Fereshte Ghahari

Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices

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.C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim Department of Electrical Engineering, Columbia University, New York, NY Department of Mechanical Engineering, Columbia University, New York, NY Department of Physics, Columbia University, New York, NY Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, FL 5 Department of Physics, Tohoku University, Sendai, Japan and National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan

Observation of the fractional quantum Hall effect in graphene

When electrons are confined in two dimensions and subject to strong magnetic fields, the Coulomb interactions between them can become very strong, leading to the formation of correlated states of matter, such as the fractional quantum Hall liquid. In this strong quantum regime, electrons and magnetic flux quanta bind to form complex composite quasiparticles with fractional electronic charge; these are manifest in transport measurements of the Hall conductivity as rational fractions of the elementary conductance quantum. The experimental discovery of an anomalous integer quantum Hall effect in graphene has enabled the study of a correlated two-dimensional electronic system, in which the interacting electrons behave like massless chiral fermions. However, owing to the prevailing disorder, graphene has so far exhibited only weak signatures of correlated electron phenomena, despite intense experimental and theoretical efforts. Here we report the observation of the fractional quantum Hall effect in ultraclean, suspended graphene. In addition, we show that at low carrier density graphene becomes an insulator with a magnetic-field-tunable energy gap. These newly discovered quantum states offer the opportunity to study correlated Dirac fermions in graphene in the presence of large magnetic fields.

Tailoring Electrical Transport Across Grain Boundaries in Polycrystalline Graphene

Going Up Against the Grain Boundaries Exfoliated graphene sheets are single crystals that exhibit excellent electronic properties, but their fabrication is too slow for large-scale device fabrication. Growth methods such as chemical vapor deposition are faster, but create polycrystalline graphene sheets that contain grain boundaries that can scatter charge carriers and decrease performance. Tsen et al. (p. 1143) found that the presence of overlapping domains within polycrystalline graphene samples could increase conductivity of samples by an order of magnitude, allowing them to rival exfoliated samples. Overlap between crystallites in vapor-grown graphene improves electronic conductivity. 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.

Measurement of the ν = 1 / 3 Fractional Quantum Hall Energy Gap in Suspended Graphene

We report on magnetotransport measurements of multiterminal suspended graphene devices. Fully developed integer quantum Hall states appear in magnetic fields as low as 2 T. At higher fields the formation of longitudinal resistance minima and transverse resistance plateaus are seen corresponding to fractional quantum Hall states, most strongly for ν=1/3. By measuring the temperature dependence of these resistance minima, the energy gap for the 1/3 fractional state in graphene is determined to be at ∼20  K at 14 T.

Enhanced Thermoelectric Power in Graphene: Violation of the Mott Relation by Inelastic Scattering

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.

An on/off Berry phase switch in circular graphene resonators

Flicking the Berry phase switch When an electron completes a cycle around the Dirac point (a particular location in graphenes electronic structure), the phase of its wave function changes by π. This so-called Berry phase is tricky to observe directly in solid-state measurements. Ghahari et al. built a graphene nanostructure consisting of a central region doped with positive carriers surrounded by a negatively doped background. Scanning tunneling spectroscopy revealed sudden jumps in conductivity as the external magnetic field was increased past a threshold value. The jumps occurred when electron orbits started encompassing the Dirac point, reflecting the switch of the Berry phase from zero to π. The tunability of conductivity by such minute changes in magnetic field is promising for future applications. Science, this issue p. 845 Scanning tunneling spectroscopy reveals a transition in the character of electron orbits in a graphene nanostructure. 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 wave function 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 relatively 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 millitesla, potentially enabling a variety of optoelectronic graphene device applications.

Electric field effects in graphene/LaAlO3/SrTiO3 heterostructures and nanostructures

We report the development and characterization of graphene/LaAlO3/SrTiO3 heterostructures. Complex-oxide heterostructures are created by pulsed laser deposition and are integrated with graphene using both mechanical exfoliation and transfer from chemical-vapor deposition on ultraflat copper substrates. Nanoscale control of the metal-insulator transition at the LaAlO3/SrTiO3 interface, achieved using conductive atomic force microscope lithography, is demonstrated to be possible through the graphene layer. LaAlO3/SrTiO3-based electric field effects using a graphene top gate are also demonstrated. The ability to create functional field-effect devices provides the potential of graphene-complex-oxide heterostructures for scientific and technological advancement.