Twisted 2D materials form complex moiré structures that spontaneously reduce symmetry through picoscale deformation within a mesoscale lattice. We show twisted 2D materials contain a torsional displacement field comprised of three transverse periodic lattice distortions (PLD). The torsional PLD amplitude provides a single order parameter that concisely describes the structural complexity of twisted bilayer moirés. Moreover, the structure and amplitude of a torsional periodic lattice distortion is quantifiable using rudimentary electron diffraction methods sensitive to reciprocal space. In twisted bilayer graphene, the torsional PLD begins to form at angles below 3.89° and the amplitude reaches 8 pm around the magic angle of 1. 1°. At extremely low twist angles (e.g. below 0.25°) the amplitude increases and additional PLD harmonics arise to expand Bernal stacked domains separated by well defined solitonic boundaries. The torsional distortion field in twisted bilayer graphene is analytically described and has an upper bound of 22.6 pm. Similar torsional distortions are observed in twisted WS2, CrI3, and WSe2/MoSe2.
Periodic lattice distortions (PLD) are at the heart of correlated electronic behaviors such as superconductivity [1], metal-insulator transitions [2], and charge density waves (CDW) [3]. PLDs are typically intrinsic to a crystal [3, 4], Fermi-surface driven [5], accompanied by a CDW, and have periodicity spanning a few unit cells (~1-2nm). Recently, extrinsic van der Waals (vdW) driven superlattices with tunable periodicity (up to a few 100nm) were discovered in twisted bilayer graphene (tBLG) [6]. tBLG has been spotlighted for extraordinary correlated electron behaviors at the so-called "magic" angle (1.1°) [7]. Therefore, a full atomistic structural understanding is key to harnessing the exotic properties of tBLG. Here, we provide an analytic description of tBLG superlattices at and near the magic angle using a torsional PLD and report the torsional PLD amplitude of 7.8 ± 0.6 pm and 6.1 ± 0.4 pm for twist angle (θ) of 1.1° and 1.2°.
Twisted 2D materials form complex moiré structures that spontaneously reduce symmetry through picoscale deformation within a mesoscale lattice. We show twisted 2D materials contain a torsional displacement field comprised of three transverse periodic lattice distortions (PLD). The torsional PLD amplitude provides a single order parameter that concisely describes the structural complexity of twisted bilayer moirés. Moreover, the structure and amplitude of a torsional periodic lattice distortion is quantifiable using rudimentary electron diffraction methods sensitive to reciprocal space. In twisted bilayer graphene, the torsional PLD begins to form at angles below 3.89°and the amplitude reaches 8 pm around the magic angle of 1.1°. At extremely low twist angles (e.g. below 0.25°) the amplitude increases and additional PLD harmonics arise to expand Bernal stacked domains separated by well defined solitonic boundaries. The torsional distortion field in twisted bilayer graphene is analytically described and has an upper bound of 22.6 pm. Similar torsional distortions are observed in twisted WS 2 , CrI 3 , and WS 2 / MoSe 2 .
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