Generic miniband structure of graphene on a hexagonal substrate

Abstract

Using a general symmetry-based approach, we provide a classification of generic miniband structures for electrons in graphene placed on substrates with the hexagonal Bravais symmetry. In particular, we identify conditions at which the first moiré miniband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density.PACS numbers: 73.22.Pr,73.21.Cd, Recently, it has been demonstrated that the electronic quality of graphene-based devices can be dramatically improved by placing graphene on an atomically flat crystal surface, such as hexagonal boron nitride (hBN) [1][2][3][4][5][6][7]. At the same time, graphene's electronic spectrum also becomes modified, acquiring a complex, energydependent form caused by incommensurability between the graphene and substrate crystal lattices [8][9][10]. In particular, for graphene placed on hBN, the difference between their lattice constants and crystallographic misalignment generate a hexagonal periodic structure known as a moiré pattern [2,3,[8][9][10]. The resulting periodic perturbation, usually referred to as a superlattice, acts on graphene's charge carriers and leads to multiple minibands and the generation of secondary Dirac-like spectra. The resulting new Dirac fermions present yet another case where graphene allows mimicking of QED phenomena under conditions that cannot be achieved in particle physics experiments. In contrast to relativistic particles in free space, the properties of secondary Dirac fermions in graphene can be affected by a periodic sublattice symmetry breaking and modulation of carbon-carbon hopping amplitudes, in addition to a simple potential modulation. The combination of different features in the modulation results in a multiplicity of possible outcomes for the moiré miniband spectrum in graphene which we systematically investigate in this article.To describe the effect of a substrate on electrons in graphene at a distance, d, much larger than the spacing, a, between carbon atoms in graphene's honeycomb lattice, we use the earlier observation [8][9][10][11][12][13][14] that, at d a the lateral variation of the wavefunctions of the p z carbon orbitals is smooth on the scale of a. This is manifested in the comparable sizes of the skew and vertical hopping in graphite and permits an elegant continuummodel description [11][12][13][14] of the interlayer coupling in twisted bilayers and the resulting band structure. A similar idea applied to graphene on a hBN substrate [8][9][10] suggests that a substrate perturbation for Dirac electrons in graphene can be described in terms of simple harmonic functions corresponding to the six smallest reciprocal lattice vectors of the moiré superlattice.Below, we shall use a similar approach to analyse the generic properties of moiré minibands for electrons in graphene subjected to a subs...

Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride

“…However, such measurements are not capable of mapping out the electronic dispersion with momentum-resolved information, and the lack of direct experimental results has led to ambiguous and even conflicting results about the electronic spectra of SDCs and the existence of gaps. Although various theoretical models have been proposed 10,11,[20][21][22][23] , the locations and dispersions of SDCs depend strongly on the parameters used to describe the inversion-symmetric and inversion-asymmetric superlattice potential modulation 10 . Different choices of inversion-symmetric perturbation could result in either isolated or overlapping SDCs 10 , and the locations of SDCs could change from the edges of the superlattice Brillouin zone (SBZ) 9,10 to the corners 10,11,21 .…”

Moiré pattern as a magnifying glass for strain and dislocations in van der Waals heterostructures

“…The perturbation produced by the moiré superlattice results in the formation of mini- bands for Dirac electrons in graphene [13][14][15][16][17][18][19][20]. The latter can be found by numerical diagonalisation of the moiré superlattice Hamiltonian,…”

“…where Pauli matrices σ i=1,2,3 act on the electron amplitudes on graphene's A and B sublattices, and parameters U 0 , U 3 and U 1 take into account a smooth potential, A-B sublattice asymmetry and the modulation of A-B hopping in graphene produced by the underlay [19]. This moiré potential contains six harmonics characterised by Bragg vectors,…”

“…1. For the unstrained heterostructure (left) the spectrum exhibits a secondary Dirac point [13][14][15][17][18][19] in the valence band. Strain, as low as w ∼ 1 − 3%…”

“…Then we numerically diagonalize the resulting Heisenberg matrix for each momentum point in the mini Brillouin zone. For numerical diagonalisation, we choose phenomeno-logical parameters U 0 = v|G |/30, U 1 = −v|G |/15, and U 3 = −v|G | √ 3/30, whose relative size corresponds to two specific microscopic models [19].…”