I. V. Grigorieva

Two-Dimensional Gas of Massless Dirac Fermions in Graphene

Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrödinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Diracs (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective ‘speed of light’ c* ≈ 106 m s-1. Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphenes conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass mc of massless carriers in graphene is described by E = mcc*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

Van der Waals heterostructures

Research on graphene and other two-dimensional atomic crystals is intense and is likely to remain one of the leading topics in condensed matter physics and materials science for many years. Looking beyond this field, isolated atomic planes can also be reassembled into designer heterostructures made layer by layer in a precisely chosen sequence. The first, already remarkably complex, such heterostructures (often referred to as ‘van der Waals’) have recently been fabricated and investigated, revealing unusual properties and new phenomena. Here we review this emerging research area and identify possible future directions. With steady improvement in fabrication techniques and using graphene’s springboard, van der Waals heterostructures should develop into a large field of their own.

Unimpeded permeation of water through helium-leak-tight graphene-based membranes

Porous Membranes Thin semi-permeable membranes are commonly used as chemical barriers or for filtration purposes. While the size of the pores will influence which molecules are able to pass, other factors—including the surface chemistry of the pore walls, electrostatic interactions, and differences in solubility—can also affect the diffusion rates. There is also a trade-off between the thickness of the membrane regarding strength and permeation rates (see the Perspective by Paul). Karan et al. (p. 444) fabricated membranes from amorphous carbon, which showed excellent strength and could be used for filtrations involving organic solvents. Nair et al. (p. 442) observed unusual behavior in graphene-based membranes which were able to prevent the diffusion of many small-molecule gases, including helium, but showed almost barrier-free movement of water. Graphite oxide membranes are impermeable to many liquids, vapors, and gases, including He, but allow evaporation of water. Permeation through nanometer pores is important in the design of materials for filtration and separation techniques and because of unusual fundamental behavior arising at the molecular scale. We found that submicrometer-thick membranes made from graphene oxide can be completely impermeable to liquids, vapors, and gases, including helium, but these membranes allow unimpeded permeation of water (H2O permeates through the membranes at least 1010 times faster than He). We attribute these seemingly incompatible observations to a low-friction flow of a monolayer of water through two-dimensional capillaries formed by closely spaced graphene sheets. Diffusion of other molecules is blocked by reversible narrowing of the capillaries in low humidity and/or by their clogging with water.

Precise and Ultrafast Molecular Sieving Through Graphene Oxide Membranes

Graphene oxide membranes allow only very small hydrated molecules and ions to pass with an accelerated transport rate. [Also see Perspective by Mi] Graphene-based materials can have well-defined nanometer pores and can exhibit low frictional water flow inside them, making their properties of interest for filtration and separation. We investigate permeation through micrometer-thick laminates prepared by means of vacuum filtration of graphene oxide suspensions. The laminates are vacuum-tight in the dry state but, if immersed in water, act as molecular sieves, blocking all solutes with hydrated radii larger than 4.5 angstroms. Smaller ions permeate through the membranes at rates thousands of times faster than what is expected for simple diffusion. We believe that this behavior is caused by a network of nanocapillaries that open up in the hydrated state and accept only species that fit in. The anomalously fast permeation is attributed to a capillary-like high pressure acting on ions inside graphene capillaries. On the Fast Track Membranes based on graphene can simultaneously block the passage of very small molecules while allowing the rapid permeation of water. Joshi et al. (p. 752; see the Perspective by Mi) investigated the permeation of ions and neutral molecules through a graphene oxide (GO) membrane in an aqueous solution. Small ions, with hydrated radii smaller than 0.45 nanometers, permeated through the GO membrane several orders of magnitude faster than predicted, based on diffusion theory. Molecular dynamics simulations revealed that the GO membrane can attract a high concentration of small ions into the membrane, which may explain the fast ion transport.

Fluorographene: A Two-Dimensional Counterpart of Teflon

A stoichiometric derivative of graphene with a fluorine atom attached to each carbon is reported. Raman, optical, structural, micromechanical, and transport studies show that the material is qualitatively different from the known graphene-based nonstoichiometric derivatives. Fluorographene is a high-quality insulator (resistivity >10(12) Ω) with an optical gap of 3 eV. It inherits the mechanical strength of graphene, exhibiting a Youngs modulus of 100 N m(-1) and sustaining strains of 15%. Fluorographene is inert and stable up to 400 °C even in air, similar to Teflon.

Cloning of Dirac fermions in graphene superlattices

Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies (see, for example, refs 1, 2, 3, 4, 5, 6, 7, 8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter’s butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate’s moiré potential acts as a superlattice and leads to profound changes in the graphene’s electronic spectrum. Second-generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene’s conduction and valence bands. Strong magnetic fields lead to Zak-type cloning of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.

Spin-half paramagnetism in graphene induced by point defects

T he possibility to induce a magnetic response in graphene by the introduction of defects has been generating much interest, as this would expand the already impressive list of its special properties and allow novel devices where charge and spin manipulation could be combined. So far there have been many theoretical studies (for reviews, see refs 1‐3) predicting that point defects in graphene should carry magnetic moments B and these can in principle couple (anti)ferromagnetically 1‐12 . However, experimental evidence for such magnetism remains both scarce and controversial 13‐16 . Here we show that point defects in graphene—(1) fluorine adatoms in concentrations x gradually increasing to stoichiometric fluorographene CFxD1:0 (ref. 17) and (2) irradiation defects (vacancies)—carry magnetic moments with spin 1=2. Both types of defect lead to notable paramagnetism but no magnetic ordering could be detected down to liquid helium temperatures. The induced paramagnetism dominates graphene’s low-temperature magnetic properties, despite the fact that the maximum response we could achieve was limited to one moment per approximately 1,000 carbon atoms. This limitation is explained by clustering of adatoms and, for the case of vacancies, by the loss of graphene’s structural stability. Our work clarifies the controversial issue of graphene’s magnetism and sets limits for other graphitic compounds. The emerging consensus that magnetism in carbon-based systems can exist is based mostly on a large body of work on magnetic measurements of highly-oriented pyrolytic graphite (HOPG) and carbon films, with many reports of weak ferromagnetic signals at room temperature (T) observed in both pristine HOPG and after itsionirradiation(see,forexample,refs18,19).However,thewhole subject remains controversial, especially concerning (1) the role of possible contamination and (2) the mechanism responsible for the strong interaction required to lead to ferromagnetic ordering at room temperature. Some observations of ferromagnetism are probably artefacts, doing little justice to the subject (one frequent artefact is identified and described in the Supplementary Information, where we show that commonly used HOPG crystals contain micrometre-sized magnetic particles). Adatom magnetism in graphite is also contentious and, for example, different studies of fluorinatedgraphitehavereportedinconsistentresults 20,21 .

Tunable metal-insulator transition in double-layer graphene heterostructures

Disordered conductors with resistivity above the resistance quantum h/e(2) should exhibit an insulating behaviour at low temperatures, a universal phenomenon known as a strong (Anderson) localization(1-3). Observed in a multitude of materials, including damaged graphene and its disordered chemical derivatives(4-10), Anderson localization has not been seen in generic graphene, despite its resistivity near the neutrality point reaching approximate to h/e(2) per carrier type(4,5). It has remained a puzzle why graphene is such an exception. Here we report a strong localization and the corresponding metal-insulator transition in ultra-high-quality graphene. The transition is controlled externally, by changing the carrier density in another graphene layer placed at a distance of several nm and decoupled electrically. The entire behaviour is explained by electron-hole puddles that disallow localization in standard devices but can be screened out in double-layer graphene. The localization that occurs with decreasing rather than increasing disorder is a unique occurrence, and the reported double-layer heterostructures presents a new experimental system that invites further studies.

Detecting topological currents in graphene superlattices.

Making use of graphenes valleys Graphene has two distinct valleys in its electronic structure, in which the electrons have the same energy. Theorists have predicted that creating an asymmetry between the two valleys will coax graphene into exhibiting the so-called valley Hall effect (VHE). In this effect, electrons from the two valleys move across the sample in opposite directions when the experimenters run current along the sample. Gorbachev et al. achieved this asymmetry by aligning graphene with an underlying layer of hexagonalboron nitride (hBN) (see the Perspective by Lundeberg and Folk). The authors measured the transport characteristics of the sample, which were consistent with the theoretical predictions for the VHE. The method may in the future lead to information processing using graphenes valleys. Science, this issue p. 448; see also p. 422 Graphene is aligned with a layer of hexagonal boron nitride to achieve the valley Hall effect. [Also see Perspective by Lundeberg and Folk] Topological materials may exhibit Hall-like currents flowing transversely to the applied electric field even in the absence of a magnetic field. In graphene superlattices, which have broken inversion symmetry, topological currents originating from graphene’s two valleys are predicted to flow in opposite directions and combine to produce long-range charge neutral flow. We observed this effect as a nonlocal voltage at zero magnetic field in a narrow energy range near Dirac points at distances as large as several micrometers away from the nominal current path. Locally, topological currents are comparable in strength with the applied current, indicating large valley-Hall angles. The long-range character of topological currents and their transistor-like control by means of gate voltage can be exploited for information processing based on valley degrees of freedom.

Phase transitions in individual sub-micrometre superconductors

The properties of a superconductor are expected to change radically when its size becomes comparable to that of the Cooper pairs, the quasiparticles responsible for superconductivity. The effect of such confinement is well understood for the case of thesuppression of superconductivity by magnetic fields (which gives rise to so-called Little–Parks oscillations of the phase boundary). But little is known about what happens in small superconductors in the zero-resistance state, which cannot be probed by resistance measurements. Here we apply a new technique of ballistic Hall magnetometry to study the magnetization of individual superconducting discs of diameters down to 100 nm. The superconducting state of these discs is found to be qualitatively different from both macroscopic and microscopic superconductors, with numerous phase transitions whose character changes rapidly with size and temperature. This exotic behaviour is due to size quantization of the Cooper-pair motion and resulting transitions between discrete states of the superconducting Bose condensate in a magnetic field.