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Intrinsically multilayer moiré heterostructures

Aaron Dunbrack and Jennifer Cano
Phys. Rev. B 107, 235425 – Published 27 June 2023
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  • INTRODUCTION
  • CONFIGURATION SPACE FOR BILAYERS: MOIRÉ…
  • MOIRÉ IN FREQUENCY SPACE
  • CONFIGURATION SPACE OF INTRINSICALLY…
  • DETECTION OF INTRINSICALLY TRILAYER…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    We introduce trilayer and multilayer moiré heterostructures that cannot be viewed from the “moiré-of-moiré” perspective of helically twisted trilayer graphene. These “intrinsically trilayer” moiré systems feature periodic modulation of a local quasicrystalline structure. They open the door to realizing moiré heterostructures with vastly more material constituents because they do not constrain the lattice constants of the layers. In this manuscript, we define intrinsically multilayer patterns, provide a recipe for their construction, derive their local configuration space, and connect the visual patterns to physical observables in material systems.

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    • Received 22 February 2023
    • Accepted 2 June 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.235425

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Crystal structureFlat bandsQuasicrystalline structureTwistronics
    1. Physical Systems
    Quasicrystals
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Aaron Dunbrack1 and Jennifer Cano1,2

    • 1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11974, USA
    • 2Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA

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    Issue

    Vol. 107, Iss. 23 — 15 June 2023

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    Images

    • Figure 1
      Figure 1

      A moiré lattice of two square layers twisted at 6.7329∘. Commensurate lattice in red, moiré lattice in blue.

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    • Figure 2
      Figure 2

      A moiré lattice of two hexagonal layers with unit-length interatomic distance stacked with a relative twist angle of 5∘. Red and blue circles indicate an “AA-stacked” region where hexagons align and an “AB-stacked” region where they are offset, respectively.

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    • Figure 3
      Figure 3

      Moiré pattern from two square lattices with side lengths 1 and 2 arranged with a relative twist angle of 42∘.

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    • Figure 4
      Figure 4

      A moiré pattern formed by two unit triangular lattices arranged with a relative twist of 22.4∘ (21.8∘+0.6∘). The resulting triangular moiré lattice has a unit cell of side length 36.1, shown in green. The moiré pattern is subtle, alternating between regions with individual sixfold-symmetric “centers” (red) and regions with triplets of “centers” connected in a triangle (blue). A larger picture of the moiré pattern is shown in Fig. S2-1 [68].

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    • Figure 5
      Figure 5

      A moiré lattice formed by two unit square lattices arranged at a relative twist of 37.5∘ (36.9∘+0.6∘), with a 42.7 side length moiré cell (green square). There is a resulting pattern of “holey regions” (red square) and “knitted regions” (blue square). A larger unannotated picture of the moiré pattern is presented in Fig. S2-2 [68].

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    • Figure 6
      Figure 6

      Reciprocal space of two square lattices stacked at a commensurate 36.9∘ twist angle. Red(blue) open circles indicate the reciprocal lattice vectors of the top(bottom) layer; black filled circles indicate shared reciprocal lattice vectors. Thick lines shows that the (1,2) mode of the blue layer coincides with the (2,1) mode of the red layer. Light gray indicates the reciprocal commensurate lattice.

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    • Figure 7
      Figure 7

      Lowest frequency modes of two square lattices at a small relative twist. Red and blue circles indicate reciprocal lattice vectors of each layer. The small difference between the lowest modes k1−k2 gives the moiré wave vector kM, from which Eq. (21) follows.

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    • Figure 8
      Figure 8

      A moiré lattice of three unit square lattices at a relative twist of 119.3∘, resulting in a moiré unit cell of side length 47 (drawn in green). Local structures are shown at right. Top right illustrates the reciprocal lattice vectors at exactly 120∘ (left) and after the 0.7∘ deviation from the singular structure (right, deviation exaggerated for illustration purposes), resulting in the moiré reciprocal lattice vector GM shown in green. A larger unannotated picture of the moiré pattern is presented in Fig. S2-3 [68].

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    • Figure 9
      Figure 9

      Starting from a particular singular structure, a small twist away combined with a corresponding strain results in another singular structure. These transformations yield a manifold of singular structures rather than an isolated point, as occurs for bilayers.

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    • Figure 10
      Figure 10

      Several singular structures of TTLG along two specific slices of the four-dimensional parameter space (θ12,θ23,δ12,δ32) indicated by solid colored lines. The dashed green line represents the constraint of helically twisted trilayer graphene. The bilayer singular structures shown in the left figure deviate from helically twisted trilayer graphene to order θ, but the trilayer singular structure shown in the right figure only deviates to order θ2. Hence, the green singular structure produces a moiré pattern at 1/θ2 scale, whereas the bilayer singular structures plotted in blue/red/purple produce (competing) moiré pattern at 1/θ scale.

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    • Figure 11
      Figure 11

      Different quadrilaterals can be formed with the same side lengths. Consequently, a stacked four-layer system can have multiple singular configurations with different large twist angles, i.e., there are different twist angles such that ∑iGi=0, where Gi is a reciprocal lattice vector in each layer. Moiré lattices formed by twisting slightly away from these configurations exhibit the same physics on the moiré length scale, provided the small twists are chosen to give the same moiré lattice vectors.

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    • Figure 12
      Figure 12

      Two layers of graphene (black) arranged with a large twist angle generate a moiré pattern with the additional outer (blue) layers on the exterior, which are aligned with each other. (Left) Physical configuration of the layers. (Right) The large black hexagons and small blue hexagons indicate the BZ of graphene and the outer layers, respectively. The reciprocal lattice vector of the outer layers couples the K points of the graphene layers, effectively compensating for the large twist.

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    • Figure 13
      Figure 13

      Inducing two periodic potentials (reciprocal lattice vectors in blue) on a layer of graphene (BZ in black) produces a moiré superlattice (reciprocal lattice vector in red). Left: sandwiching graphene between two nearly aligned layers produces an effective superlattice potential. The alignment of the graphene layer is unimportant. Right: if the two other layers are twisted at a large angle so their reciprocal lattice vector adds to one of graphene, intrinsically trilayer moiré can arise.

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    • Figure 14
      Figure 14

      Frequency modes of triangular (red circles) and square (blue circles) lattices with the same lattice constants at zero twist. The black filled circles indicate shared modes that yield moiré modes after a small twist or mismatch. Note since they only align along a 1D subspace, the resulting moiré is also 1D.

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    • Figure 15
      Figure 15

      A unit square lattice on a unit triangular lattice at a relative twist angle of 7∘ exhibits a 1D moiré pattern.

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