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Journal ArticleDOI

Quasiperiodicity, band topology, and moiré graphene

04 Mar 2021-Physical Review B (American Physical Society (APS))-Vol. 103, Iss: 11, pp 115110
TL;DR: In this article, the effect of the periodic potential induced by hexagonal boron nitride (hBN) substrate on the low energy physics of twisted bilayer graphene was investigated.
Abstract: A number of moir\'e graphene systems have nearly flat topological bands where electron motion is strongly correlated. Though microscopically these systems are only quasiperiodic, they can typically be treated as translation invariant to an excellent approximation. Here we reconsider this question for magic angle twisted bilayer graphene that is nearly aligned with a hexagonal boron nitride (hBN) substrate. We carefully study the effect of the periodic potential induced by hBN on the low energy physics. The combination of this potential and the moir\'e lattice produced by the twisted graphene generates a quasiperiodic term that depends on the alignment angle between hBN and the moir\'e graphene. We find that the alignment angle has a significant impact on both the band gap near charge neutrality and the behavior of electrical transport. We also introduce and study toy models to illustrate how a quasiperiodic potential can give rise to localization and change in transport properties of topological bands.
Citations
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Journal ArticleDOI
TL;DR: The quantum anomalous Hall effect occurs in twisted bilayer graphene when it is nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer as discussed by the authors.
Abstract: The quantum anomalous Hall effect sometimes occurs in twisted bilayer graphene when it is nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer. The authors argue that the quantum anomalous Hall effect is likely only when the graphene/graphene and graphene/hBN moir\'e patterns are nearly commensurate. This picture gives rise to the series of ``Hall windows'' in the twist-angle space illustrated here.

50 citations

Journal ArticleDOI
TL;DR: In this article, the role of long-range electron-electron interactions near the first magic angle was analyzed in the band structure of twisted trilayer graphene, and superconducting phases with either spin-singlet/valley-triplet or spin-triplets/valleyssinglet symmetry were found with critical temperatures up to a few Kelvin.
Abstract: We study the symmetries of twisted trilayer graphene's band structure under various extrinsic perturbations, and analyze the role of long-range electron-electron interactions near the first magic angle. The electronic structure is modified by these interactions in a similar way to twisted bilayer graphene. We analyze electron pairing due to long-wavelength charge fluctuations, which are coupled among themselves via the Coulomb interaction and additionally mediated by longitudinal acoustic phonons. We find superconducting phases with either spin-singlet/valley-triplet or spin-triplet/valley-singlet symmetry, with critical temperatures up to a few Kelvin for realistic choices of parameters.

32 citations

Journal ArticleDOI
TL;DR: In this article, the effect of twisting on bilayer bilayers was studied, and the effects of lattice relaxation on the electronic structure, piezoelectric charges, and spontaneous polarization were investigated.
Abstract: We study the effect of twisting on bilayer $h\text{\ensuremath{-}}\mathrm{BN}$. The effect of lattice relaxation is included; we look at the electronic structure, piezoelectric charges, and spontaneous polarization. We show that the electronic structure without lattice relaxation shows a set of extremely flat in-gap states similar to Landau levels, where the spacing scales with twist angle. With lattice relaxation we still have flat bands, but now the spectrum becomes independent of twist angle for sufficiently small angles. We describe in detail the nature of the bands and we study appropriate continuum models, at the same time explaining the structure of the in-gap states. We find that even though the spectra for both parallel and antiparallel alignment are very similar, the spontaneous polarization effects only occur for parallel alignment. We argue that this suggests a large interlayer hopping between boron and nitrogen.

23 citations

Journal ArticleDOI
TL;DR: In this article , the anomalous Hall effect was observed in twisted bilayer graphene (tBLG) at half filling of both the electron and hole moiré bands, suggesting that the ground state is valley-polarized.
Abstract: Magic-angle twisted bilayer graphene (tBLG) displays a variety of symmetry-broken phases, correlated Chern insulators, orbital magnetism and superconductivity1–8. In particular, the anomalous Hall effect has been observed when the bands are filled with an odd number of electrons per moiré unit cell5,6,9, indicating the emergence of a zero-field orbital magnetic state with spontaneously broken time-reversal symmetry10–12. Here we present measurements of two tBLG devices with twist angles slightly away from the magic angle and report the observation of the anomalous Hall effect at half filling of both the electron and hole moiré bands. We suggest that two factors—the increased band dispersion away from the magic angle, and substrate potentials from the encapsulating boron nitride—probably play critical roles in stabilizing a valley-polarized ground state at half filling. Our findings further expand the rich correlated phase diagram of tBLG, and indicate the need to develop a more complete understanding of its manifold of closely competing symmetry-breaking orders. The anomalous Hall effect can signify that a material has a spontaneous magnetic order. Now, twisted bilayer graphene shows this effect at half filling, suggesting that the ground state is valley-polarized.

20 citations

Journal ArticleDOI
20 Apr 2022
TL;DR: In this article , the authors used an atomistic tight-binding model together with semi-classical molecular dynamics to consider relaxation effects of hexagonal boron nitride (hBN) substrate.
Abstract: Abstract Twisted bilayer graphene (TBG) has taken the spotlight in the condensed matter community since the discovery of correlated phases. In this work, we study heterostructures of TBG and hexagonal boron nitride (hBN) using an atomistic tight-binding model together with semi-classical molecular dynamics to consider relaxation effects. The hBN substrate has significant effects on the band structure of TBG even in the case where TBG and hBN are not aligned. Specifically, the substrate induces a large mass gap and strong pseudo-magnetic fields that break the layer degeneracy. Interestingly, such degeneracy can be recovered with a second hBN layer. Finally, we develop a continuum model that describes the tight-binding band structure. Our results show that a real-space tight-binding model in combination with semi-classical molecular dynamics is a powerful tool to study the electronic properties of moiré heterostructures, and to explain experimental results in which the effect of the substrate plays an important role.

12 citations

References
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Journal ArticleDOI
05 Mar 2018-Nature
TL;DR: The realization of intrinsic unconventional superconductivity is reported—which cannot be explained by weak electron–phonon interactions—in a two-dimensional superlattice created by stacking two sheets of graphene that are twisted relative to each other by a small angle.
Abstract: The behaviour of strongly correlated materials, and in particular unconventional superconductors, has been studied extensively for decades, but is still not well understood. This lack of theoretical understanding has motivated the development of experimental techniques for studying such behaviour, such as using ultracold atom lattices to simulate quantum materials. Here we report the realization of intrinsic unconventional superconductivity-which cannot be explained by weak electron-phonon interactions-in a two-dimensional superlattice created by stacking two sheets of graphene that are twisted relative to each other by a small angle. For twist angles of about 1.1°-the first 'magic' angle-the electronic band structure of this 'twisted bilayer graphene' exhibits flat bands near zero Fermi energy, resulting in correlated insulating states at half-filling. Upon electrostatic doping of the material away from these correlated insulating states, we observe tunable zero-resistance states with a critical temperature of up to 1.7 kelvin. The temperature-carrier-density phase diagram of twisted bilayer graphene is similar to that of copper oxides (or cuprates), and includes dome-shaped regions that correspond to superconductivity. Moreover, quantum oscillations in the longitudinal resistance of the material indicate the presence of small Fermi surfaces near the correlated insulating states, in analogy with underdoped cuprates. The relatively high superconducting critical temperature of twisted bilayer graphene, given such a small Fermi surface (which corresponds to a carrier density of about 1011 per square centimetre), puts it among the superconductors with the strongest pairing strength between electrons. Twisted bilayer graphene is a precisely tunable, purely carbon-based, two-dimensional superconductor. It is therefore an ideal material for investigations of strongly correlated phenomena, which could lead to insights into the physics of high-critical-temperature superconductors and quantum spin liquids.

5,613 citations

Journal ArticleDOI
TL;DR: In this article, the Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential, where the Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap.
Abstract: The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential $U$. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small $\frac{U}{\ensuremath{\hbar}{\ensuremath{\omega}}_{c}}$.

4,811 citations

Journal ArticleDOI
05 Mar 2018-Nature
TL;DR: It is shown experimentally that when this angle is close to the ‘magic’ angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling, and these flat bands exhibit insulating states at half-filling, which are not expected in the absence of correlations between electrons.
Abstract: 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.

3,005 citations

Journal ArticleDOI
TL;DR: In this paper, an effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing the operator of the Schr\"odinger equation with a matrix method, and the graph of the spectrum over a wide range of "rational" fields is plotted.
Abstract: An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

2,656 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a survey of the use of Wannier functions in the context of electronic-structure theory, including their applications in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization.
Abstract: The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

2,217 citations