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Moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic potential modulation on a length scale ideally suited to studying the fractal features of the Hofstadter energy spectrum in large magnetic fields. Hofstadter's butterfly emerges in graphene superlattices In 1976 Douglas Hofstadter predicted that electrons in a lattice subjected to electrostatic and magnetic fields would show a characteristic energy spectrum determined by the interplay between two quantizing fields. The expected spectrum would feature a repeating butterfly-shaped motif, known as Hofstadter's butterfly. The experimental realization of the phenomenon has proved difficult because of the problem of producing a sufficiently disorder-free superlattice where the length scales for magnetic and electric field can truly compete with each other. Now that goal has been achieved — twice. Two groups working independently produced superlattices by placing ultraclean graphene (Ponomarenko et al .) or bilayer graphene (Kim et al .) on a hexagonal boron nitride substrate and crystallographically aligning the films at a precise angle to produce moiré pattern superstructures. Electronic transport measurements on the moiré superlattices provide clear evidence for Hofstadter's spectrum. The demonstrated experimental access to a fractal spectrum offers opportunities for the study of complex chaotic effects in a tunable quantum system. Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum 1 . Known as Hofstadter’s butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely 11 , 12 , 13 , 14 , 15 , 16 , 17 . Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.

The recently discovered flat electronic bands and strongly correlated and superconducting phases in magic-angle twisted bilayer graphene (MATBG) 1 , 2 crucially depend on the interlayer twist angle, θ . Although control of the global θ with a precision of about 0.1 degrees has been demonstrated 1 – 7 , little information is available on the distribution of the local twist angles. Here we use a nanoscale on-tip scanning superconducting quantum interference device (SQUID-on-tip) 8 to obtain tomographic images of the Landau levels in the quantum Hall state 9 and to map the local θ variations in hexagonal boron nitride (hBN)-encapsulated MATBG devices with relative precision better than 0.002 degrees and a spatial resolution of a few moiré periods. We find a correlation between the degree of θ disorder and the quality of the MATBG transport characteristics and show that even state-of-the-art devices—which exhibit correlated states, Landau fans and superconductivity—display considerable local variation in θ of up to 0.1 degrees, exhibiting substantial gradients and networks of jumps, and may contain areas with no local MATBG behaviour. We observe that the correlated states in MATBG are particularly fragile with respect to the twist-angle disorder. We also show that the gradients of θ generate large gate-tunable in-plane electric fields, unscreened even in the metallic regions, which profoundly alter the quantum Hall state by forming edge channels in the bulk of the sample and may affect the phase diagram of the correlated and superconducting states. We thus establish the importance of θ disorder as an unconventional type of disorder enabling the use of twist-angle gradients for bandstructure engineering, for realization of correlated phenomena and for gate-tunable built-in planar electric fields for device applications. SQUID-on-tip tomographic imaging of Landau levels in magic-angle graphene provides nanoscale maps of local twist-angle disorder and shows that its properties are fundamentally different from common types of disorder.

van der Waals heterostructures constitute a new class of artificial materials formed by stacking atomically thin planar crystals. We demonstrated band structure engineering in a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally aligned hexagonal boron nitride substrate. The spatially varying interlayer atomic registry results in both a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. In our samples, this interplay between short-and long-wavelength effects resulted in a band structure described by isolated superlattice minibands and an unexpectedly large band gap at charge neutrality. This picture is confirmed by our observation of fractional quantum Hall states at ±5/3 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.

The electronic properties of graphene depends on how many layers are involved. Monolayer graphene is a zero-gapped semi-metal. Bilayer graphene is a small-gapped semiconductor. Magnetotransport measurements indicate trilayer graphene can be both, depending on its stacking. Graphene 1 , 2 , 3 is an extraordinary two-dimensional (2D) system with chiral charge carriers and fascinating electronic, mechanical and thermal properties 4 , 5 . In multilayer graphene 6 , 7 , stacking order provides an important yet rarely explored degree of freedom for tuning its electronic properties 8 . For instance, Bernal-stacked trilayer graphene (B-TLG) is semi-metallic with a tunable band overlap, and rhombohedral-stacked trilayer graphene (r-TLG) is predicted to be semiconducting with a tunable band gap 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 . These multilayer graphenes are also expected to exhibit rich novel phenomena at low charge densities owing to enhanced electronic interactions and competing symmetries. Here we demonstrate the dramatically different transport properties in TLG with different stacking orders, and the unexpected spontaneous gap opening in charge neutral r-TLG. At the Dirac point, B-TLG remains metallic, whereas r-TLG becomes insulating with an intrinsic interaction-driven gap ∼6 meV. In magnetic fields, well-developed quantum Hall (QH) plateaux in r-TLG split into three branches at higher fields. Such splitting is a signature of the Lifshitz transition, a topological change in the Fermi surface, that is found only in r-TLG. Our results underscore the rich interaction-induced phenomena in trilayer graphene with different stacking orders, and its potential towards electronic applications.

The Hofstadter energy spectrum provides a uniquely tunable system to study emergent topological order in the regime of strong interactions. Previous experiments, however, have been limited to low Bloch band fillings where only the Landau level index plays a role. We report measurements of high-mobility graphene superlattices where the complete unit cell of the Hofstadter spectrum is accessible. We observed coexistence of conventional fractional quantum Hall effect (QHE) states together with the integer QHE states associated with the fractal Hofstadter spectrum. At large magnetic field, we observed signatures of another series of states, which appeared at fractional Bloch filling index. These fractional Bloch band QHE states are not anticipated by existing theoretical pictures and point toward a distinct type of many-body state.

Extended abstract of a paper presented at Microscopy and Microanalysis 2009 in Richmond, Virginia, USA, July 26 – July 30, 2009

Extended abstract of a paper presented at Microscopy and Microanalysis 2009 in Richmond, Virginia, USA, July 26 – July 30, 2009

Extended abstract of a paper presented at Microscopy and Microanalysis 2006 in Chicago, Illinois, USA, July 30 – August 3, 2006