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).

We introduce configuration space as a natural representation for calculating the mechanical relaxation patterns of incommensurate two-dimensional (2D) bilayers, bypassing supercell approximations to encompass aperiodic relaxation patterns. The approach can be applied to a wide variety of 2D materials through the use of a continuum model in combination with a generalized stacking fault energy for interlayer interactions. We present computational results for small-angle twisted bilayer graphene and molybdenum disulfide (MoS2), a representative material of the transition metal dichalcogenide (TMDC) family of 2D semiconductors. We calculate accurate relaxations for MoS2 even at small twist-angle values, enabled by the fact that our approach does not rely on empirical atomistic potentials for interlayer coupling. The results demonstrate the efficiency of the configuration space method by computing relaxations with minimal computational cost for twist angles down to 0.05 • , which is smaller than what can be explored by any available real space techniques. We also outline a general explanation of domain formation in 2D bilayers with nearly-aligned lattices, taking advantage of the relationship between real space and configuration space.Layered materials consist of 2D atomically thin sheets which are weakly coupled by the van der Waals force. For understanding the electronic and mechanical properties of multilayered structures of such materials, it is useful to view them as a series of conventional crystals with a weak perturbative interaction between sheets 1 . Bilayer systems with slight lattice misalignment due to differing lattice constants or relative twist-angle are of interest in optical and transport experiments 2-5 . In smallangle twisted bilayer graphene (tBLG), highly regular domain-wall patterns have been observed experimentally and studied theoretically 6-8 The appearance of domain walls is the result of atomic relaxation which serves to minimize the additional energy due to misalignment. Under electric-field gating the domain walls give rise to interesting topologically-protected edge states 9-11 . Understanding this relaxation and predicting its behavior in other nearly-aligned bilayers may be useful in the search for topological edge states and quantum information applications.To this end, we chose to study three different bilayer systems, graphene and the two high-symmetry alignments of MoS 2 , which is a standard representative of the transition-metal dichalcogenide family of 2D materials. A unit-cell with basis vectors a 1 = a(1, 0) and a 2 = a( √ 3/2, 1/2) is used, where the lattice parameter a for graphene is 2.47Å and for MoS 2 is 3.18Å . Insight into the mechanical domain-wall formation can be gained by paying special attention to the relationship between intralayer bonding energies and interlayer stacking energies, the latter arising from the much weaker van der Waals force. To do this, a consistent model must be chosen for both types of energy. We will assume smooth and slowly-varying relaxation...

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.

We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation. Our model, based on k · p perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models. The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist. We show that minimal models for the low-energy states of tBLG can be easily modified to capture the changes in electronic states as a function of twist angle.The discovery of correlated phases in twisted bilayer graphene (tBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring of the nature of unconventional superconductivity 1,2 . The challenge in modeling these phenomena from an atomistic perspective is that the actual structure of tBLG near the magic-angle twist (∼ 1.1 • ) where correlated behavior is observed, consists of a large number of atoms, exceeding 10 4 . To make progress from the theoretical point of view, a minimal model is needed that can capture the essence of singleparticle states near the Fermi level ("low-energy" states). Such a model should reproduce the energy spectrum as a function of their relative twist angle with reasonable accuracy and with the required fidelity in capturing the nature of low-energy states. The appearance of correlated behavior is related to bands with very low dispersion ("flat" bands) caused by interlayer hybridization between the two Dirac cones from the different layers 3-6 .Existing models based on DFT calculations 7,8 or large supercell tight-binding Hamiltonians 9-11 are too complex to form the basis of a realistic many-body theory. At the other extreme, simplified continuum models allow for efficient calculations, but are based on heuristic arguments about the nature of the relevant electronic states [12][13][14][15] . An important feature of the physical system is the presence of atomic relaxation near the magicangle twist, which has significant effects on the lowenergy bandstructure 10,[16][17][18][19] . Many simplified models for the flat bands of magic-angle tBLG have been proposed based on symmetry analysis, but they rely on empirical parameterization and are designed for only the magicangle twist configuration 20-22 , typically ignoring atomic relaxation.Here, we present an ab initio k · p perturbation continuum model for tBLG which accurately accounts for the effects of atomic relaxation. Our model reproduces the results of DFT-quality tight-binding hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value. Such a singl...

The recently demonstrated unconventional superconductivity 1 in twisted bilayer graphene (tBLG) opens the possibility for interesting applications of two-dimensional layers that involve correlated electron states. Here we explore the possibility of modifying electronic correlations by the application of uniaxial pressure on the weakly interacting layers, which results in increased interlayer coupling and a modification of the magic angle value and associated density of states. Our findings are based on first-principles calculations that accurately describe the height-dependent interlayer coupling through the combined use of Density Functional Theory and Maximally localized Wannier functions. We obtain the relationship between twist angle and external pressure for the magic angle flat bands of tBLG. This may provide a convenient method to tune electron correlations by controlling the length scale of the superlattice.Recent experimental results in twisted bilayer graphene (tBLG) have shown it to be an important system for understanding unconventional superconductivity 1 , and more generally correlated physics in two-dimensional (2D) materials 2 . This discovery comes after systematic development of experimental techniques which at present allow for twist angle control in stacked 2D heterostructures with a remarkable precision of 0.1 •3-6 . In bilayer graphene, a relative twist between the layers by a "magic" angle produces just the right amount of band hybridization to form flat bands near the Fermi level 7-13 . The flat bands have the majority of their electron density located at the AA-stacking regions of the moiré supercell. As the Fermi velocity goes to zero, the scale of the electron kinetic energy falls below the scale of the two-particle Coulomb interaction, producing correlated behavior, although the precise mechanism for these effects is still a topic of active research. Understanding the nature of the flat bands induced by the magic angle twist in tBLG is vital in studies of correlated electrons in 2D, and could lead to the discovery of other systems with similar behavior, generally referred to as "twistronics" 14 . We present here an ab-initio study of how the interlayer electronic coupling in tBLG depends on external uniaxial pressure in the direction perpendicular to the layers, and how this pressure could act as a tuning parameter for correlated physics.Manipulating superconductivity in tBLG by external pressure would follow the historic trend of using pressure to probe the nature of the superconducting T c 15,16 . The T c in conventional BCS superconductors usually decreases with pressure, but in unconventional superconductors pressure often increases T c . This is attributed to strong dependence of electronic correlation on external pressure, although the exact mechanism is not well understood and may vary between materials. 2D materials are particularly sensitive to pressure along the direction perpendicular to the layers, as they are coupled through weak van der Waals interactions. The mechanical ...