The ability in experiments to control the relative twist angle between successive layers in two-dimensional (2D) materials offers an 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 ${\mathrm{MoS}}_2$, a representative of the transition-metal dichalcogenide family of 2D semiconductors, to illustrate the application of our method, which enables fast and efficient simulation of multilayered stacks in the presence of local disorder and external fields. We comment on the interesting features of their density of states as a function of twist angle and local configuration and on how these features can be experimentally observed.

We introduce configuration space as a natural representation for calculating the mechanical relaxation patterns of incommensurate two-dimensional (2D) bilayers. 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 (${\mathrm{MoS}}_2$), a representative material of the transition-metal dichalcogenide family of 2D semiconductors. We calculate accurate relaxations for ${\mathrm{MoS}}_2$ 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. 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. The configuration space approach also enables calculation of relaxations in incommensurate multilayer systems.

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 [Coherent and Dissipative Transport in Aperiodic Solids, Lecture Notes in Physics (Springer, 2003), Vol. 597, pp. 413–486 and J. Math. Phys. 35(10), 5373–5451 (1994)] 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.

The tensor-train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certa...