Pilkyung Moon

Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure

Graphene, Gapped and Butterflied The remarkable transport properties of graphene, such as the high electron mobility, make it a promising material for electronics. However, unlike semiconductors such as silicon, graphenes electronic structure lacks a band gap, and a transistor made out of graphene would not have an “off” state. Hunt et al. (p. 1427, published online 16 May; see the Perspective by Fuhrer) modulated the electronic properties of graphene by building a heterostructure consisting of a graphene flake resting on hexagonal boron nitride (hBN), which has the same honeycomb structure as graphene, but consists of alternating boron and nitrogen atoms instead of carbons. The natural mismatch between the graphene and hBN lattices led to a moire pattern with a large wavelength, causing the opening of a band gap, the formation of an elusive fractional quantum Hall state, and, at high magnetic fields, a fractal phenomenon in the electronic structure called the Hofstadter butterfly. A band gap is observed in a monolayer graphene–hexagonal boron nitride heterostructure. [Also see Perspective by Fuhrer] 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 ±53 filling and features associated with the Hofstadter butterfly at ultrahigh magnetic fields.

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

Optical absorption in twisted bilayer graphene

We theoretically study the optical absorption property of twisted bilayer graphenes with various stacking geometries, and demonstrate that the spectroscopic characteristics serve as a fingerprint to identify the rotation angle between two layers. We find that the absorption spectrum almost continuously evolves in changing the rotation angle, regardless of the lattice commensurability. The spectrum is characterized by series of peaks associated with the van Hove singularity, and the peak energies systematically shift with the rotation angle. We calculate the optical absorption in two different frameworks; the tight-binding model and the effective continuum model based on the Dirac equation. For small rotation angles less than

Energy Spectrum and Quantum Hall Effect in Twisted Bilayer Graphene

10^\circ

Charge Inversion and Topological Phase Transition at a Twist Angle Induced van Hove Singularity of Bilayer Graphene.

, the effective model well reproduces the low-energy band structure and the optical conductivity of the tight-binding model, and also explains the optical selection rule analytically in terms of the symmetry of the effective Hamiltonian.

Complete suppression of large InAs island formation on GaAs by metal organic chemical vapor deposition with periodic AsH3 interruption

We investigate the electronic structure and the quantum Hall effect in twisted bilayer graphenes with various rotation angles in the presence of magnetic field. Using a low-energy approximation, which incorporates the rigorous interlayer interaction, we computed the energy spectrum and the quantized Hall conductivity in a wide range of magnetic field from the semiclassical regime to the fractal spectrum regime. In weak magnetic fields, the low-energy conduction band is quantized into electronlike and holelike Landau levels at energies below and above the van Hove singularity, respectively, and the Hall conductivity sharply drops from positive to negative when the Fermi energy goes through the transition point. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadters butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy. The typical electron density and magnetic field amplitude characterizing the spectrum monotonically decrease as the rotation angle is reduced, indicating that the rich electronic structure may be observed in a moderate condition.

Optical absorption of twisted bilayer graphene with interlayer potential asymmetry

van Hove singularities (VHSs) in the density of states play an outstanding and diverse role for the electronic and thermodynamic properties of crystalline solids. At the critical point the Fermi surface connectivity changes, and topological properties undergo a transition. Opportunities to systematically pass a VHS at the turn of a voltage knob and study its diverse impact are however rare. With the advent of van der Waals heterostructures, control over the atomic registry of neighboring graphene layers offers an unprecedented tool to generate a low energy VHS easily accessible with conventional gating. Here we have addressed magnetotransport when the chemical potential crosses the twist angle induced VHS in twisted bilayer graphene. A topological phase transition is experimentally disclosed in the abrupt conversion of electrons to holes or vice versa, a loss of a nonzero Berry phase and distinct sequences of integer quantum Hall states above and below the singularity.

Optical properties of the Hofstadter butterfly in the moiré superlattice

Self-assembled InAs quantum dots (QDs) on GaAs substrates were grown by metal organic chemical vapor deposition with periodic AsH3 interruption. In contrast to the conventional InAs QD growth method, AsH3 was interrupted periodically while TMIn was introduced into the reactor continuously. By interrupting AsH3 periodically, the growth surface is modulated between As-stabilized surface and In-stabilized one, resulting in complete suppression of relaxed large island formation and significant improvement in photoluminescence intensity. With further optimization of growth parameters, the authors obtained the emission at 1.32μm and narrow linewidth of 32meV at room temperature.

Effects of thin GaAs insertion layer on InAs∕(InGaAs)∕InP(001) quantum dots grown by metalorganic chemical vapor deposition

We investigate the band structure and the optical absorption spectrum of twisted bilayer graphenes with changing interlayer bias and Fermi energy simultaneously. We show that the interlayer bias lifts the degeneracy of the superlattice Dirac point, while the amount of the Dirac point shift is significantly suppressed in small rotation angles, and even becomes opposite to the applied bias. We calculate the optical absorption spectrum in various asymmetric potentials and Fermi energies, and associate the characteristic spectral features with the band structure. The spectroscopic features are highly sensitive to the interlayer bias and the Fermi energy, and widely tunable by the external field effect.

Electronic Properties of Incommensurate Atomic Layers

We investigate the optical absorption spectrum and the selection rule for the Hofstadter butterfly in twisted bilayer graphene under magnetic fields. We demonstrate that the absorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spectrum. We find that the optical selection rule has a nested self-similar structure as well, and it is governed by the conservation of the total angular momentum summed over different hierarchies.