Abstract
The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a super lattice, called a moir\'e pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as "moir\'e molecules" or "moir\'e crystals". The molecules generate localized electronic states and moir\'e flat bands, while the crystals are sometimes unconventional and produce electronic banding in the configuration basis. The underpinning of this classification is the duality between interlayer configuration and monolayer Bloch momentum in moir\'e Hamiltonians. We apply this understanding to diagrams of local electron density in untwisted geometries to produce intuitive and quantitative predictions of twistronic properties. We provide a conceptual introduction to this framework through a one-dimensional model, and then apply it to 2D twisted bilayers of the semi-metal graphene, and of MoS$_2$, a representative material of the transition metal dichalcogenide (TMDC) family of semiconductors. This level of thorough understanding of twistronic phenomena is vital in the search for new material platforms for localized moir\'e electrons.
Highlights
The number of two-dimensional (2D) materials that have been experimentally isolated as single layers of atomic-scale thickness has been growing at a rapid pace [1,2,3,4,5,6]
We show that instead of relying on the bands of Bloch theory in momentum space, the local density of states (LDOS) in either the space of atomic configurations or Bloch states provides a surprising amount of clarity in the study of twisted electronic structure
For the second and third levels of the oscillator, this provides critical values of θc(1) ≈ 3.5◦ and θc(2) ≈ 2.8◦. These are all in excellent agreement with the results of the tight-binding calculations, namely, that when θ < θc(n) the bandwidth for the nth level is below 5 meV. Comparing this result to full density functional theory (DFT) calculations [23] we find good agreement to the unrelaxed case: the flat bands are associated with AA stacking spots at the valence band maximum
Summary
The number of two-dimensional (2D) materials that have been experimentally isolated as single layers of atomic-scale thickness has been growing at a rapid pace [1,2,3,4,5,6]. The LDOS can form sharply defined features in configuration space and energy, which we call “configuration banding.” These two regimes of interesting twistronic features, hosting localized modes or banding, are analogous to the electronic structure observed in conventional molecules (localized states) or crystals (extended states or bands). This pattern arises because of a duality between the momentum and position operators that can occur in specific scenarios for moiré Hamiltonians. Configuration banding occurs when an infinite number of Bloch states are needed to accurately capture the interlayer interaction between two band structures; this corresponds to the “momentum crystal,” which often exists alongside conventional Bloch bands in the moiré systems. The appendix examines how configuration banding manifests in the real space LDOS
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