This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

Fabrication techniques developed for graphene research allow the disassembly of many layered crystals (so-called van der Waals materials) into individual atomic planes and their reassembly into designer heterostructures, which reveal new properties and phenomena. Andre Geim and Irina Grigorieva offer a forward-looking review of the potential of layering two-dimensional materials into novel heterostructures held together by weak van der Waals interactions. Dozens of these one-atom- or one-molecule-thick crystals are known. Graphene has already been well studied but others, such as monolayers of hexagonal boron nitride, MoS2, WSe2, graphane, fluorographene, mica and silicene are attracting increasing interest. There are many other monolayers yet to be examined of course, and the possibility of combining graphene with other crystals adds even further options, offering exciting new opportunities for scientific exploration and technological innovation. 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.

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Van der Waals heterostructures

Graphene’s success has shown that it is possible to create stable, single and few-atom-thick layers of van der Waals materials, and also that these materials can exhibit fascinating and technologically useful properties. Here we review the state-of-the-art of 2D materials beyond graphene. Initially, we will outline the different chemical classes of 2D materials and discuss the various strategies to prepare single-layer, few-layer, and multilayer assembly materials in solution, on substrates, and on the wafer scale. Additionally, we present an experimental guide for identifying and characterizing single-layer-thick materials, as well as outlining emerging techniques that yield both local and global information. We describe the differences that occur in the electronic structure between the bulk and the single layer and discuss various methods of tuning their electronic properties by manipulating the surface. Finally, we highlight the properties and advantages of single-, few-, and many-layer 2D materials in...

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Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene

Since the discovery of mechanically exfoliated graphene in 2004, research on ultrathin two-dimensional (2D) nanomaterials has grown exponentially in the fields of condensed matter physics, material science, chemistry, and nanotechnology. Highlighting their compelling physical, chemical, electronic, and optical properties, as well as their various potential applications, in this Review, we summarize the state-of-art progress on the ultrathin 2D nanomaterials with a particular emphasis on their recent advances. First, we introduce the unique advances on ultrathin 2D nanomaterials, followed by the description of their composition and crystal structures. The assortments of their synthetic methods are then summarized, including insights on their advantages and limitations, alongside some recommendations on suitable characterization techniques. We also discuss in detail the utilization of these ultrathin 2D nanomaterials for wide ranges of potential applications among the electronics/optoelectronics, electrocat...

Recent Advances in Ultrathin Two-Dimensional Nanomaterials

A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked two-dimensional building blocks held together by the van der Waals forces between the layers. This design means that the properties of van der Waals heterostructures can be engineered precisely, even more so than those of two-dimensional materials. One such property is the 'twist' angle between different layers in the heterostructure. This angle has a crucial role in the electronic properties of van der Waals heterostructures, but does not have a direct analogue in other types of heterostructure, such as semiconductors grown using molecular beam epitaxy. For small twist angles, the moire pattern that is produced by the lattice misorientation between the two-dimensional layers creates long-range modulation of the stacking order. So far, studies of the effects of the twist angle in van der Waals heterostructures have concentrated mostly on heterostructures consisting of monolayer graphene on top of hexagonal boron nitride, which exhibit relatively weak interlayer interaction owing to the large bandgap in hexagonal boron nitride. Here we study a heterostructure consisting of bilayer graphene, in which the two graphene layers are twisted relative to each other by a certain angle. We show experimentally that, as predicted theoretically, when this angle is close to the 'magic' angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling. These flat bands exhibit insulating states at half-filling, which are not expected in the absence of correlations between electrons. We show that these correlated states at half-filling are consistent with Mott-like insulator states, which can arise from electrons being localized in the superlattice that is induced by the moire pattern. These properties of magic-angle-twisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic many-body quantum phases in two dimensions in the absence of a magnetic field. The accessibility of the flat bands through electrical tunability and the bandwidth tunability through the twist angle could pave the way towards more exotic correlated systems, such as unconventional superconductors and quantum spin liquids.

Correlated insulator behaviour at half-filling in magic-angle graphene superlattices

When a Van Hove singularity exists near the Fermi energy of a solid’s density of states, it can cause a variety of exotic phenomena to emerge. Scanning tunnelling microscope measurements indicate that when graphite’s graphene sheets are rotated out of their usual alignment, it can generate low-energy Van Hove singularities for which the position is controlled by the angle of rotation.

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Observation of Van Hove singularities in twisted graphene layers

We report high magnetic field scanning tunneling microscopy and Landau level spectroscopy of twisted graphene layers grown by chemical vapor deposition For twist angles exceeding ~3° the low energy carriers exhibit Landau level spectra characteristic of massless Dirac fermions Above 20° the layers effectively decouple and the electronic properties are indistinguishable from those in single-layer graphene, while for smaller angles we observe a slowdown of the carrier velocity which is strongly angle dependent At the smallest angles the spectra are dominated by twist-induced van Hove singularities and the Dirac fermions eventually become localized An unexpected electron-hole asymmetry is observed which is substantially larger than the asymmetry in either single or untwisted bilayer graphene

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Single-Layer Behavior and Its Breakdown in Twisted Graphene Layers

We report low temperature high magnetic field scanning tunneling microscopy and spectroscopy of graphene flakes on graphite that exhibit the structural and electronic properties of graphene decoupled from the substrate. Pronounced peaks in the tunneling spectra develop with increasing field revealing a Landau level sequence that provides a direct way to identify graphene and to determine the degree of its coupling to the substrate. The Fermi velocity and quasiparticle lifetime, obtained from the positions and width of the peaks, provide access to the electron-phonon and electron-electron interactions.

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Scanning tunneling spectroscopy of graphene on graphite.

We report high magnetic field scanning tunneling microscopy and Landau level spectroscopy of twisted graphene layers grown by chemical vapor deposition. For twist angles exceeding ~3° the low energy carriers exhibit Landau level spectra characteristic of massless Dirac fermions. Above 20° the layers effectively decouple and the electronic properties are indistinguishable from those in single-layer graphene, while for smaller angles we observe a slowdown of the carrier velocity which is strongly angle dependent. At the smallest angles the spectra are dominated by twist-induced van Hove singularities and the Dirac fermions eventually become localized. An unexpected electron-hole asymmetry is observed which is substantially larger than the asymmetry in either single or untwisted bilayer graphene.

The unique electronic behaviour of monolayer and bilayer graphene1,2 is a result of the unusual quantum-relativistic characteristics of the so-called ‘Dirac fermions’ (DFs) that carry charge in these materials. Although DFs in monolayer graphene move as if they were massless, and in bilayer graphene they do so with non-zero mass, all DFs show chirality, which gives rise to an unusual Landau level (LL) energy spectrum3,4,5,6,7,8,9,10,11 and the observation of an anomalous quantum Hall effect in both types of graphene4,5,8. Here we report low-temperature scanning tunnelling spectra of graphite subjected to a magnetic field of up to 12 T, which provide the first direct observations of the LLs that produce such behaviour. Unexpectedly, we find evidence for the coexistence of both massless and massive DFs in graphite, and confirm the quantum-relativistic nature of these quasiparticles through the appearance of a zero-energy LL.

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Guohong Li

Papers

Observation of Van Hove singularities in twisted graphene layers

Single-Layer Behavior and Its Breakdown in Twisted Graphene Layers

Scanning tunneling spectroscopy of graphene on graphite.

Single-Layer Behavior and Its Breakdown in Twisted Graphene Layers

Observation of Landau levels of Dirac fermions in graphite