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Graphene Bilayers with a Twist FOCUS | REVIEW ARTICLE FOCUS | https://doi.org/10.1038/s41563-020-00840-0REVIEW ARTICLE Graphene bilayers with a twist Eva Y. Andrei! !1 and Allan H. MacDonald2 Near a magic twist angle, bilayer graphene transforms from a weakly correlated Fermi liquid to a strongly correlated two-dimensional electron system with properties that are extraordinarily sensitive to carrier density and to controllable envi- ronmental factors such as the proximity of nearby gates and twist-angle variation. Among other phenomena, magic-angle twisted bilayer graphene hosts superconductivity, interaction-induced insulating states, magnetism, electronic nematic- ity, linear-in-temperature low-temperature resistivity and quantized anomalous Hall states. We highlight some key research results in this field, point to important questions that remain open and comment on the place of magic-angle twisted bilayer graphene in the strongly correlated quantum matter world. he two-dimensional (2D) materials field can be traced back to zone. Since σ·pK is the chirality operator, and σy pK σ pK À Á 0 ¼TðÞÁ the successful isolation of single-layer graphene sheets exfoli- is its time-reversal partner ( is the time-reversalI operator), states T ated from bulk graphite1. Galvanized by this breakthrough, in the K and Kʹ Dirac cones haveI opposite chirality. K and Kʹ define T 2 researchers have by now isolated and studied dozens of 2D crystals a valley degree of freedom, which together with spin produces a and predicted the existence of many more3. Because all the atoms in four-fold band degeneracy. The Dirac points separate the conduc- any 2D crystal are exposed to our 3D world, this advance has made tion and valence bands so that at charge neutrality the Fermi sur- it possible to tune the electronic properties of a material without face (pink plane in Fig. 1c) consists of just these two points. Owing changing its chemical composition, for example by introducing a to the conical energy–momentum relationships, the electron- and 4–11 12–14 15,16 6 −1 large strain , plucking out atoms , stacking 2D crystals or hole-like quasiparticles move at a constant speed, vF ≈ 10 m s , simply using electrical gates to add or remove electrons. One of the which is determined by the slope of the cone35–37. As a result, the simplest techniques, changing the relative orientation between 2D charge carriers behave as if they were massless ultra-relativistic par- crystals15–20, has been especially impactful following the discovery ticles35, albeit with a non-universal velocity that is approximately of interaction-induced insulating states and superconductivity in 300 times smaller than the speed of light. The band crossings defin- 21–27 38 twisted bilayer graphene (TBG) with a special magic twist angle . ing the Dirac points are protected by three discrete symmetries , C2z In this Review, we survey some of the progress that has been made (in-plane inversion), Tr and C3 (three-fold rotation), which prevent a toward understanding these phenomena. Our coverage is necessar- gap from opening in graphene’s spectrum unless one of the symme- ily incomplete owing both to limitations in space and to the rap- tries is broken. The absence of a gap stands as an obstacle to appli- idly evolving research landscape. We limit our attention to TBG, cations of graphene that require switching, such as transistors. The although twist-angle control is now also proving its value in other conical band structure produces a density of states (DOS) which, 2D crystal systems28–34. at low energies, is linear in energy E, and vanishes at the Dirac 2 E point, ρ E jj2, making it easy to change the Fermi energy by ðÞπ ℏvF Energy bands of an isolated graphene sheet electrostaticI doping.ðÞ Because the band structure of graphene is a We first consider monolayer graphene. Its electronic properties derive direct consequence of its geometry, graphene-like band structures strictly from geometry: a 2D honeycomb lattice with identical circu- have also been realized in other systems satisfying these conditions, larly symmetric orbitals on interpenetrating triangular Bravais sub- including artificial structures and cold atoms39–41. lattices, labelled A and B (Fig. 1a), and identical nearest-neighbour hopping parameters t. These conditions lead to a unique band Moiré superlattices and electronic properties structure that at low energy, and in the absence of interactions, is Moiré patterns created by superposing 2D meshes have a long his- described by a one-parameter 2D Dirac–Weyl Hamiltonian: tory in textiles, art and mathematics. In surface science, they made their debut shortly after the invention of the scanning tunnelling σ pK 0 microscope (STM), which enabled the observation of anomalously H vF Á ¼ 0 σy p large super-periodic patterns on the surface of graphite42. More than À Á K0 two decades elapsed before scientists observed that, beyond having 3 where vF at is the Fermi (or Dirac) velocity, ħ is the reduced aesthetic appeal, moiré patterns can greatly alter electronic prop- ¼ 2ℏ Planck constant,I a = 0.142 nm is the carbon–carbon distance, t ≈ erties15. The STM moiré pattern obtained by superposing two gra- 2.7 eV is the hopping parameter, σ = (σx, σy) are the Pauli matrixes phene crystals with a twist angle θ between the two layers consists operating on the sublattice degree of freedom, pK=ħ(k−K) and of an array of alternating bright and dark spots with period L≈a0/(2 pK ℏ k K0 are the momenta measured relative to the K and Kʹ sin(θ/2)) (Figs. 1d and 2a). The bright spots correspond to regions 0 ¼ ðÞÀ cornersI (K=4π/(3a0)) of the hexagonal Brillouin zone of graphene referred to as AA, where every atom in the top layer has a partner (Fig. 1b), respectively, and a0 p3a is graphene’s lattice constant. directly underneath it in the bottom layer. In the darker regions, ¼ The low-energy band structureI consists of a pair of inequivalent the local stacking is approximately A on B or B on A, referred to as Dirac cones (linear crossings betweenffiffiffi the conduction and valence AB or BA respectively, and also known as Bernal stacking. In the bands), shown in Fig. 1c. The apices of the Dirac cones, referred to as AB case, each top-layer A atom sits directly above an atom in the Dirac points, are located at the three equivalent K corners (full cir- bottom layer while top-layer B atoms have no partner in the bottom cles in Fig. 1b) or Kʹ corners (open circles in Fig. 1b) of the Brillouin layer, and vice versa for BA stacking. 1Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ, USA. 2Department of Physics, The University of Texas at Austin, Austin, TX, USA. e-mail: [email protected] NATURE MATERIALS | VOL 19 | DECEMBER 2020 | 1265–1275 | www.nature.com/naturematerials 1265 REVIEW ARTICLE | FOCUS NATURE MATERIALS momentum space at which an energy band reaches energy minima a b G c 1 and maxima along orthogonal directions in momentum space. In B G 2D systems, saddle points create greatly enhanced DOS peaks that a 2 2 A are easily identified in scanning tunnelling spectroscopy (STS) stud- a 15 1 a K K K K ies and are referred to as van Hove singularities (VHSs). At large θ, the Dirac cones are widely separated, and the low-energy states in one layer are only weakly influenced by tunnel coupling to the adja- cent layer. The energy separation between the conduction-band and valence-band VHSs at large twist angles, E E 2w, is equal to d e f Δ ≈ Δ 0 − kθ twice the isolated layer energies at the Dirac cone intersection points, K1 less an avoided-crossing interlayer tunnelling energy shift, 2w. An θ estimate of the interlayer tunnelling strength can be obtained from K 2 Bernal-stacked bilayer graphene, w ≈ 0.1 eV. For θ > 10°, where ΔE > 1 eV, the low-energy electronic properties of the TBG are nearly K1 indistinguishable from those of isolated graphene15,17 layers. Still, K2 the presence of the second graphene layer has an influence, since it efficiently screens random potential fluctuations introduced by the Fig. 1 | Structure of monolayer graphene and TBG. a, Honeycomb lattice of substrate46, allowing access to low-energy electronic properties that graphene with the two sublattices, A and B, represented by white and blue would otherwise be obscured by the substrate-induced potential 47 circles. a1, a2 are the honeycomb lattice vectors and a is the carbon–carbon fluctuations . The same screening effect also acts on electron–elec- distance. b, Graphene’s hexagonal Brillouin zone (shaded area) showing the tron interactions, of course, and may have the tendency to sup- two inequivalent sets of K (full circles) and K’ (open circles) Brillouin-zone press strong correlation physics. When the twist angle is reduced, corners. G1 and G2 are the reciprocal lattice vectors. c, Low-energy band the VHSs come closer together, tunnelling between layers couples structure of graphene showing the two inequivalent Dirac cones and the the low-energy Dirac cone states, the hybridization between layers corresponding Dirac points located at the K and K’ corners of the Brillouin becomes strong, and the bilayer can no longer be viewed as consist- zone (red hexagon). The pink plane defines the Fermi surface at charge ing of two weakly coupled layers. All electronic wavefunctions are neutrality. d, Moiré pattern formed by two superposed honeycomb lattices. then coherent superpositions of components on the four sublattices The bright spots correspond to local AA stacking where each atom in the of the bilayer with intricate correlated position dependences.
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