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 10 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.

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 moiré 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 moiré 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.

2 Control of the interlayer twist angle in two-dimensional (2D) van der Waals (vdW)heterostructures enables one to engineer a quasiperiodic moiré superlattice of tunable length scale 1-7 . In twisted bilayer graphene (TBG), the simple moiré superlattice band description suggests that the electronic band width can be tuned to be comparable to the vdW interlayer interaction at a 'magic angle' 8 , exhibiting strongly correlated behavior. However, the vdW interlayer interaction can also cause significant structural reconstruction at the interface by favoring interlayer commensurability, which competes with the intralayer lattice distortion 9-15 . Here we report the atomic scale reconstruction in TBG and its effect on the electronic structure. We find a gradual transition from incommensurate moiré structure to an array of commensurate domain structures as we decrease the twist angle across the characteristic crossover angle, θc ~1°. In the twist regime smaller than θc where the atomic and electronic reconstruction become significant, a simple moiré band description breaks down. Upon applying a transverse electric field, we observe electronic transport along the network of onedimensional (1D) topological channels that surround the alternating triangular gapped domains, providing a new pathway to engineer the system with continuous tunability.In the absence of atomic scale reconstruction, a small rigid rotation of the vdW layers relative to each other results in a moiré pattern, whose long wavelength periodicity is determined by the twist angle. For unreconstructed TBG, atomic registry varies continuously across the moiré period between three distinct types of symmetric stacking configurations: energetically favorable AB and BA Bernal stacking and unfavorable AA stacking (Fig. 1a). This quasiperiodic moiré superlattice, associated with the incommensurability of the twisted layers, modifies the band structure significantly. In the small twist regime, low-energy flat bands appear at a series of magic angles ( ≤ 1.1°) where the diverging density of states (DOS) and vanishing Fermi velocity, associated with strong electronic correlation, are predicted 8 . The recent experiment demonstrated the presence of the first magic angle near ~1.1° where Mott insulator and unconventional superconductivity were observed 6,7 . The TBG moiré band calculation, however, assumes a rigid rotation of layers ignoring atomic scale reconstruction. Despite the weak nature of vdW interaction and the absence of dangling bonds, recent experimental works on similar material systems suggestthere is substantial lattice reconstruction at vdW interfaces, especially at small twist angle close to global commensuration between two adjacent layers 9,10 . Atomic scale reconstruction at vdW B 92, 155438 (2015).

2The electrical Hall effect is the production of a transverse voltage under an out-of-plane magnetic field [1]. Historically, studies of the Hall effect have led to major breakthroughs including the discoveries of Berry curvature and the topological Chern invariants [2, 3]. In magnets, the internal magnetization allows Hall conductivity in the absence of external magnetic field [3]. This anomalous Hall effect (AHE) has become an important tool to study quantum magnets [3][4][5][6][7][8]. In nonmagnetic materials without external magnetic fields, the electrical Hall effect is rarely explored because of the constraint by time-reversal symmetry.However, strictly speaking, only the Hall effect in the linear response regime, i.e., the Hall voltage linearly proportional to the external electric field, identically vanishes due to time-reversal symmetry [9]. The Hall effect in the nonlinear response regime, on the other hand, may not be subject to such symmetry constraints [10][11][12]. Here, we report the observation of the nonlinear Hall effect (NLHE) [12] in the electrical transport of the nonmagnetic 2D quantum material, bilayer WTe 2 . Specifically, flowing an electrical current in bilayer WTe 2 leads to a nonlinear Hall voltage in the absence of magnetic field. The NLHE exhibits unusual properties sharply distinct from the AHE in metals: The NLHE shows a quadratic I -V characteristic; It strongly dominates the nonlinear longitudinal response, leading to a Hall angle of ∼ 90 • . We further show that the NLHE directly measures the "dipole moment" [12] of the Berry curvature, which arises from layer-polarized Dirac fermions in bilayer WTe 2 . Our results demonstrate a new Hall effect and provide a powerful methodology to detect Berry curvature in a wide range of nonmagnetic quantum materials in an energy-resolved way.In 1879 Edwin H. Hall observed that, when an electrical current passes through a gold film under a magnetic field, a transverse voltage develops [1]. This effect, known as the Hall effect, forms the basis of both fundamental research and practical applications such as magnetic field measurements and motion detectors. In contrast to the classical Hall effect where the Lorentz force bends the trajectory of the charge carriers, quantum mechanics describes the "bending" by the intrinsic geometry of the quantum electron wavefunctions under time-reversal symmetry breaking. This crucial theoretical understanding eventually led to the seminal discoveries of the Berry curvature and the topological Chern number, which have become pillars of modern condensed matter physics [2, 3]. One important cur-3 rent frontier is to identify AHE with quantized or topological character in unconventional magnetic quantum materials, where spin-orbit coupling (SOC), geometrical frustration and electronic correlations coexist [3][4][5][6][7][8]. These extensive studies [1,[3][4][5][6][7][8] have established a paradigm for the electrical Hall effect: (1) A non-vanishing Hall conductivity arises from the momentum-integrated Berry curva...

Twisted bilayer graphene (TBLG) is one of the simplest van der Waals heterostructures, yet it yields a complex electronic system with intricate interplay between moiré physics and interlayer hybridization effects. We report on electronic transport measurements of high mobility small angle TBLG devices showing clear evidence for insulating states at the superlattice band edges, with thermal activation gaps several times larger than theoretically predicted. Moreover, Shubnikov-de Haas oscillations and tight binding calculations reveal that the band structure consists of two intersecting Fermi contours whose crossing points are effectively unhybridized. We attribute this to exponentially suppressed interlayer hopping amplitudes for momentum transfers larger than the moiré wave vector.

The kagome lattice based on 3d transition metals is a versatile platform for novel topological phases hosting symmetry-protected electronic excitations and exotic magnetic ground states. However, the paradigmatic states of the idealized two-dimensional (2D) kagome lattice -Dirac fermions and topological flat bands -have not been simultaneously observed, partly owing to the complex stacking structure of the kagome compounds studied to date. Here, we take the approach of examining FeSn, an antiferromagnetic single-layer kagome metal with spatially-decoupled kagome planes. Using polarization-and termination-dependent angleresolved photoemission spectroscopy (ARPES), we detect the momentum-space signatures of coexisting flat bands and Dirac fermions in the vicinity of the Fermi energy. Intriguingly, when complemented with bulk-sensitive de Haas-van Alphen (dHvA) measurements, our data reveal an even richer electronic structure that exhibits robust surface Dirac fermions on specific crystalline terminations. Through band structure calculations and matrix element simulations, we demonstrate that the bulk Dirac bands arise from in-plane localized Fe-3d orbitals under kagome symmetry, while the surface state realizes a rare example of fully spin-polarized 2D Dirac fermions when combined with spin-layer locking in FeSn. These results highlight FeSn as a prototypical host for the emergent excitations of the kagome lattice. The prospect to harness these excitations for novel topological phases and spintronic devices is a frontier of great promise at the confluence of topology, magnetism, and strongly-correlated electron physics.

Atomically thin molybdenum disulfide (MoS) is an ideal semiconductor material for field-effect transistors (FETs) with sub-10 nm channel lengths. The high effective mass and large bandgap of MoS minimize direct source-drain tunneling, while its atomically thin body maximizes the gate modulation efficiency in ultrashort-channel transistors. However, no experimental study to date has approached the sub-10 nm scale due to the multiple challenges related to nanofabrication at this length scale and the high contact resistance traditionally observed in MoS transistors. Here, using the semiconducting-to-metallic phase transition of MoS, we demonstrate sub-10 nm channel-length transistor fabrication by directed self-assembly patterning of mono- and trilayer MoS. This is done in a 7.5 nm half-pitch periodic chain of transistors where semiconducting (2H) MoS channel regions are seamlessly connected to metallic-phase (1T') MoS access and contact regions. The resulting 7.5 nm channel-length MoS FET has a low off-current of 10 pA/μm, an on/off current ratio of >10, and a subthreshold swing of 120 mV/dec. The experimental results presented in this work, combined with device transport modeling, reveal the remarkable potential of 2D MoS for future sub-10 nm technology nodes.

The ability in experiments to control the relative twist angle between successive layers in twodimensional (2D) materials offers a new approach to manipulating their electronic properties; we refer to this approach as "twistronics". A major challenge to theory is that, for arbitrary twist angles, the resulting structure involves incommensurate (aperiodic) 2D lattices. Here, we present a general method for the calculation of the electronic density of states of aperiodic 2D layered materials, using parameter-free hamiltonians derived from ab initio density-functional theory. We use graphene, a semimetal, and MoS2, a representative of the transition metal dichalcogenide (TMDC) family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multi-layered stacks in the presence of local disorder and external fields. We comment on the interesting features of their Density of States (DoS) as a function of twist-angle and local configuration and on how these features can be experimentally observed.