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. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, 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. 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.

When electrons are confined in two dimensions (2D) and subjected to strong magnetic fields, the Coulomb interactions between them become dominant and can lead to novel states of matter such as fractional quantum Hall (FQH) liquids 1 . In these liquids electrons linked to magnetic flux quanta form complex composite quasipartices, which are manifested in the quantization of the Hall conductivity as rational fractions of the conductance quantum. The recent experimental discovery of an anomalous integer quantum Hall effect in graphene has opened up a new avenue in the study of correlated 2D electronic systems, in which the interacting electron wavefunctions are those of massless chiral fermions 2,3 . However, due to the prevailing disorder, graphene has thus far exhibited only weak signatures of correlated electron phenomena 4,5 , despite concerted experimental efforts and intense theoretical interest 6-12 . Here, we report the observation of the fractional quantum Hall effect in ultraclean suspended graphene, supporting the existence of strongly correlated electron states in the presence of a magnetic field. In addition, at low carrier density graphene becomes an insulator with an energy gap tunable by magnetic field. These newly discovered quantum states offer the opportunity to study a new state of matter of strongly correlated Dirac fermions in the presence of large magnetic fields.In a perpendicular magnetic field, the energy spectrum of a clean two-dimensional electron system (2DES) splits into a fan of Landau levels (LLs). When the Fermi energy is tuned to lie between the LLs, the system enters the integer quantum Hall (IQH) regime, in which current is carried by states at the edge of the sample and the overall conductance is quantized as G=νe 2 /h, where ν is an integer Landau level filling factor. Already the first observation of the IQH effect in graphene demonstrated an unusual sequence of filling factors ν=±2,±6,±10..., which differs from previously studied 2DESs. This sequence originates from two peculiar features of graphene: the four-fold spin and pseudo-spin (valley) degeneracy of LLs and the existence of a non-trivial Berry phase associated with the pseudo-spin of Dirac quasiparticles 13 .In clean samples and under very strong magnetic fields, additional IQH states emerge at filling factors ν=0, ±1 4,5 . These fragile IQH states are conjectured to result from electron-electron (e-e) interactions lifting the pseudospin and spin degeneracy of the zeroth LL 14 . The nature of these states, and in particular the unusual ν=0 state, responsible for the divergent resistivity of graphene at high magnetic fields 15 , has raised considerable interest in the effect of e-e interactions among Dirac quasiparticles. In conventional semiconductor heterojunctions such correlation effects are spectacularly manifested in the FQH effect, where new electronic ground states are formed in which the elementary excitations are composite particles with fractional charge 16 . The possibility of a FQH effect in graphene, and t...

Graphene produced by chemical vapor deposition (CVD) is polycrystalline, and scattering of charge carriers at grain boundaries (GBs) could degrade its performance relative to exfoliated, single-crystal graphene. However, the electrical properties of GBs have so far been addressed indirectly without simultaneous knowledge of their locations and structures. We present electrical measurements on individual GBs in CVD graphene first imaged by transmission electron microscopy. Unexpectedly, the electrical conductance improves by one order of magnitude for GBs with better interdomain connectivity. Our study suggests that polycrystalline graphene with good stitching may allow for uniformly high electrical performance rivaling that of exfoliated samples, which we demonstrate using optimized growth conditions and device geometry.

We report the enhancement of the thermoelectric power (TEP) in graphene with extremely low disorder. At high temperature we observe that the TEP is substantially larger than the prediction of the Mott relation, approaching to the hydrodynamic limit due to strong inelastic scattering among the charge carriers. However, closer to room temperature the inelastic carrier-optical-phonon scattering becomes more significant and limits the TEP below the hydrodynamic prediction. We support our observation by employing a Boltzmann theory incorporating disorder, electron interactions, and optical phonons.

The phase of a quantum state may not return to its original value after the system's parameters cycle around a closed path; instead, the wavefunction may acquire a measurable phase difference called the Berry phase. Berry phases typically have been accessed through interference experiments. Here, we demonstrate an unusual Berry-phase-induced spectroscopic feature: a sudden and large increase in the energy of angular-momentum states in circular graphene p-n junction resonators when a small critical magnetic field is reached. This behavior results from turning on a π-Berry phase associated with the topological properties of Dirac fermions in graphene. The Berry phase can be switched on and off with small magnetic field changes on the order of 10 mT, potentially enabling a variety of optoelectronic graphene device applications.