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.

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We give an exact formulation for the transport coefficients of incommensurate two-dimensional atomic multilayer systems in the tight-binding approximation. This formulation is based upon the C˚algebra framework introduced by Bellissard and collaborators [2,3] to study aperiodic solids (disordered crystals, quasicrystals, and amorphous materials), notably in the presence of magnetic fields (quantum Hall effect). We also present numerical approximations and test our methods on a one-dimensional incommensurate bilayer system.