientation angle. Then, as an example, we concentrate on the h-BN-lattice induced deformation of the GE lattice using atomistic simulations and compare with our analytic results. Using experimental height defo...

Moiré superlattice effects in graphene/boron‐nitride van der Waals heterostructures

“…). Finally, it should be mentioned that the strain‐induced vector and scalar potentials in Hamiltonian produce a significant non‐trivial contribution to the mSL potential .…”

“…Many theoretical works have investigated the influence of one or two‐dimensional electrostatic potentials on graphene electrons , with the former situation realisable using patterned gates . Magnetic and pseudo‐magnetic field superlattices (the latter arising from periodically strained graphene) have also been extensively studied , with steps towards experimental realisation . There has also been significant work on the aligned heterostructures of bilayer graphene with hBN, including the observation of Hofstadter's butterfly in transport measurements .…”

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

“…Therefore, we begin by modeling h-BN as stretched graphene. Pair potentials like Lennard-Jones underestimate the corrugation of the interplanar potential energy surface [26,34]. For this reason, we describe the interaction between graphene and h-BN by a registry-dependent potential for graphene [33], which is scaled to the lattice constant of h-BN.…”

“…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].…”

Harmonic model of corrugations of incommensurate two-dimensional layers

“…We expect the properties of such high-quality graphene heterostructures to be particularly sensitive to the sub-nanometer corrugations arising from the interaction between graphene and substrate lattices. Indeed, the moiré superlattices of graphene on BN have been implicated in the opening of an electronic gap at the Dirac point [13,14,15,16,17] and the emergence of electronic minibands [2,14,18], and are believed to affect optical [2,18] and magnetic properties [2].…”