# van der Waals heterostructures combining graphene and hexagonal boron nitride

## Abstract

As the first in a large family of 2D van der Waals (vdW) materials, graphene has attracted enormous attention owing to its remarkable properties. The recent development of simple experimental techniques for combining graphene with other atomically thin vdW crystals to form heterostructures has enabled the exploration of the properties of these so-called vdW heterostructures. Hexagonal boron nitride is the second most popular vdW material after graphene, owing to the new physics and device properties of vdW heterostructures combining the two. Hexagonal boron nitride can act as a featureless dielectric substrate for graphene, enabling devices with ultralow disorder that allow access to the intrinsic physics of graphene, such as the integer and fractional quantum Hall effects. Additionally, under certain circumstances, hexagonal boron nitride can modify the optical and electronic properties of graphene in new ways, inducing the appearance of secondary Dirac points or driving new plasmonic states. Integrating other vdW materials into these heterostructures and tuning their new degrees of freedom, such as the relative rotation between crystals and their interlayer spacing, provide a path for engineering and manipulating nearly limitless new physics and device properties.

## Key points

• Atomically thin flakes of van der Waals materials such as graphene and hexagonal boron nitride (hBN) can be mixed and matched into heterostructures with fundamentally new optoelectronic properties.

• Graphene encapsulated in hBN has very high mobility, with very low charge carrier inhomogeneity and ballistic transport characteristics over micrometre length scales at low temperature.

• High-mobility graphene devices exhibit well-developed multicomponent integer and fractional quantum Hall effects, as well as additional exotic correlated electronic phases in a magnetic field.

• When the graphene and hBN crystals are rotationally aligned, a long-wavelength moiré superlattice emerges, which creates new, finite-energy Dirac points in the graphene bandstructure and leads to the Hofstadter butterfly spectrum.

• Graphene–hBN heterostructures host new hybrid polaritons, as well as plasmonic excitations with exceptionally long lifetimes that can be tuned with a moiré superlattice.

## Introduction

The isolation of atomically thin crystals in 2004 (Ref.1) has driven intense research efforts into the physics of 2D materials. Graphene has proved to be the most popular 2D material, but over 100 different materials can in principle be similarly isolated into atomic monolayers. These so-called van der Waals (vdW) materials are characterized by strong interatomic bonds within the 2D plane and weak vdW interactions between the layers. Even in the monolayer limit, this family of materials spans a wide spectrum of electronic behaviours: graphene is a semimetal2 (Box 1), hexagonal boron nitride (hBN; Box 1) is a wide-gap insulator3, MoS2 is a semiconductor4, NbSe2 is a superconductor5 (and it exhibits charge-density waves6), WTe2 is a quantum spin Hall insulator7,8,9 and CrI3 is a ferromagnet10.

Two-dimensional vdW materials are of great interest on their own, but their true potential lies in the possibility of mixing and matching atomically thin layers of various crystals to form a heterostructure. These composite materials can be assembled with on-demand geometries without the typical interfacial lattice-matching constraints encountered in the layer-by-layer epitaxial growth of conventional heterostructures. A great variety of new optoelectronic device properties and physical phenomena has emerged as a consequence of this ability to create designer heterostructures of various 2D materials, with the promise of much more to come. Additionally, vdW heterostructures offer entirely new mechanisms for tuning emergent device properties resulting from interfacial interactions between the stacked materials. For example, the relative rotation between graphene crystals11,12,13,14 and their interlayer spacing, which sets the strength of interlayer electronic interactions15, can be used as experimental knobs to control material properties.

The rise of vdW heterostructures also addressed a critical challenge facing the field of 2D materials: the poor electronic quality of 2D materials resting on amorphous SiO2 substrates. Because 2D materials are entirely composed of exposed surfaces, environmental disorder couples strongly to the material and degrades its electronic mobility. Although suspending graphene by selective etching of the SiO2 substrate was found to solve this problem16,17, the suspended devices were extremely fragile, could not be doped to high carrier densities and suffered from substantial strain. Hence, a more flexible substrate-supported geometry was needed. Foundational work in vdW heterostructures demonstrated that the quality of graphene devices was considerably enhanced by the addition of an hBN substrate18. More recently, graphene devices of electronic properties comparable to those of suspended devices have been fabricated by fully encapsulating them with hBN and then graphite. Graphite screens environmental disorder but also acts as a gate electrode to tune the carrier density and electrical displacement field. Although numerous 2D vdW materials have benefited from this encapsulation technique, in this Review, we restrict our attention to vdW heterostructures composed of graphene and hBN. We discuss in particular the physical phenomena that can be observed in graphene–hBN heterostructures thanks to the very high quality of the devices, ranging from the quantum Hall effect (QHE) and fractional QHE (FQHE) to the emergence of more exotic correlated states. In addition to acting as a passive dielectric substrate, hBN can be used as an active component of the heterostructure, as it forms a moiré superlattice when aligned with graphene. In this case, secondary Dirac points arise, accompanied by the appearance of the Hofstadter butterfly spectrum, and plasmon and phonon polaritons with exceptional properties can be observed.

## Improving the quality of graphene devices

The first graphene devices were fabricated directly on silicon oxide substrates, because identifying graphene monolayers on SiO2 using optical contrast measurements is relatively easy1. However, these devices suffered from low mobility and high charge disorder, and eventually it became clear through scanning probe microscopy measurements that the substrate was an important source of disorder owing to charged impurities native to the oxide19,20,21. Furthermore, even without impurities, the surface phonons in SiO2 set a rather low upper limit on the mobility of graphene22. These issues sparked the search for a new substrate for graphene, which would ideally be an atomically flat insulator free of dangling bonds, with minimal charged impurities and a large surface phonon energy. In 2010, hBN was identified18 as a candidate material meeting these criteria.

However, placing graphene on hBN substrates proved challenging, because isolated graphene and hBN flakes tend to be much less than 100 μm wide and are quite fragile owing to their atomic thinness. In the first demonstration of the use of hBN as a substrate for graphene, a water-based transfer method was used, in which flakes were cleaved directly onto sacrificial polymers and aligned onto one another under an optical microscope. These initial graphene–hBN devices showed an increase in mobility of roughly one order of magnitude compared with the mobility of the best devices on SiO2 (Ref.18). They also showed a very narrow resistance peak as a function of carrier density, implying that they had relatively small charge inhomogeneity. However, these devices still displayed obvious signatures of disorder owing to contaminants trapped between the layers during the transfer process.

More recently, a transfer process in which flakes are assembled into an arbitrary heterostructure using a dry, contamination-free transfer procedure was developed23 (Fig. 1a). Graphite and hBN crystals are separately exfoliated on SiO2 substrates, and suitable flakes are identified optically24,25. A polypropylene carbonate (PPC) polymer on a polydimethylsiloxane (PDMS) elastomer stamp is then used to peel an hBN flake directly off the substrate under an optical microscope. Next, the hBN is positioned over a graphene flake and brought into contact with it while the whole system is heated to promote adhesion between the flakes. The two flakes develop a strong vdW adhesion, and, as a result, the graphene is peeled off its substrate as the elastomer stamp is removed. A second, encapsulating hBN layer can then be added to the stack following the same procedure. The completed stack of flakes is deposited on a pristine SiO2 wafer for device processing (Fig. 1b). Selective ion etching using CHF3 and O2 plasma is typically used to shape the heterostructure as desired and to expose a 1D graphene edge that can be electrically contacted with standard metal deposition techniques. Surprisingly, these 1D edge contacts have substantially lower contact resistance than standard 2D surface contacts23.

This technique was initially developed for hBN-encapsulated graphene devices, but it has been equally successful for building more intricate and varied heterostructures and device geometries of arbitrary vdW materials, with the possibility of including a large number of different flakes in the stack26. Encapsulation with hBN results in extremely clean graphene devices exhibiting ballistic transport over tens of micrometres23,27,28. This improvement arises naturally because no surface of any flake in the heterostructure (except the outer surface of the top hBN) is exposed to polymers during the transfer procedure, limiting the intrinsic disorder in the channel. As a secondary benefit, remnant hydrocarbon contamination from the exfoliation procedure tends to self-aggregate into 'bubbles' at the interface between many pairs of 2D materials (including graphene and hBN), which can be squeezed out of the channel during the transfer procedure29,30. This leaves a pristine, contamination-free interface between graphene and hBN (Fig. 1c). State-of-the-art devices exhibit signatures of ultralow disorder with charge carrier fluctuations of ~5 × 109 cm−2, ballistic transport at zero magnetic field and a well-resolved QHE response in high magnetic fields.

The dry transfer procedure has proved to be a remarkably successful and adaptable method for fabricating vdW heterostructures with on-demand geometries. With careful lamination, individual flakes can be added to the stack in well-controlled locations and with a rotational accuracy that is better than 1° (Refs31,32,33). Air-sensitive vdW materials, such as NbSe2, can be encapsulated between flakes of hBN in a nitrogen glove box, which protects them against degradation caused by exposure to air34. Flakes of graphite can then be used to encapsulate the entire heterostructure on the top and bottom, providing screening from the outside environment but also acting as top and bottom gates to electrostatically dope the active channel in the stack and tune the perpendicular displacement field35. Recently, it was realized that using graphite flakes as top and bottom gates substantially improves the device quality compared with using metal gates35, probably because of their atomic flatness, lack of atomic-scale grains and near-work-function matching with graphene. The remaining disorder in these heterostructures likely arises from intrinsic defects in the hBN crystals.

It is interesting to compare structures of graphene encapsulated with hBN and gated with graphite on the top and bottom surfaces (and comparable variants) and conventional 2D electron gases in semiconducting quantum well heterostructures. In the latter, the various dielectric and channel layers are sequentially grown epitaxially in a single vacuum environment, and electrical contacts to the channel are typically made by diffusing metal into the quantum well. Although this vacuum growth procedure protects the heterostructure from environmental contamination, it inherently limits the variety of device geometries that can be realized. The layer-by-layer assembly technique used for vdW heterostructures analogously buries the channel, protecting it from the environment. Contacts are made either by exposing the 1D channel edges with a plasma etch23, by using graphite flakes36 or by embedding metal into flakes of hBN to make 2D interfacial contacts37, which are often useful for contacting air-sensitive materials. The vdW stacking technique also makes it possible to mix and match different crystals that cannot be grown epitaxially owing to interfacial constraints. For example, interfaces between graphene and superconductors, semiconductors with strong spin–orbit coupling or magnetic insulators can easily be integrated into vdW heterostructures. This stacking method additionally offers flexibility in the design of the shape, position and rotational alignment of individual crystals, allowing for considerable variations in device properties even among structures composed of identical materials.

## Atomic-scale device characterization

Scanning probe microscopy has proved to be a powerful tool for characterizing graphene–hBN devices on the atomic scale, yielding sensitive measures of both the surface roughness and the spatial charge inhomogeneity. Because graphene is a single atomic monolayer, it tends to conform to its substrate38, leading to a surface roughness of ~0.5 nm for graphene–SiO2 owing to the amorphous nature of the oxide. By contrast, the surface roughness of graphene on hBN is greatly reduced — typically less than 50 pm (Refs39,40) — which helps to substantially enhance the mobility of graphene. The use of hBN as a substrate also enables new scanning probe microscopy experiments owing to the flatness of the substrate (see Ref.41 for a related review).

hBN substrates have a dramatic effect on the characteristic charge inhomogeneity in graphene. Local charge fluctuations in graphene devices can be imaged using scanning tunnelling spectroscopy by measuring the relative energy shift of the Dirac point (the charge neutrality point) as a function of position in the device (Fig. 2). Typical graphene–SiO2 devices have charge fluctuations on the order of 1012 cm−2 owing to trapped charges in the oxide layer19,20,21 (Fig. 2a). Because hBN can be grown as a high-quality crystal42 rather than an amorphous material like SiO2, the number of charge traps it hosts is typically orders of magnitude lower43 than in SiO2. Thus, adding a layer of hBN isolates the graphene from the SiO2, increasing the distance between the charged impurities and the graphene, such that the influence of the impurities is reduced and the size of the resulting charge puddles increases (Fig. 2b). By using a layer of graphite under the hBN, the charged impurities in the oxide are further screened, and the charge fluctuations are suppressed even more efficiently (Fig. 2c). With these improved device designs, charge fluctuations have been reduced by at least three orders of magnitude compared with those of early graphene–SiO2 devices, enabling the experimental probing of intrinsic properties of graphene that were previously obscured by disorder.

## Quantum Hall effect in graphene

In a magnetic field, the band structure of a 2D material develops into a series of discrete, highly degenerate energy levels known as Landau levels (LLs), which gives rise to the integer QHE (IQHE). Starting from the linear band structure of graphene, the energy dispersion of the LLs can be calculated as $${E}_{N}={\rm{sgn}}\left(N\right)\sqrt{(2e\hslash {v}_{{\rm{F}}}^{2}NB)}$$, where e is the charge of the electron, vF is the Fermi velocity, $$\hslash$$ is Planck’s reduced constant, B is the magnetic field and N is an integer representing the LL index. This Landau quantization is unique to graphene and is a result of its linear band dispersion; typical III–V quantum wells have E k2 dispersions, which result in a linear (EN NB) LL sequence. Additionally, the valley degree of freedom in graphene adds an extra twofold degeneracy, whereas in most materials, the degeneracy of each LL is multiplied by a factor of two owing to the spin degree of freedom. The π Berry curvature in graphene offsets the LL energy spectrum such that a shared electron–hole LL exists at the Dirac point (that is, at zero energy). Assuming no lifting of the fourfold isospin degeneracy, the Hall conductivity of graphene is $${\sigma }_{xy}=\pm 4\frac{{e}^{2}}{h}(N+{\rm{1/2}})$$. Early measurements of graphene–SiO2 already confirmed this unique quantization sequence of the IQHE44,45.

Although the most robust IQH states in graphene arise at filling factors v = ±2, ±6, ±10 and so on (so-called main sequence states), high-quality devices exhibit states at every integer v (Fig. 3a,b). States outside the main sequence arise from breaking the spin and/or valley degeneracies. Experimental measurements of the energy gaps of IQH states reveal a non-trivial competition between various isospin-polarized ground states46. In a single-particle picture, the energy of the spin Zeeman effect can be written as EZ,s = gsBB, where μB is the Bohr magneton, $$s=\pm \frac{1}{2}$$ is the spin quantum number and gs = 2 is the spin g-factor. The cyclotron energy is over three orders of magnitude larger than EZ,s owing to the large Fermi velocity of graphene (inset in Fig. 3b); thus, the relatively large energy gaps experimentally observed outside the main sequence when the spin degeneracy is lifted are surprising. A similar Zeeman-like energy can be used to describe the coupling of the magnetic field to the valley degree of freedom as EZ,v = gvτμBB, where τ ± 1 is the valley index and gv is the valley g-factor. In the single-particle picture, gv = 0 owing to the perfect symmetry between the graphene sublattices, which makes it even more surprising to observe full four-fold symmetry breaking at any magnetic field. Experimental measurements of the spin and valley g-factors revealed that they are substantially larger than the single-particle predictions46, indicating that the IQH states observed outside the main sequence are ferromagnetic; they arise primarily owing to spontaneous breaking of the isospin symmetries owing to many-body exchange interactions.

The spin and valley polarization of the broken-symmetry IQH states can be inferred experimentally by applying an additional in-plane magnetic field46, which enhances the spin Zeeman energy but does not modify the valley Zeeman energy owing to atomic-scale confinement of the electronic wavefunctions to the graphene plane. Whereas LLs are spin-polarized at half-filling for all $$N\ne 0$$, a valley-textured canted antiferromagnet is favoured at N = 0. This result arises from a special feature of the N = 0 LL wavefunctions of graphene, in which the valley degree of freedom corresponds exactly to the real-space sublattice degree of freedom, tuning the competition between exchange interactions and favouring valley (lattice) polarization. By contrast, in the case of $$N\ne 0$$, because all wavefunctions are equally weighted on the A and B sublattices, the exchange interactions favour spontaneous spin polarization at half-filling.

A critical feature of the IQHE is as follows: whereas the bulk of the 2D material is insulating within a LL gap, chiral edge modes appear at the boundary of the sample. Their number corresponds to the Hall conductance σxy in units of the fundamental conductance quantum $$\frac{{e}^{2}}{h}$$. Remarkably, graphene-based heterostructures have also been shown to host helical edge modes, in which counter-propagating edge modes lie on top of one another but are protected from backscattering by their opposite spin polarization. Such an edge mode configuration results in the so-called quantum spin Hall effect. Edge modes in both monolayer and bilayer graphene can be tuned dynamically from chiral to helical in the N = 0 LL by adding an in-plane component to the magnetic field47,48. This drives the ground state from a canted antiferromagnet, which acts as both a bulk and an edge insulator, to a spin ferromagnet, in which edge modes of opposite spins and direction become degenerate at the boundaries of the sample. Proposals for realizing novel Majorana and parafermionic bound modes in topological superconductors could be tested by coupling these graphene-based systems exhibiting the quantum spin Hall effect with superconducting electrodes. These devices may have exciting potential applications in fault-tolerant quantum computation49,50.

## Fractional quantum Hall effect in graphene

In partially filled LLs, interactions between electrons can drive exotic excitations with fractional charge at certain rational filling factors. At exactly half-filling of the lowest LL, the quasiparticles can be pictured as composite fermions composed of a single electron coupled to two quanta of magnetic flux51. This flux pinning effectively resets the magnetic field of the composite fermion to zero, simplifying the system of highly interacting electrons to one of weakly interacting quasiparticles. The FQHE — marked by plateaux of the Hall conductivity at fractional values of $$\frac{{e}^{2}}{h}$$ — can be viewed as analogous to the IQHE but arises because of the integer quantization of the energy spectrum of the composite fermions that happens when the effective magnetic field of the composite fermions is tuned away from zero. FQH states emerge at filling factors $$\nu =\frac{n}{2pn\pm 1}$$, where n and p are non-zero integers; n is analogous to the IQHE filling factor, and 2p describes the number of flux quanta attached to each electron. Whereas the IQHE is a single-particle phenomenon, the FQHE arises owing to strong correlations between electrons.

The FQHE was not observed in graphene–SiO2 devices owing to the large disorder in these samples, and although it can be observed in transport measurements on suspended graphene devices52,53, strain in the graphene lattice and limitations in the device geometry prevent detailed studies of these states. By contrast, graphene–hBN devices are clean enough to manifest the FQHE and can easily be doped to high carrier densities. Early studies of graphene–hBN identified FQH states at filling factors with denominators of three and revealed the critical role played by the multicomponent (spin and valley) nature of graphene54. State-of-the-art graphene devices now reveal a full hierarchical sequence of FQH states at all filling factors, with odd denominators as large as 15 (Refs55,56), as well as more fragile states in the 4-flux sequence, in which composite fermions consist of an electron coupled to 4 magnetic flux quanta (Fig. 3c). Measurements of the energy gaps of these FQH states display transitions between various spin and valley orderings as a function of magnetic field. Additionally, signatures of electron solid states have been observed in graphene in excited LLs57 (in contrast to FQH states, which are described as electron liquids). Charge-density wave phases connected with the ordering of the electron solid are observed in transport experiments and give rise to a re-entrant IQHE in which σxy re-quantizes to an integer value at certain fractional filling factors of excited LLs. Such an effect has been previously observed only in very high-mobility GaAs quantum wells, further demonstrating the ultrahigh quality achievable in graphene–hBN heterostructures.

Several new device geometries and measurement schemes can be used to realize ultrahigh-quality measurements of FQH states. The simplest is the Hall bar geometry (Fig. 3a) with encapsulating top and bottom graphite gates, which shows notable improvements over devices with evaporated metal or degenerately doped silicon gates36. Additionally, patterned top gates can electrostatically define Hall bar geometries by using the insulating v = 0 state at high magnetic field57. Measurements of the bulk device conductivity, which are not complicated by the presence of edge modes, have so far proved to be the most sensitive probes of FQH states and are typically free of the artefacts that often arise in devices with Hall bar geometries. Devices with the Corbino geometry, in which current passes through a disk-shaped sample with metallic leads at the edge and at the centre, such that there are no edge modes coupling the inner and outer electrodes, have recently been realized as well55,56. Capacitance measurements using a dual-graphite-gating geometry have also been used as probes of bulk electronic compressibility at high magnetic field35 and achieve similarly high resolutions of fragile FQH states.

## Exotic correlated states

Correlated states outside the usual FQH sequence have recently been observed in ultrahigh-quality graphene heterostructures. One notable example is the observation of FQH states with even-denominator filling factors in both monolayer and bilayer graphene. Because the quasiparticles that form an effective Fermi surface at half-filling of an LL obey fermionic exchange statistics, they are bound by the Pauli exclusion principle and are not anticipated to condense into a gapped ground state. Nevertheless, even-denominator FQH states have been observed over a small range of magnetic fields in monolayer graphene (Fig. 4a). These states are thought to originate from the competition between spin-ordered and valley-ordered phases that arise owing to the multicomponent nature of the system58 and can be understood within the framework of the usual composite fermion picture with the inclusion of multiple degrees of freedom.

Recently observed even-denominator FQH states in the N = 1 LL of bilayer graphene might be described by even more exotic excitations (see Fig. 4b)35,36. Whereas these states may potentially be described by multicomponent ground states, similar to the states in monolayer graphene, theoretical calculations suggest that they are more likely described by a Moore–Read Pfaffian wavefunction (a spin-polarized, p-wave triplet wavefunction with Bardeen–Cooper–Schrieffer-like pairing of quasiparticles). The Pfaffian wavefunction notably hosts excitations with non-Abelian exchange statistics59; hence, exchanging the positions of two quasiparticles results in a unitary matrix operation on the ground-state wavefunction rather than the usual accumulation of a scalar phase that is obtained for electrons or typical composite fermions. Such a state is expected to arise owing to residual interactions between composite fermions in excited LLs, which can result in an effective attractive potential that causes composite fermions to pair as in a Bardeen–Cooper–Schrieffer-like transition. States that might be described by a Pfaffian wavefunction have previously been observed in GaAs quantum wells at $$\nu =\frac{5}{2}$$ (Refs60,61); however, direct experimental confirmation of their ground-state wavefunction remains elusive. Bilayer graphene provides an especially attractive platform to study these exotic even-denominator FQH states, because they are expected to be tuneable by both the transverse displacement field and the magnetic field. Future interferometry experiments in graphene-based platforms may help to shed light on these states and provide a potential roadmap for their use in fault-tolerant quantum computing applications.

Experiments probing the Coulomb drag between double wells of monolayer or bilayer graphene are now made possible by the clean device structures and high-quality dielectric properties of hBN (Refs26,62,63,64,65,66). Double-layer quantum well systems can host spatially indirect excitons composed of a strongly Coulomb-bound electron and hole in each layer. At high charge carrier density, these effective bosons condense into a superfluid Bose–Einstein condensate ground state. The use of very thin hBN spacers enables the fabrication of high-quality graphene double well structures with layers an order of magnitude closer than in GaAs double wells67. As a result, graphene-based devices can be studied in the strongly coupled regime — in which the magnetic length lB is larger than the interlayer spacing — which is dominated by mutual Coulomb repulsions between electrons. Studies of double-layer structures of bilayer graphene separated by hBN spacers ~2.5–5 nm thick have identified Bose–Einstein condensates in the quantum Hall regime; their characteristic energy scale is almost an order of magnitude higher than that of Bose–Einstein condensates in GaAs because of the stronger interlayer coupling26,66. The Bose–Einstein condensate is stabilized when the total filling factor vT, which is the sum of the filling factors of the top (vtop) and bottom (vbot) layers, is equal to one independent of the relative weights of vtop and vbot (Fig. 4c). In such cases, every filled (electron-like) state in one layer is balanced by an empty (hole-like) state in the other, thus allowing an interlayer excitonic pairing of each of the charge carriers. Ongoing studies extend the investigation of these novel double-layer states to the FQH regime140.

## Modifying graphene with moiré superlattices

Heterostructures of 2D vdW materials are very flexible platforms for engineering novel optoelectronic properties. In addition to realizing new effective material properties simply by combining different crystals and changing their order within a heterostructure, the properties of vdW heterostructures can be manipulated by tuning the relative rotation between crystals, as well as their interlayer spacing. The case of graphene–hBN provides a striking demonstration of the manipulation of properties that can be achieved by tuning each of these novel degrees of freedom, which are absent in conventional 2D quantum wells. Whereas hBN typically acts as a featureless dielectric substrate for graphene, the near-lattice matching of the two crystals results in a long-wavelength geometric interference pattern, called a moiré pattern, if their rotational mismatch is small. This moiré pattern effectively acts as a periodic superlattice electrical potential for graphene, strongly modifying the device properties.

Scanning tunnelling microscopy images of graphene with a small rotational misalignment relative to an hBN substrate reveal the presence of the substrate-induced moiré superlattice: in addition to the hexagonal atomic lattice of the graphene, a larger-wavelength hexagonal superstructure is visible31,39,40,68,69,70,71,72,73 (Fig. 5a). The wavelength of the moiré pattern depends on the lattice mismatch δ between graphene and hBN and on their relative rotational orientation and is given by

$$\lambda =\frac{\left(1+\delta \right)a}{\sqrt{2\left(1+\delta \right)\left(1-cos\theta \right)+{\delta }^{2}}}$$
(1)

where a is the graphene lattice constant and θ is the twist angle between the two lattices (inset in Fig. 5b). The resulting periodic potential enables scattering processes along the directions of its reciprocal lattice vectors which are otherwise forbidden in graphene. Assuming that the periodic potential does not break the sublattice symmetry in graphene, and considering only the scalar part of the potential, a bandgap is not opened, but new Dirac points are created in the conduction and valence bands where the periodic potential connects the k and –k bands. For the linear band structure of graphene, this occurs at an energy of

$$\begin{array}{c}E=\frac{\hslash {v}_{f}| {\boldsymbol{G}}| }{2}=\frac{2\pi \hslash {v}_{f}}{\sqrt{3}\lambda }\end{array}$$
(2)

where G is a reciprocal superlattice vector. Similar to the behaviour of the original Dirac point, the density of states around these new Dirac points is expected to be reduced or vanish. Indeed, scanning tunnelling spectroscopy measurements reveal new density of states minima at energies symmetric with respect to the original Dirac point68, and the extracted energy dispersion of these secondary Dirac points as a function of the moiré wavelength confirms the relationship in equation 2 (Fig. 5b). In transport measurements, these secondary Dirac points manifest themselves as additional peaks in the resistance symmetrically flanking the original Dirac point74,75,76 and occur when the induced charge density of the secondary Dirac points nsdp equals four electrons per moiré unit cell, corresponding to the spin and valley degeneracies (Fig. 5c). Explicitly, this condition is met $${n}_{{\rm{sdp}}}=\frac{8}{\sqrt{3}{\lambda }^{2}}$$ when; hence, secondary Dirac points are observed only in devices with twist angles $$\lesssim$$2° ($$\lambda > \sim$$ 7 nm), corresponding to carrier densities accessible through field-effect doping ($$\lesssim$$1013 cm−2).

In contrast to the most simple theoretical expectations, electrical transport and optical measurements show signatures of sizeable bandgaps at the Dirac point and secondary Dirac points in the valence band15,31,76,77,78,79,80. These bandgaps are understood as arising primarily owing to structural relaxations of the graphene lattice — observed in scanning probe measurements31,81 — occurring to minimize the stacking potential energy with the hBN substrate. The graphene locally strains in the in-plane direction and corrugates in the out-of-plane direction to match the electronic moiré potential. The resulting imbalance of carbon-on-boron and carbon-on-nitrogen stacking within the graphene breaks the sublattice symmetry and opens a bandgap. Theoretical calculations suggest that many-body interactions further enhance the size of the bandgap82,83. Bandgaps at the Dirac point in devices with near 0° alignment have been shown in transport measurements to be as large as ~40 meV and can survive up to twist angles of 5° (Ref.76), whereas the gap at the valence-band secondary Dirac points is typically smaller and vanishes at ~1° misalignment79,84. However, disagreement remains in the literature regarding how to properly model these gaps82,83,85,86,87,88,89,90.

Numerous techniques have been developed for experimentally controlling the rotational alignment in graphene–hBN heterostructures, including the intentional alignment of straight crystalline edges74, self-rotation of graphene flakes to an aligned position during thermal annealing84, epitaxial growth of graphene on hBN (Ref.70) and dynamic rotational alignment of the crystals using an atomic force microscope tip79. Whereas the twist angle between crystals cannot be modified in devices fabricated using the first three techniques, aligning the crystals with the tip of an atomic force microscope enables in situ control of the twist angle within a single device, providing powerful dynamic control over the device properties by exploiting the rotational degree of freedom. The magnitude of interlayer interactions, which sets the strength of the moiré coupling, is also critical in determining the device properties and can be tuned in situ by applying hydrostatic pressure to modify the interlayer spacing between graphene and hBN (Ref.15). Devices with bandgaps that can be tuned through dynamic rotational control or pressure can help to constrain theoretical models by elucidating the nature of the gaps as a function of the twist angle and moiré coupling strength. In particular, measurements as a function of the twist angle reveal a non-monotonic dependence of the gap on the rotation79 (Fig. 5d), whereas measurements under pressure demonstrate that the gap can be enhanced by nearly a factor of two15. The potential for further enhancement of the gap is especially appealing for next-generation digital logic applications, which rely on the high electronic quality of semiconducting graphene.

The long-wavelength moiré patterns observed in aligned graphene–hBN devices provide a unique platform to explore the interplay between electrical potentials and magnetic fields that interact over similar length scales. The secondary Dirac points arising as a result of the moiré superlattice potential break up into sequences of LLs upon application of a magnetic field, similarly to the Dirac point. At low magnetic fields, effectively independent sequences of LLs emerging from each of these points in the band structure are observed in transport experiments. As the magnetic field increases, these LLs intersect when

$$\frac{\phi }{{\phi }_{0}}=\frac{1}{q}$$
(3)

where ϕ is the flux through one moiré unit cell, ϕ0 is the flux quantum and q is an integer number. The resulting series of energy levels is known as a Hofstadter butterfly spectrum. Previous efforts to directly engineer superlattices in conventional III–V quantum wells proved challenging, as the minimum pitch size achievable through direct electron-beam lithography patterning is typically $$> \sim$$30 nm, and the processing tends to substantially increase the disorder in the 2D electron gas91,92,93,94,95,96,97. In these systems, the superlattice features are too close to the original band edge and become obscured by disorder. The Hofstadter butterfly was predicted in 1976 (Ref.98), but its first true discovery came in relation to aligned graphene–hBN in 2013 (Refs74,75,76). Graphene on hBN provides a unique system that makes this spectrum visible in laboratory-scale magnetic fields, as the field needed to fill one flux quantum per moiré unit cell is approximately 30 T for a 10 nm moiré.

Recent experimental observations have demonstrated that recurring Bloch bands form at magnetic fields corresponding to rational filling of the moiré unit cell (equation 3); at these fillings, the charge carriers experience effectively zero magnetic field. The system becomes metallic at these special values of the magnetic field, resulting in quantum oscillations that persist well above room temperature99. A rich sequence of quantum Hall states also develops in high magnetic fields, and their trajectories can be tracked using the following equation:

$$\frac{n}{{n}_{0}}=t\frac{\phi }{{\phi }_{0}}+s$$
(4)

where n/n0 is the charge carrier density per superlattice unit cell, s is an integer or rational fraction representing the Bloch band-filling index and t is an integer or rational fraction related to the gap structure (it is identical to the LL filling fraction in the case of zero superlattice potential). Five distinct types of state are observable, depending on whether t and s are integers or fractional and whether they are zero or non-zero. Surprising behaviour is observed even in the simplest case of the IQHE ($$t\in {\mathbb{Z}}$$ and s = 0), in which the gaps of symmetry-broken IQH states close near rational flux filling (equation 3), suggestive of a reverse Stoner transition in which the quantum Hall ferromagnetic order is destroyed100. This behaviour is thought to arise as a result of a competition between the moiré coupling potential and Coulomb interactions, which scale as $${E}_{c}=\frac{{e}^{2}}{\varepsilon \,{l}_{B}}\propto \sqrt{B}$$.

States with $$t\in {\mathbb{Z}}$$ and $$s\ne 0$$ have also been observed84,100,101 and arise from the filling of the moiré Bloch bands (Fig. 6). Such states can be interpreted more generally as resulting from the filling of topological Chern bands of the moiré superlattice, which are decoupled from the low-energy graphene band structure101. The observation of states with integer t and fractional s further demonstrates the importance of interactions within the Hofstadter butterfly, as these states are not predicted within a single-particle picture and most likely arise as a result of interaction-driven spontaneous symmetry breaking of the superlattice potential. For example, the observation of states with $$s=\frac{m}{3}$$ suggests a tripling of the moiré unit cell area, which may arise owing to the formation of a $$\sqrt{3}\times \sqrt{3}$$ Kekulé charge-density wave commensurate with the moiré potential84,101,102,103. Evidence of tertiary Dirac points at $$n\approx 6.6{n}_{0}$$ is also consistent with the formation of such a charge-density wave, and they persist even to very low magnetic fields103. States with both fractional t and s have also been recently observed and belong to a new class of fractional Chern insulator states101. More work is needed to understand the rich spectrum of many-body ground states within the Hofstadter butterfly.

## Long-lived plasmon and phonon polaritons

The development of hBN substrates also enabled rapid progress in the study of enhanced light–matter interactions in graphene through dipole-type polaritonic excitations104,105,106. Polaritonic phenomena are very rich in such systems, as graphene is a metal with free electrons and hosts surface plasmon polaritons (SPPs), whereas hBN is a polar insulator and hosts phonon polaritons. These two distinct modes can merge into hybrid polaritons, leading to exotic properties. Furthermore, graphene and hBN are 2D systems with exposed surfaces and well-defined edges, which makes them ideal platforms to explore intriguing polaritonic physics with advanced near-field imaging techniques.

SPPs are a mix of oscillating charge motions and electromagnetic fields. SPPs in graphene are highly unconventional owing to the linear Dirac dispersion. They can be detected as a resonant absorption peak in the far infrared107 or directly imaged in real space through scattering-type scanning near-field optical microscopy (s-SNOM) measurements108,109 (Fig. 7a). In an s-SNOM experiment, a metal-coated atomic microscopy tip can directly launch SPPs in graphene, as it provides the extra momentum needed to bridge the mismatch between free-space photons and confined polaritons. Within its propagation length, the SPP travels circularly towards the edge of the 2D sheet, where it is reflected back and detected as out-scattered light. Spatial scanning of the tip reveals characteristic fringes with a period of half the wavelength of the SPP, arising owing to the interference between launched and reflected waves. This technique provides a direct visualization of the polariton wavelength and propagation length. Initial studies using graphene–SiO2 observed propagating SPPs as well as their modulation with electrical gating, but the propagation length was relatively short108,109, which hindered further investigation of the intrinsic plasmon properties and potential applications.

hBN provides an exceptional environment for graphene plasmons not only because its flatness and cleanliness as a substrate substantially reduce impurity scattering but also because the hBN lattice provides a highly anisotropic dielectric environment for SPPs, which further enhances the confinement of optical fields. Encapsulating graphene in hBN improves the lifetime of SPPs owing to unprecedentedly low plasmon damping combined with high field confinement110,111. These advantages enable the probing of more intricate properties of graphene plasmons, including non-local or wavevector-dependent electromagnetic responses112 and ultrafast sub-picosecond switching113. At liquid-nitrogen temperatures, the plasmonic propagation length can exceed 10 μm in hBN-encapsulated graphene samples, setting a record for highly confined polariton modes114.

The moiré superlattice arising when graphene and hBN crystals are aligned can further modify the properties of collective plasmonic responses115. When graphene is grown on hBN (Fig. 7b), there are regions with precise crystalline alignment (moiré pattern with λ ≈ 14 nm) adjacent to regions of large misalignment (λ < 0.5 nm). At the boundary between these regions, two different types of plasmon are observed: one in the moiré superlattice and one in pristine graphene116. Compared with plasmons in pristine graphene, moiré plasmons exhibit higher damping because interband transitions associated with the superlattice mini-bands open additional loss channels. The moiré plasmons are also predicted to generate an additional low-energy branch in the terahertz and far-infrared regions117, but this has not yet been observed. Moreover, graphene placed on hBN crystals in a commensurate way can have a topologically non-trivial band structure with Berry curvature hot spots, which are responsible for large non-local transport signals from valley currents118. Valley plasmons and chiral plasmons arising after optical pumping have therefore been predicted in such systems119, but so far, signatures of these effects have remained elusive.

As a layered material with strong anisotropy between the in-plane and out-of-plane directions, hBN itself hosts intriguing phonon polaritons with a hyperbolic dispersion, in which the in-plane and out-of-plane dielectric constants have opposite signs in certain frequency ranges. These hyperbolic phonon polaritons (HPPs) are relevant to many photonic technologies, including sub-diffraction imaging. Mid-infrared polaritonic waves were launched, detected and imaged in hBN using s-SNOM; they exhibit deep sub-wavelength confinement yet low damping, which allows them to travel over substantially longer distances than covered by plasmons in graphene120. Notably, the polariton wavelength can be effectively tuned through the thickness of hBN, and the hyperbolic dispersion is observed in two frequency ranges, opening the way to engineer vdW heterostructures with controlled plasmonic responses. HPPs enabled the fabrication of a hyper-focusing lens with sub-diffractional focusing by using metallic plates underneath the hBN (Fig. 7c)121,122. Even more remarkably, HPPs in hBN can intertwine with graphene plasmons to form hybrid polaritons that can be effectively modulated by both the hBN thickness and electrical gating of graphene (Fig. 7d)123. These hybrid polaritons have very low loss and propagate over longer distances than covered by either of their individual constituents, enabling the use of graphene–hBN heterostructures as electromagnetic metamaterials for nanophotonic applications.

The interplay between hot carriers in graphene and HPPs in hBN leads to improved optoelectronic performance of graphene–hBN devices. If a mid-infrared laser is used to illuminate a graphene p–n junction encapsulated in hBN, the light is first converted into HPPs in hBN and then propagates as confined rays heating up the graphene, leading to a strong thermoelectric current124. This unique energy conversion path overcomes the limitation of graphene p–n junction photodetectors, which suffer from weak electronic absorption, without compromising the operation speed. Time-resolved photocurrent measurements have also identified a reverse energy transfer channel, where charge carriers in graphene cool down efficiently through transferring energy to the HPPs in hBN (ref.125). The effective near-field energy transfer between carriers in graphene and HPPs in hBN can enable new design principles for optoelectronic devices.

## Outlook

The realization of ultrahigh-quality graphene–hBN heterostructures paves the way for a variety of new experiments. The fabrication of these devices has become simple enough to be commonplace, and the flexibility of the technique enables the realization of almost any desired combination of vdW materials with on-demand gating structures. The ultralow charge inhomogeneity of less than 1010 cm−2 opens the way to experiments probing the intrinsic physics of these devices, largely unobstructed by the uncontrolled and undesired effects of disorder. These improvements have already led to the realization of a number of novel device geometries and physical phenomena in graphene, including electronic confinement using electrostatically defined quantum point contacts126,127, realization of Mach–Zehnder interferometers in electrostatically defined p–n junctions128, integration of superconducting electrodes on graphene in the quantum Hall regime129,130 and studies of hydrodynamic electronic transport131,132.

Future experiments with vdW heterostructures are likely to rely heavily on the degrees of freedom that are unique to these systems, including the interlayer rotation between crystals and their interlayer spacing, and that have no direct analogues in conventional 2D quantum wells. In particular, a number of systems exhibiting strongly correlated electronic phases have recently emerged, owing to uniquely flat electronic dispersions driven by the interlayer rotation degree of freedom. In these cases, the mutual Coulomb repulsion between electrons dominates over their kinetic energy. Bilayer graphene with a twist angle of precisely 1.1° has recently driven tremendous excitement owing to the observation of superconductivity and correlated insulating states12,13. Rhombohedrally stacked trilayer graphene with a 0° alignment relative to hBN has also shown signatures of correlated insulating states133. These phases can be further manipulated by tuning the interlayer coupling with pressure, as has recently been demonstrated for twisted bilayer graphene134.

More generally, the integration of a variety of atomically thin vdW crystals into heterostructures has already substantially widened the range of achievable optoelectronic properties, with promise for further future advances. Despite the development of the Scotch-tape method for exfoliating vdW materials into atomic monolayers being over a decade old, the community is still finding surprising new properties even in isolated crystals. Within the past 2 years alone, magnetism has been realized in CrI3 monolayers10, and monolayer WTe2 has been found to be a quantum spin Hall insulator7,9 (with the additional ability to enter a superconducting phase when electrostatically doped135,136). Combining these materials may help to achieve properties that were previously thought to be impossible, such as a quantum anomalous Hall phase in graphene sandwiched between WSe2 (with strong spin–orbit coupling) and CrI3 (with a strong magnetic exchange field). In addition to the fascinating new physics remaining to be discovered within isolated 2D materials, the potential for engineering new optoelectronic properties in heterostructures by combining various materials from the large vdW family and manipulating them using the new degrees of freedom available in these structures offers countless possibilities for future growth in this field.

## References

1. 1.

Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).

2. 2.

Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

3. 3.

Britnell, L. et al. Electron tunneling through ultrathin boron nitride crystalline barriers. Nano. Lett. 12, 1707–1710 (2012).

4. 4.

Mak, K., Lee, C., Hone, J., Shan, J. & Heinz, T. Atomically thin MoS2: a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010).

5. 5.

Xi, X. et al. Ising pairing in superconducting NbSe2 atomic layers. Nat. Phys. 12, 139–143 (2015).

6. 6.

Xi, X. et al. Strongly enhanced charge-density-wave order in monolayer NbSe2. Nat. Nanotechnol. 10, 765–769 (2015).

7. 7.

Tang, S. et al. Quantum spin Hall state in monolayer 1T’-WTe2. Nat. Phys. 13, 683–687 (2017).

8. 8.

Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677–682 (2017).

9. 9.

Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76–79 (2018).

10. 10.

Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).

11. 11.

Kim, K. et al. Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene. Proc. Natl Acad. Sci. USA 114, 3364–3369 (2017).

12. 12.

Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

13. 13.

Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).

14. 14.

Huang, S. et al. Topologically protected helical states in minimally twisted bilayer graphene. Phys. Rev. Lett. 121, 037702 (2018).

15. 15.

Yankowitz, M. et al. Dynamic band-structure tuning of graphene moiré superlattices with pressure. Nature 557, 404–408 (2018).

16. 16.

Bolotin, K. et al. Ultrahigh electron mobility in suspended graphene. Solid State Commun. 146, 351–355 (2008).

17. 17.

Du, X., Skachko, I., Barker, A. & Andrei, E. Y. Approaching ballistic transport in suspended graphene. Nat. Nanotechnol. 3, 491–495 (2008).

18. 18.

Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotechnol. 5, 722–726 (2010).

19. 19.

Martin, J. et al. Observation of electron-hole puddles in graphene using a scanning single-electron transistor. Nat. Phys. 4, 144–148 (2008).

20. 20.

Zhang, Y., Brar, V. W., Girit, C., Zettl, A. & Crommie, M. F. Origin of spatial charge inhomogeneity in graphene. Nat. Phys. 5, 722–726 (2009).

21. 21.

Deshpande, A., Bao, W., Miao, F., Lau, C. N. & LeRoy, B. J. Spatially resolved spectroscopy of monolayer graphene on SiO2. Phys. Rev. B 79, 205411 (2009).

22. 22.

Chen, J.-H., Jang, C., Xiao, S., Ishigami, M. & Fuhrer, M. S. Intrinsic and extrinsic performance limits of graphene devices on SiO2. Nat. Nanotechnol. 3, 206–209 (2008).

23. 23.

Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).

24. 24.

Blake, P. et al. Making graphene visible. Appl. Phys. Lett. 91, 063124 (2007).

25. 25.

Golla, D. et al. Optical thickness determination of hexagonal boron nitride flakes. Appl. Phys. Lett. 102, 161906 (2013).

26. 26.

Li, J. I. A., Taniguchi, T., Watanabe, K., Hone, J. & Dean, C. R. Excitonic superfluid phase in double bilayer graphene. Nat. Phys. 13, 751–755 (2017).

27. 27.

Mayorov, A. S. et al. Micrometer-scale ballistic transport in encapsulated graphene at room temperature. Nano. Lett. 11, 2396–2399 (2011).

28. 28.

Taychatanapat, T., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Electrically tunable transverse magnetic focusing in graphene. Nat. Phys. 9, 225–229 (2013).

29. 29.

Haigh, S. J. et al. Cross-sectional imaging of individual layers and buried interfaces of graphene-based heterostructures and superlattices. Nat. Mater. 11, 764–767 (2012).

30. 30.

Kretinin, A. V. et al. Electronic properties of graphene encapsulated with different two-dimensional atomic crystals. Nano. Lett. 14, 3270–3276 (2014).

31. 31.

Woods, C. R. et al. Commensurate-incommensurate transition in graphene on hexagonal boron nitride. Nat. Phys. 10, 451–456 (2014).

32. 32.

Kim, K. et al. van der Waals heterostructures with high accuracy rotational alignment. Nano. Lett. 16, 1989–1995 (2016).

33. 33.

Cao, Y. et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016).

34. 34.

Cao, Y. et al. Quality heterostructures from two-dimensional crystals unstable in air by their assembly in inert atmosphere. Nano. Lett. 15, 4914–4921 (2015).

35. 35.

Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature 549, 360–364 (2017).

36. 36.

Li, J. I. A. et al. Even-denominator fractional quantum Hall states in bilayer graphene. Science 358, 648–652 (2017).

37. 37.

Telford, E. J. et al. Via method for lithography free contact and preservation of 2D materials. Nano. Lett. 18, 1416–1420 (2018).

38. 38.

Cullen, W. et al. High-fidelity conformation of graphene to SiO2 topographic features. Phys. Rev. Lett. 105, 215504 (2010).

39. 39.

Xue, J. et al. Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nat. Mater. 10, 282–285 (2011).

40. 40.

Decker, R. et al. Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy. Nano. Lett. 11, 2291–2295 (2011).

41. 41.

Yankowitz, M., Xue, J. & LeRoy, B. J. Graphene on hexagonal boron nitride. J. Phys. Condens. Matter 26, 303201 (2014).

42. 42.

Taniguchi, T. & Watanabe, K. Synthesis of high-purity boron nitride single crystals under high pressure by using Ba–BN solvent. J. Cryst. Growth 303, 525–529 (2007).

43. 43.

Wong, D. et al. Characterization and manipulation of individual defects in insulating hexagonal boron nitride using scanning tunnelling microscopy. Nat. Nanotechnol. 10, 949–953 (2015).

44. 44.

Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).

45. 45.

Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

46. 46.

Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 8, 550–556 (2012).

47. 47.

Maher, P. et al. Evidence for a spin phase transition at charge neutrality in bilayer graphene. Nat. Phys. 9, 154–158 (2013).

48. 48.

Sanchez-Yamagishi, J. D. et al. Helical edge states and fractional quantum Hall effect in a graphene electron–hole bilayer. Nat. Nanotechnol. 12, 118–122 (2016).

49. 49.

San-Jose, P., Lado, J. L., Aguado, R., Guinea, F. & Fernández-Rossier, J. Majorana zero modes in graphene. Phys. Rev. X 5, 041042 (2015).

50. 50.

Alicea, J. & Fendley, P. Topological phases with parafermions: theory and blueprints. Annu. Rev. Condens. Matter Phys. 7, 119–139 (2016).

51. 51.

Jain, J. K. Composite-fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. 63, 199–202 (1989).

52. 52.

Du, X., Skachko, I., Duerr, F., Luican, A. & Andrei, E. Y. Fractional quantum Hall effect and insulating phase of Dirac electrons in graphene. Nature 462, 192–195 (2009).

53. 53.

Bolotin, K. I., Ghahari, F., Shulman, M. D., Stormer, H. L. & Kim, P. Observation of the fractional quantum Hall effect in graphene. Nature 462, 196–199 (2009).

54. 54.

Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene. Nat. Phys. 7, 693–696 (2011).

55. 55.

Polshyn, H. et al. Quantitative transport measurements of fractional quantum Hall energy gaps in edgeless graphene devices. Phys. Rev. Lett. 121, 226801 (2018).

56. 56.

Zeng, Y. et al. High quality magnetotransport in graphene using the edge-free Corbino geometry. Preprint at arXiv https://arxiv.org/abs/1805.04904 (2018).

57. 57.

Chen, S. et al. Competing fractional quantum hall and electron solid phases in graphene. Preprint at arXiv https://arxiv.org/abs/1807.10523 (2018).

58. 58.

Zibrov, A. A. et al. Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene. Nat. Phys. 14, 930–935 (2018).

59. 59.

Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

60. 60.

Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 1776–1779 (1987).

61. 61.

Pan, W. et al. Exact quantization of the even-denominator fractional quantum Hall state at ν=5/2 Landau level filling factor. Phys. Rev. Lett. 83, 3530–3533 (1999).

62. 62.

Gorbachev, R. V. et al. Strong Coulomb drag and broken symmetry in double-layer graphene. Nat. Phys. 8, 896–901 (2012).

63. 63.

Li, J. I. A. et al. Negative Coulomb drag in double bilayer graphene. Phys. Rev. Lett. 117, 046802 (2016).

64. 64.

Lee, K. et al. Giant frictional drag in double bilayer graphene heterostructures. Phys. Rev. Lett. 117, 046803 (2016).

65. 65.

Liu, X., Wang, L., Fong, K. C., Gao, Y. & Maher, P. Frictional magneto-Coulomb drag in graphene double-layer heterostructures. Phys. Rev. Lett. 119, 056802 (2017).

66. 66.

Liu, X., Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P. Quantum Hall drag of exciton condensate in graphene. Nat. Phys. 13, 746–750 (2017).

67. 67.

Eisenstein, J. P. Exciton condensation in bilayer quantum Hall systems. Annu. Rev. Condens. Matter Phys. 5, 159–181 (2014).

68. 68.

Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).

69. 69.

Chae, J. et al. Renormalization of the graphene dispersion velocity determined from scanning tunneling spectroscopy. Phys. Rev. Lett. 109, 116802 (2012).

70. 70.

Yang, W. et al. Epitaxial growth of single-domain graphene on hexagonal boron nitride. Nat. Mater. 12, 792–797 (2013).

71. 71.

Järvinen, P. et al. Molecular self-assembly on graphene on SiO2 and h-BN substrates. Nano. Lett. 13, 3199–3204 (2013).

72. 72.

Wong, D., Wang, Y., Jung, J., Pezzini, S. & DaSilva, A. M. Local spectroscopy of moiré-induced electronic structure in gate-tunable twisted bilayer graphene. Phys. Rev. B 92, 155409 (2015).

73. 73.

Jiang, Y. et al. Visualizing strain-induced pseudomagnetic fields in graphene through an hBN magnifying glass. Nano. Lett. 17, 2839–2843 (2017).

74. 74.

Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices. Nature 497, 594–597 (2013).

75. 75.

Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

76. 76.

Hunt, B. et al. Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science 340, 1427–1430 (2013).

77. 77.

Chen, Z.-G. et al. Observation of an intrinsic bandgap and Landau level renormalization in graphene/boron-nitride heterostructures. Nat. Commun. 5, 4461 (2014).

78. 78.

Wang, E. et al. Gaps induced by inversion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride. Nat. Phys. 12, 1111–1115 (2016).

79. 79.

Ribeiro-Palau, R. et al. Twistable electronics with dynamically rotatable heterostructures. Science 361, 690–693 (2018).

80. 80.

Kim, H. et al. Accurate gap determination in monolayer and bilayer graphene/h-BN moiré superlattices. Nano Lett. 18, 7732–7741 (2018).

81. 81.

Yankowitz, M., Watanabe, K., Taniguchi, T., San-Jose, P. & LeRoy, B. J. Pressure-induced commensurate stacking of graphene on boron nitride. Nat. Commun. 7, 13168 (2016).

82. 82.

Song, J. C. W., Shytov, A. V. & Levitov, L. S. Electron interactions and gap opening in graphene superlattices. Phys. Rev. Lett. 111, 266801 (2013).

83. 83.

Jung, J., DaSilva, A. M., MacDonald, A. H. & Adam, S. Origin of band gaps in graphene on hexagonal boron nitride. Nat. Commun. 6, 6308 (2015).

84. 84.

Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).

85. 85.

Bokdam, M., Amlaki, T., Brocks, G. & Kelly, P. J. Band gaps in incommensurable graphene on hexagonal boron nitride. Phys. Rev. B 89, 201404 (2014).

86. 86.

Moon, P. & Koshino, M. Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice. Phys. Rev. B 90, 155406 (2014).

87. 87.

San-Jose, P., Gutiérrez-Rubio, A., Sturla, M. & Guinea, F. Electronic structure of spontaneously strained graphene on hexagonal boron nitride. Phys. Rev. B 90, 115152 (2014).

88. 88.

Wallbank, J. R., Mucha-Kruczyński, M., Chen, X. & Fal’ko, V. I. Moiré superlattice effects in graphene/boron-nitride van der Waals heterostructures. Ann. der Phys. 527, 359–376 (2015).

89. 89.

Slotman, G. J. et al. Effect of structural relaxation on the electronic structure of graphene on hexagonal boron nitride. Phys. Rev. Lett. 115, 186801 (2015).

90. 90.

Jung, J. et al. Moiré band model and band gaps of graphene on hexagonal boron nitride. Phys. Rev. B 96, 085442 (2017).

91. 91.

Esaki, L. & Tsu, R. Superlattice and negative differential conductivity in semiconductors. IBM J. Res. Dev. 14, 61–65 (1970).

92. 92.

Weiss, D., Klitzing, K. V. & Ploog, K. Magnetoresistance oscillations in a two-dimensional electron gas induced by a submicrometer periodic potential. Europhys. Lett. 8, 179–184 (1989).

93. 93.

Ismail, K., Chu, W., Yen, A., Antoniadis, D. A. & Smith, H. I. Negative transconductance and negative differential resistance in a grid-gate modulation-doped field-effect transistor. Appl. Phys. Lett. 54, 460–462 (1989).

94. 94.

Fang, H. & Stiles, P. J. Novel magnetoresistance oscillations in a two-dimensional superlattice potential. Phys. Rev. B 41, 10171–10174 (1990).

95. 95.

Schlösser, T., Ensslin, K. & Kotthaus, J. P. Landau subbands generated by a lateral electrostatic superlattice-chasing the Hofstadter butterfly. Semicond. Sci. Tech. 11, 1582–1585 (1996).

96. 96.

Albrecht, C. et al. Evidence of Hofstadter’s fractal energy spectrum in the quantized Hall conductance. Phys. Rev. Lett. 86, 147–150 (2001).

97. 97.

Melinte, S. et al. Laterally modulated 2D electron system in the extreme quantum limit. Phys. Rev. Lett. 92, 683–684 (2004).

98. 98.

Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

99. 99.

Kumar, R. K. et al. High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices. Science 357, 181–184 (2017).

100. 100.

Yu, G. L. et al. Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices. Nat. Phys. 10, 525–529 (2014).

101. 101.

Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).

102. 102.

DaSilva, A. M., Jung, J. & MacDonald, A. H. Fractional Hofstadter states in graphene on hexagonal boron nitride. Phys. Rev. Lett. 117, 036802 (2016).

103. 103.

Chen, G. et al. Emergence of tertiary Dirac points in graphene Moiré superlattices. Nano. Lett. 17, 3576–3581 (2017).

104. 104.

Koppens, F. H. L., Chang, D. E. & García de Abajo, F. J. Graphene plasmonics: a platform for strong light–matter interactions. Nano. Lett. 11, 3370–3377 (2011).

105. 105.

Basov, D. N., Fogler, M. M. & de Abajo, F. J. G. Polaritons in van der Waals materials. Science 354, 195 (2016).

106. 106.

Low, T. et al. Polaritons in layered two-dimensional materials. Nat. Mater. 16, 182–194 (2016).

107. 107.

Ju, L. et al. Graphene plasmonics for tunable terahertz metamaterials. Nat. Nanotechnol. 6, 630–634 (2011).

108. 108.

Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 487, 82–85 (2012).

109. 109.

Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77–81 (2012).

110. 110.

Woessner, A. et al. Highly confined low-loss plasmons in graphene–boron nitride heterostructures. Nat. Mater. 14, 421–425 (2014).

111. 111.

Iranzo, D. A. et al. Probing the ultimate plasmon confinement limits with a van der Waals heterostructure. Science 360, 291–295 (2018).

112. 112.

Lundeberg, M. B. et al. Tuning quantum nonlocal effects in graphene plasmonics. Science 357, 187–191 (2017).

113. 113.

Ni, G. X. et al. Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene. Nat. Photonics 10, 244–247 (2016).

114. 114.

Ni, G. X. et al. Fundamental limits to graphene plasmonics. Nature 557, 530–533 (2018).

115. 115.

Polini, M. & Koppens, F. H. L. Graphene: plasmons in moiré superlattices. Nat. Mater. 14, 1187–1188 (2015).

116. 116.

Ni, G. X. et al. Plasmons in graphene moiré superlattices. Nat. Mater. 14, 1217–1222 (2015).

117. 117.

Tomadin, A., Guinea, F. & Polini, M. Generation and morphing of plasmons in graphene superlattices. Phys. Rev. B 90, 161406 (2014).

118. 118.

Gorbachev, R. V., Song, J., Yu, G. L. & Kretinin, A. V. Detecting topological currents in graphene superlattices. Science 346, 448–451 (2014).

119. 119.

Song, J. C. W. & Rudner, M. S. Chiral plasmons without magnetic field. Proc. Natl Acad. Sci. USA 113, 4658–4663 (2016).

120. 120.

Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride. Science 343, 1125–1129 (2014).

121. 121.

Dai, S. et al. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. Nat. Commun. 6, 6963 (2015).

122. 122.

Li, P. et al. Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing. Nat. Commun. 6, 7507 (2015).

123. 123.

Dai, S. et al. Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial. Nat. Nanotechnol. 10, 682–686 (2015).

124. 124.

Woessner, A. et al. Electrical detection of hyperbolic phonon-polaritons in heterostructures of graphene and boron nitride. npj 2D Mater. Appl. 1, 25 (2017).

125. 125.

Tielrooij, K.-J. et al. Out-of-plane heat transfer in van der Waals stacks through electron–hyperbolic phonon coupling. Nat. Nanotechnol. 13, 41–46 (2017).

126. 126.

Zimmermann, K. et al. Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices. Nat. Commun. 8, 14983 (2017).

127. 127.

Overweg, H. et al. Electrostatically induced quantum point contacts in bilayer graphene. Nano. Lett. 18, 553–559 (2017).

128. 128.

Wei, D. S. et al. Mach-Zehnder interferometry using spin- and valley-polarized quantum Hall edge states in graphene. Sci. Adv. 3, e1700600 (2017).

129. 129.

Amet, F. et al. Supercurrent in the quantum Hall regime. Science 352, 966–969 (2016).

130. 130.

Lee, G.-H. et al. Inducing superconducting correlation in quantum Hall edge states. Nat. Phys. 13, 693–698 (2017).

131. 131.

Bandurin, D. A. et al. Negative local resistance caused by viscous electron backflow in graphene. Science 351, 1055–1058 (2016).

132. 132.

Crossno, J. et al. Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene. Science 351, 1058–1061 (2016).

133. 133.

Chen, G. et al. Gate-tunable Mott insulator in trilayer graphene-boron nitride Moiré superlattice. Preprint at arXiv https://arxiv.org/abs/1803.01985 (2018).

134. 134.

Yankowitz, M. et al. Tuning superconductivity in twisted bilayer graphene. Preprint at arXiv https://arxiv.org/abs/1808.07865 (2018).

135. 135.

Fatemi, V. et al. Electrically tunable low-density superconductivity in a monolayer topological insulator. Science 362, 926–929 (2018).

136. 136.

Sajadi, E. et al. Gate-induced superconductivity in a monolayer topological insulator. Science 362, 922–925 (2018).

137. 137.

Zunger, A., Katzir, A. & Halperin, A. Optical properties of hexagonal boron nitride. Phys. Rev. B 13, 5560–5573 (1976).

138. 138.

Watanabe, K., Taniguchi, T. & Kanda, H. Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal. Nat. Mater. 3, 404–409 (2004).

139. 139.

Lee, M. et al. Ballistic miniband conduction in a graphene superlattice. Science 353, 1526–1529 (2016).

140. 140.

Li, J. I. A. et al. Evidence for pairing states of composite fermions in double-layer graphene. Preprint at arXiv https://arxiv.org/abs/1901.03480 (2019).

## Acknowledgements

B.J.L. acknowledges the support of the US Army Research Office under grant W911NF-14-1-0653. Research reviewed by P.J.-H. and Q.M. has been supported by the Center for Excitonics, an Energy Frontier Research Center funded by the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), under award number DE-SC0001088, US Air Force Office of Scientific Research (AFOSR) grant FA9550-16-1-0382, the National Science Foundation under award DMR-1405221, the Gordon and Betty Moore Foundation’s Emergent Phenomena in Quantum Systems (EPiQS) Initiative through grant GBMF4541 and the US DOE, Office of BES, Division of Materials Sciences and Engineering, under award number DE-SC0001819.

## Author information

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Correspondence to Brian J. LeRoy.

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Yankowitz, M., Ma, Q., Jarillo-Herrero, P. et al. van der Waals heterostructures combining graphene and hexagonal boron nitride. Nat Rev Phys 1, 112–125 (2019) doi:10.1038/s42254-018-0016-0

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