Mapping the twist-angle disorder and Landau levels in magic-angle graphene.

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 demonstrated1-7, 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 state9 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.

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.

Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure.

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.

Hofstadter butterfly and integer quantum hall effect in three dimensions.

For a three-dimensional (3D) lattice in magnetic fields we have shown that the hopping along the third direction, which normally smears out the Landau quantization gaps, can rather give rise to a Hofstadter's butterfly specific to 3D when a criterion is fulfilled by anisotropic (quasi-one-dimensional) systems. In 3D the angle of the magnetic field plays the role of the field intensity in 2D, so that the butterfly can occur in much smaller fields. We have also calculated the Hall conductivity in terms of the topological invariant in the Kohmoto-Halperin-Wu formula, and each of sigma(xy),sigma(zx) is found to be quantized.

Branching ratio and L2 + L3 intensities of 3d-transition metals in phthalocyanines and the amine complexes

L(2,3) inner-shell excitation spectra were obtained by electron energy-loss spectroscopy (EELS) for the divalent first transition series metals in phthalocyanine complexes (MPc) such as titanium oxide phthalocyanine (TiOPc), fluoro-chromium phthalocyanine (CrFPc), manganese phthalocyanine (MnPc), iron phthalocyanine (FePc), cobalt phthalocyanine (CoPc), nickel phthalocyanine (NiPc) and copper phthalocyanine (CuPc). It was found that the value of normalized total intensity of I(L2 + L3) was nearly proportional to the formal electron vacancies of each 3d-state, and the values of the branching ratio, I(L3)/I((L2 + L3), represented a high-spin-state rather than low-spin-state for MnPc, FePc and NiPc. EELS was also applied to charge-transfer complexes of FePc with an amine such as pyridine or gamma-picoline. It was concluded that their I(L2 + L3) intensity of Fe showed the decrease in vacancies of 3d-states on the formation of the charge-transfer complex with these amines, which suggests some electron transfer from the amine to Fe in phthalocyanine. The EELS study provides beneficial information for investigating the electronic states of the specific metal sites in organic materials.

The effect of ionic strength on the lipid peroxidation of porcine intestinal brush-border membrane vesicles.

The effects of salt concentration gradient (inside to outside) on the lipid peroxidation of porcine intestinal brush-border membrane vesicles have been studied and several interesting features of the peroxidation have been elucidated. The addition of dithiothreitol and Fe2+ is far more effective in induction of the lipid peroxidation than any of the other metal ion species tested (Fe3+, Cu2+, Ni2+, Zn2+ and Cr3+). The peroxidation rate of the membrane vesicles induced by dithiothreitol plus Fe2+ was sensitive for the incubation temperature and was increased with increase of the temperature. Imposition of an inward salt concentration gradient on the membrane vesicles preloaded with 300 mM mannitol by addition of 100 mM chloride of K+, Na+, Li+, Rb+, NH4+ or choline to medium produces a very large reduction of the lipid peroxidation induced by dithiothreitol plus Fe2+. The membrane peroxidation is depressed more with the mannitol (300 mM)-preloaded vesicles than with the K2SO4 (100 mM)-preloaded vesicles when they are incubated in medium containing 20-100 mM of K2SO4. Addition of membrane-permeant anions such as SCN- and I-, but not addition of NO3-, to incubation medium has been found to decrease markedly the lipid peroxidation of the mannitol-preloaded vesicles. From these results it is suggested that the lipid peroxidation of the brush-border membranes by addition of dithiothreitol plus Fe2+ is sensitively changed with change in ionic strength.