In chapters 1 and 2, we described one of our experimental systems, an ultra-clean graphene channel formed by encapsulation between two hBN layers. In these samples, the effects of the underlying potential due to Van der Waals interaction between graphene and hBN are negligible. This is because the individual layers are usually stacked in a random manor without paying attention to the relative crystallographic directions of each material. If however, graphene is positioned on top of hBN in such a way that there is only a small misalignment angle between their crystallographic axes, the situation is quite different. In this case, beating of the two crystal periodicities produces a Moiré pattern (Fig. 16 a), which creates an additional periodic potential on the nm scale. This Moiré potential strongly modifies the electronic spectrum in graphene and is therefore referred to as a superlattice. Specifically, the additional periodic potential imposes zone folding of graphene’s energy dispersion in to a smaller superlattice Brillouin zone, which creates a number of interesting features such as Van Hove singularities and secondary Dirac points (inset in the top panel of Fig. 16c). The secondary Dirac points were first observed in scanning tunnelling microscopy experiments at energies around 0.3 eV away from the main Dirac point54. If the crystal layers are close to perfect alignment, the superlattice period is about 14 nm (Fig. 16b) and the secondary Dirac points occur at
energies around 0.2 eV from the main Dirac point in graphene. Such energies are easily accessible by electrostatic gating, and, shortly after STM experiments, signatures of the graphene/hBN superlattice were measured in transport55,48.
Figure 16c shows typical gate dependences of the resistivity xx and hall resistance Rxy in a graphene/hBN superlattice measured by the Manchester group in 201348. At zero doping, a sharp peak in resistivity is observed corresponding to the main Dirac point. At large n, neighbouring satellite peaks in resistivity are observed for holes and electrons, at equal distances from charge neutrality point. These sharp peaks in resistivity occur when the Fermi energy moves through the secondary Dirac points. Notably, the secondary Dirac point is much more pronounced for holes than electrons. Further evidence for the presence of secondary Dirac points is found also when measuring the hall resistance Rxy, because the sign of Rxy reflects the type of majority charge carrier in the conducting channel. In graphene, Rxy diverges as the Fermi-level moves closer to the Dirac point and exhibits a sign reversal when moving through it, corresponding to a change in carrier type (electrons or holes). In the graphene/hBN superlattice, this feature is mimicked around the secondary Dirac points (bottom panel of Fig. 16c). We note however that Rxy changes in a non-trivial manor when tuning the Fermi-energy, changing three times in total as we move away from the main Dirac point (for both holes and electrons). This is due to the presence of Van Hove singularities in the density of states56.
The underlying Moiré potential can also strongly influence the zero energy dispersion (charge neutrality point). This is because the superlattice potential breaks inversion symmetry within graphene’s A-B sub-lattice, creating a global band gap at the Dirac point56 . Aside from technological interest (for year’s research efforts have focussed on trying to induce a band gap in graphene to operate as a transistor device), the band gap opening also has fundamental, topological implications. For example, this creates Berry curvature hot spots close to the Dirac point, allowing the study of topological currents57 in graphene which has in turn sparked a new field in the electronics industry coined “valleytronics”.
In this Thesis, we are concerned with the magneto transport properties of the graphene/hBN superlattice. This type of experiment has gained intense interest over the past few years because the superlattices’ energy spectrum hosts the long sought Hofstadter butterfly described in Chapter 3.5. This physics becomes accessible because of the large lattice spacing in graphene/hBN superlattices; For a superlattice with a = 14 nm, we only need B = 26 T to reach the condition 0 = 1. In 2013, Hofstadter butterflies were observed in graphene/hBN superlattices48,55. Figure 16d plots
Fig. 16e shows theoretical calculations of the Hofstadter butterfly spectrum in graphene/hBN superlattices, where black regions show states and white spaces show gaps. In experiment, we can see the conductivity accurately maps out gaps and states where it is minimum (white) and maximum (black), respectively.
Figure 16| Graphene/hexagonal boron nitride superlattices. a, Schematic of a graphene on hBN heterostructure. The magnified region shows the Moiré pattern produced with a wavelength when the crystal layers are aligned to within a few degrees. b, STM images54 of heterostructures with different alignment angle that produce superlattices with 6 and 11 nm respectively (left to right). Scale bar: 5nm. c, longitudinal resistivity (xx)and hall resistance (xy) as a function of carrier density n. d, longitudinal conductivity xx (B,n) for the same device as in c. Grey scale: white, 0 k; black, 8,5 k. e, The energy dispersion calculated for graphene/hBN superlattices as a function of energy and B. The black regions are available states whilst the white regions show gaps in the energy spectrum. Data from c –e is taken from ref. 48.