Minimum model for the electronic structure of twisted bilayer graphene and related structures
TL;DR: In this article, a minimum tight-binding model with only three parameters extracted from untwisted bilayer graphene and carbon nanophotonics was proposed. But the model is not suitable for the case of carbon nanosmide and cannot be applied to other twisted layered systems.
Abstract: We introduce a minimum tight-binding model with only three parameters extracted from graphene and untwisted bilayer graphene. This model reproduces quantitatively the electronic structure of not only these two systems and bulk graphite near the Fermi level, but also that of twisted bilayer graphene including the value of the first magic angle, at which bands at ${E}_{F}$ flatten without overlap and two gaps open, one above and one below ${E}_{F}$. Our approach also predicts the second and third magic angle. The Hamiltonian is sufficiently transparent and flexible to be adopted to other twisted layered systems.
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TL;DR: This work identifies states favored by Coulomb interactions projected onto the Wannier basis of the four narrow bands of the "magic angle" twisted bilayer graphene, and finds extended excited states, the gap to which decreases in the magnetic field.
Abstract: We identify states favored by Coulomb interactions projected onto the Wannier basis of the four narrow bands of the "magic angle" twisted bilayer graphene. At the filling of 2 electrons/holes per moire unit cell, such interactions favor an insulating SU(4) ferromagnet. The kinetic terms select the ground state in which the two valleys with opposite spins are equally mixed, with a vanishing magnetic moment per particle. We also find extended excited states, the gap to which decreases in the magnetic field. An insulating stripe ferromagnetic phase is favored at 1 electron/hole per unit cell.
260 citations
TL;DR: It is suggested that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures, after calculating the maximally localized superlattice Wannier wave functions.
Abstract: We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with fourfold degeneracy. After calculating the maximally localized superlattice Wannier wave functions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures.
181 citations
TL;DR: In this paper, a strong-coupling analysis of the two-orbital extended Hubbard model on the honeycomb lattice was performed for both strong and weak coupling regimes, showing a rich intertwinement between ferromagnetic and antiferromagnetic orders with different types of nematic and magnetic orbital orders.
Abstract: The recent observation of superconductivity in proximity to an insulating phase in twisted bilayer graphene (TBG) at small ``magic'' twist angles has been linked to the existence of nearly flat bands, which make TBG a fresh playground to investigate the interplay between correlations and superconductivity The low-energy narrow bands were shown to be well described by an effective tight-binding model on the honeycomb lattice (the dual of the triangular Moir\'e superlattice) with a local orbital degree of freedom In this paper, we perform a strong-coupling analysis of the proposed $\left({p}_{x},\phantom{\rule{016em}{0ex}}{p}_{y}\right)$ two-orbital extended Hubbard model on the honeycomb lattice By decomposing the interacting terms in the particle-particle and particle-hole channels, we classify the different possible superconducting, magnetic, and charge instabilities of the system In the pairing case, we pay particular attention to the two-component ($d\text{\ensuremath{-}}\mathrm{wave}$) pairing channels, which admit vestigial phases with nematic or chiral orders, and study their phenomenology Furthermore, we explore the strong-coupling regime by obtaining a simplified spin-orbital exchange model which may describe a putative Mott-type insulating state at quarter-filling Our mean-field solution reveals a rich intertwinement between ferromagnetic and antiferromagnetic orders with different types of nematic and magnetic orbital orders Overall, our work provides a solid framework for further investigations of the phase diagram of the two-orbital extended Hubbard model in both strong- and weak-coupling regimes
161 citations
TL;DR: When single layers of 2D materials are stacked on top of one another with a small twist in orientation, the resulting structure often involves incommensurate moire patterns as mentioned in this paper.
Abstract: When single layers of 2D materials are stacked on top of one another with a small twist in orientation, the resulting structure often involves incommensurate moire patterns. In these patterns, the loss of angstrom-scale periodicity poses a significant theoretical challenge, and the new moire length scale leads to emergent physical phenomena. The range of physics arising from twisted bilayers has led to significant advances that are shaping into a new field, twistronics. At the moire scale, the large number of atoms in these systems can make their accurate simulation daunting, necessitating the development of efficient multiscale methods. In this Review, we summarize and compare such modelling methods — focusing in particular on density functional theory, tight-binding Hamiltonians and continuum models — and provide examples spanning a broad range of materials and geometries. When single layers of 2D materials are stacked on top of one another with a small twist, the resulting moire pattern introduces new electronic properties. This Review surveys and compares the modelling techniques used in this emerging field of twistronics.
150 citations
TL;DR: In this article, it was shown that the energy gaps and band dispersion are captured correctly only when the atomic corrugation, arising as a result of the interlayer interaction, is considered.
Abstract: The electronic properties of twisted bilayer graphene are being intensely investigated. The flat bands emerging at the first ``magic angle'' are responsible for anomalous insulating behavior at half-filling and superconductivity. Using state-of-the-art $a\phantom{\rule{0}{0ex}}b\phantom{\rule{0.333em}{0ex}}i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ calculations, it is shown that the energy gaps and band dispersion are captured correctly only when the atomic corrugation, arising as a result of the interlayer interaction, is considered. The atomic corrugation shows a wave profile with maximum atomic out-of-plane displacements of about 0.1 angstrom.
140 citations
References
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TL;DR: In this paper, the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations, are discussed.
Abstract: This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.
20,824 citations
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
TL;DR: In this paper, the structure of the electronic energy bands and Brillouin zones for graphite was developed using the "tight binding" approximation, and it was found that graphite is a semi-conductor with zero activation energy, but they are created at higher temperatures by excitation to a band contiguous to the highest one which is normally filled.
Abstract: The structure of the electronic energy bands and Brillouin zones for graphite is developed using the "tight binding" approximation. Graphite is found to be a semi-conductor with zero activation energy, i.e., there are no free electrons at zero temperature, but they are created at higher temperatures by excitation to a band contiguous to the highest one which is normally filled. The electrical conductivity is treated with assumptions about the mean free path. It is found to be about 100 times as great parallel to as across crystal planes. A large and anisotropic diamagnetic susceptibility is predicted for the conduction electrons; this is greatest for fields across the layers. The volume optical absorption is accounted for.
4,395 citations
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
TL;DR: This work addresses the electronic structure of a twisted two-layer graphene system, showing that in its continuum Dirac model the moiré pattern periodicity leads to moirÉ Bloch bands.
Abstract: A moire pattern is formed when two copies of a periodic pattern are overlaid with a relative twist. We address the electronic structure of a twisted two-layer graphene system, showing that in its continuum Dirac model the moire pattern periodicity leads to moire Bloch bands. The two layers become more strongly coupled and the Dirac velocity crosses zero several times as the twist angle is reduced. For a discrete set of magic angles the velocity vanishes, the lowest moire band flattens, and the Dirac-point density-of-states and the counterflow conductivity are strongly enhanced.
2,323 citations