Benjamin E. Feldman

Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2

Condensed-matter systems provide a rich setting to realize Dirac and Majorana fermionic excitations as well as the possibility to manipulate them for potential applications. It has recently been proposed that chiral, massless particles known as Weyl fermions can emerge in certain bulk materials or in topological insulator multilayers and give rise to unusual transport properties, such as charge pumping driven by a chiral anomaly. A pair of Weyl fermions protected by crystalline symmetry effectively forming a massless Dirac fermion has been predicted to appear as low-energy excitations in a number of materials termed three-dimensional Dirac semimetals. Here we report scanning tunnelling microscopy measurements at sub-kelvin temperatures and high magnetic fields on the II-V semiconductor Cd3As2. We probe this system down to atomic length scales, and show that defects mostly influence the valence band, consistent with the observation of ultrahigh-mobility carriers in the conduction band. By combining Landau level spectroscopy and quasiparticle interference, we distinguish a large spin-splitting of the conduction band in a magnetic field and its extended Dirac-like dispersion above the expected regime. A model band structure consistent with our experimental findings suggests that for a magnetic field applied along the axis of the Dirac points, Weyl fermions are the low-energy excitations in Cd3As2.

Broken-symmetry states and divergent resistance in suspended bilayer graphene

D 0 state, the devices show extremely high magnetoresistance that scales as magnetic field divided by temperature. This resistance is predominantly affected by the perpendicular component of the applied field, and the extracted energy gap is significantly larger than expected for Zeeman splitting. These findings indicate that the broken-symmetry states arise from manybody interactions and underscore the important part that Coulomb interactions play in bilayer graphene.

Unconventional sequence of fractional quantum Hall states in suspended graphene.

Skipping the Odds When confined to a plane and placed in a magnetic field at low temperatures, electrons are separated by energy into the so-called Landau levels; adding an extra electron after a Landau level that has been filled is costly. In some systems, electron-electron interactions cause the appearance of sublevels, in a phenomenon known as the fractional quantum Hall effect (FQHE). This effect has been observed in graphene, but the number of levels that had been resolved was limited. Feldman et al. (p. 1196) directly measured the change in the chemical potential caused by varying electron density, which is controlled by gate voltage. Once the FQH states were identified, the Landau levels with odd-numerator fractional fillings were found to be missing between filling factors 1 and 2, because of the broken and preserved symmetries of graphene. These observations help to explain how the FQHE in graphene is different from that observed in conventional semiconductors, and the technique will also allow local measurements to be made; hence, monitoring of spatial variations in sample behavior is possible. A scanning single-electron transistor is used to measure the local compressibility of graphene’s electronic states. Graphene provides a rich platform to study many-body effects, owing to its massless chiral charge carriers and the fourfold degeneracy arising from their spin and valley degrees of freedom. We use a scanning single-electron transistor to measure the local electronic compressibility of suspended graphene, and we observed an unusual pattern of incompressible fractional quantum Hall states that follows the standard composite fermion sequence between filling factors ν = 0 and 1 but involves only even-numerator fractions between ν = 1 and 2. We further investigated this surprising hierarchy by extracting the corresponding energy gaps as a function of the magnetic field. The sequence and relative strengths of the fractional quantum Hall states provide insight into the interplay between electronic correlations and the inherent symmetries of graphene.

Electron-hole asymmetric integer and fractional quantum Hall effect in bilayer graphene

Breaking down graphene degeneracy Bilayer graphene has two layers of hexagonally arranged carbon atoms stacked on top of each other in a staggered configuration. This spatial arrangement results in degenerate electronic states: distinct states that have the same energy. Interaction between electrons can cause the states to separate in energy, and so can external fields (see the Perspective by LeRoy and Yankowitz). Kou et al., Lee et al., and Maher et al. used three distinct experimental setups that clarify different parameter regimes of bilayer graphene. Science, this issue p. 55, p. 58, p. 61; see also p. 31 An unusual sequence of electron-interaction–driven states is observed by using a single-electron transistor. [Also see Perspective by LeRoy and Yankowitz] The nature of fractional quantum Hall (FQH) states is determined by the interplay between the Coulomb interaction and the symmetries of the system. The distinct combination of spin, valley, and orbital degeneracies in bilayer graphene is predicted to produce an unusual and tunable sequence of FQH states. Here, we present local electronic compressibility measurements of the FQH effect in the lowest Landau level of bilayer graphene. We observe incompressible FQH states at filling factors ν = 2p + 2/3, with hints of additional states appearing at ν = 2p + 3/5, where p = –2, –1, 0, and 1. This sequence breaks particle-hole symmetry and obeys a ν → ν + 2 symmetry, which highlights the importance of the orbital degeneracy for many-body states in bilayer graphene.

Local Compressibility Measurements of Correlated States in Suspended Bilayer Graphene

Bilayer graphene has attracted considerable interest due to the important role played by many-body effects, particularly at low energies. Here we report local compressibility measurements of a suspended graphene bilayer. We find that the energy gaps at filling factors ν= ± 4 do not vanish at low fields, but instead merge into an incompressible region near the charge neutrality point at zero electric and magnetic field. These results indicate the existence of a zero-field ordered state and are consistent with the formation of either an anomalous quantum Hall state or a nematic phase with broken rotational symmetry. At higher fields, we measure the intrinsic energy gaps of broken-symmetry states at ν=0, ± 1, and ± 2, and find that they scale linearly with magnetic field, yet another manifestation of the strong Coulomb interactions in bilayer graphene.

Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene

Graphene and its multilayers have attracted considerable interest because their fourfold spin and valley degeneracy enables a rich variety of broken-symmetry states arising from electron-electron interactions, and raises the prospect of controlled phase transitions among them. Here we report local electronic compressibility measurements of ultraclean suspended graphene that reveal a multitude of fractional quantum Hall states surrounding filling factors ν=-1/2 and -1/4. Several of these states exhibit phase transitions that indicate abrupt changes in the underlying order, and we observe many additional oscillations in compressibility as ν approaches -1/2, suggesting further changes in spin and/or valley polarization. We use a simple model based on crossing Landau levels of composite fermions with different internal degrees of freedom to explain many qualitative features of the experimental data. Our results add to the diverse array of many-body states observed in graphene and demonstrate substantial control over their order parameters.

Layer-dependent quantum cooperation of electron and hole states in the anomalous semimetal WTe2

The behaviour of electrons and holes in a crystal lattice is a fundamental quantum phenomenon, accounting for a rich variety of material properties. Boosted by the remarkable electronic and physical properties of two-dimensional materials such as graphene and topological insulators, transition metal dichalcogenides have recently received renewed attention. In this context, the anomalous bulk properties of semimetallic WTe2 have attracted considerable interest. Here we report angle- and spin-resolved photoemission spectroscopy of WTe2 single crystals, through which we disentangle the role of W and Te atoms in the formation of the band structure and identify the interplay of charge, spin and orbital degrees of freedom. Supported by first-principles calculations and high-resolution surface topography, we reveal the existence of a layer-dependent behaviour. The balance of electron and hole states is found only when considering at least three Te–W–Te layers, showing that the behaviour of WTe2 is not strictly two dimensional.

High-resolution studies of the Majorana atomic chain platform

High-resolution scanning tunnelling microscopy measurements show that chains of magnetic atoms on the surface of a superconductor provide a promising platform for realizing and manipulating Majorana fermion quasiparticles.

Bulk crystal growth and electronic characterization of the 3D Dirac semimetal Na3Bi

High quality hexagon plate-like Na3Bi crystals with large (001) plane surfaces were grown from a molten Na flux. The freshly cleaved crystals were analyzed by low temperature scanning tunneling microscopy and angle-resolved photoemission spectroscopy, allowing for the characterization of the three-dimensional (3D) Dirac semimetal (TDS) behavior and the observation of the topological surface states. Landau levels were observed, and the energy-momentum relations exhibited a linear dispersion relationship, characteristic of the 3D TDS nature of Na3Bi. In transport measurements on Na3Bi crystals, the linear magnetoresistance and Shubnikov-de Haas quantum oscillations are observed for the first time.

Fractional and integer quantum Hall effects in the zeroth Landau level in graphene

Experiments on the fractional quantized Hall effect in the zeroth Landau level of graphene have revealed some striking differences between filling factors in the ranges 0 1. The effective absence of valley anisotropy for |ν| > 1 means that states with an odd numerator, such as |ν |= 5/ 3o r 7/5, can accommodate charged excitations in the form of large-radius valley skyrmions, which should have a low energy cost and may be easily induced by coupling to impurities. The absence of observed quantum Hall states at these fractions is likely due to the effects of valley skyrmions. For |ν| < 1, the anisotropy terms favor phases in which electrons occupy states with opposite spins, similar to the antiferromagnetic state previously hypothesized to be the ground state at ν = 0. The anisotropy and Zeeman energies suppress large-area skyrmions, so that quantized Hall states can be observable at a set of fractions similar to those in GaAs two-dimensional electron systems. In a perpendicular magnetic field B, the competition between the Coulomb energy, which varies as B 1/2 , and the Zeeman energy, which varies as B, can explain the observation of apparent phase transitions as a function of B for fixed ν, as transitions between states with different degrees of spin polarization. In addition to an analysis of various fractional states from this point of view and an examination of the effects of disorder on valley skyrmions, we present new experimental data for the energy gaps at integer fillings ν = 0a ndν =− 1, as a function of magnetic field, and we examine the possibility that valley skyrmions may account for the smaller energy gaps observed at ν =− 1.