Geometric control of the moire twist angle in heterobilayer flakes

TL;DR

The paper addresses the challenge of precisely controlling twist angles in lattice-mismatched 2D heterobilayers by proposing a geometry-driven locking mechanism where finite-edge geometry of flakes aligns with the moiré pattern. Through large-scale atomistic simulations of graphene on hBN, complemented by an analytical framework, the authors identify robust metastable angles near $0.61^\circ$ (armchair) and $1.89^\circ$ (zigzag) that are stabilized by large interlayer energy barriers and reinforced by lattice mismatch via in-plane heterostrain. Relaxation effects and substrate-engineering considerations show that these alignment angles persist and can be tuned continuously, offering a static, strain-tunable route to precision moiré engineering. The work provides a general geometric principle applicable to other van der Waals heterostructures and helps explain observed macroscopic self-orientation phenomena in heteroflakes.

Abstract

We demonstrate a finite twist-angle stabilization mechanism in lattice-mismatched 2D heterobilayers, which results from the geometric alignment between the flake edges and its moire pattern. Using atomistic simulations of graphene on hexagonal boron nitride flakes with diameters of up to $\sim 2500$Å, we identify robust metastable angles at $\sim 0.61^\circ$ for armchair and $\sim1.89^\circ$ for zigzag-edged flakes, tunable via in-plane heterostrain. This locking mechanism, which relies on energy barriers that are an order of magnitude larger than those of nearby metastable twist angles, provides a geometric route to precision twist-angle control of two-dimensional heterostructures and to understand the self-orientation of macroscopic flakes.

Figures (5)

• Figure 1: Hexagonal graphene flakes on an extended hBN substrate, illustrating the twist angle $\theta$ and the resulting moiré angle $\varphi$. The dotted blue line in (a) marks an armchair edge; the dotted maroon line in (b) highlights a zigzag edge. Panels (c) and (d) show configurations at $\theta_A = 0.61^\circ$ and $\theta_Z = 1.89^\circ$, respectively, where maximal alignment between the moiré pattern and the flake edge is achieved, based on Eqs. (\ref{['main_v2:eq:Moireangle30']}) and (\ref{['main_v2:eq:Moireangle60']}). $R$ denotes the flake radius for both hexagonal (shown) and circular (not shown) geometries, shown here for $R= 250$ Å.
• Figure 2: Rigid interlayer energy per atom as a function of twist angle $\theta$ for G/hBN flakes, obtained using Eq. (\ref{['main_v2:eq:Etot']}). Panels (a) and (b) show hexagons with armchair and zigzag edges, respectively; panel (c) shows circular flakes with mixed edges. Energies are referenced to the unrotated AA-stacked configuration. Vertical dashed lines indicate analytical predictions for alignment angles $\theta_A$ and $\theta_Z$ based on Eqs. (\ref{['main_v2:eq:Moireangle30']}) and (\ref{['main_v2:eq:Moireangle60']}). (d) and (e) illustrate how the alignment angles from (a) and (b), respectively, can be modified by changing the lattice mismatch. (f) shows the moiré angle $\varphi$ versus $\theta$ for different lattice mismatches $\varepsilon$. Without mismatch, $\varphi$ varies weakly at small angles, while for $\varepsilon \neq 0$, it crosses $\varphi = 30^\circ$ or $0^\circ$ at $\varepsilon$-dependent twist angles, depending on flake orientation (see Fig. \ref{['main_v2:fig:fig0M']}).
• Figure 3: Twist-angle-dependent energy oscillations near the alignment angles $\theta_A$ and $\theta_Z$ for various flake sizes $R/a$, for armchair- (a) and zigzag-terminated (b) hexagonal flakes. The rotation center axis is chosen to lie at the AA stacking. Bottom panels show how these angles converge toward the analytical predictions (vertical dashed orange lines) as flake size increases, confirming the robustness of geometric locking. The green-shaded region emphasizes this trend. Arbitrary units are used for visual clarity. The top panels present the scaling behavior of the energy barrier, $\Delta E_\text{inter}(\theta) = E_\text{inter}(\theta) - E_\text{inter}(\theta^-)$, evaluated at $\theta^\star_A$ and $\theta^\star_Z$ whose maxima are close to the analytical values for $R/a=1000$ flakes. These angles are highlighted by red curved dashed lines in the bottom panels, where $\theta^-$ denotes the adjacent energy minimum. The comparatively much smaller barrier scaling at a nearby angle extremum away from alignment (blue), labeled $\theta^*$, highlights the enhanced stability of the alignment angles.
• Figure 4: Interlayer energy $\Delta E_\text{inter} (\theta) = E_\text{inter}(\theta) - E_\text{inter}^\text{AB}(0)$ as a function of twist angle for flakes with $R/a = 100$ and $500$ for AA-stacking rotation center (black) and AB-stacking center (red). Dashed lines correspond to rigid configurations; solid lines show relaxed energies. Each curve is individually referenced to its reference $E_\text{inter}^\text{AB}(0)$ for rigid or relaxed geometries at zero twist angle, see Table II in Sect. VI of the Supplemental material. Plateaus in the relaxed curves indicate spontaneous rotation into nearby metastable angles. Final angles, extracted using the Kabsch algorithm kabsch1976solution, are labeled in red above each plateau. The dashed blue vertical lines indicate the analytical predictions of $\theta_A$ and $\theta_Z$. AB curves are overall more stable than corresponding AA curves.
• Figure 5: Alignment twist angle $\theta_A$ as a function of lattice constant $a$ for various 2D materials stacked on a set substrate, valid for armchair edged flakes based on Eq. (\ref{['main_v2:eq:Moireangle30']}). Curves indicate the angles at which each material forms an aligned moiré superlattice. Vertical dashed lines and color-coded labels mark experimental lattice constants of various materials, from graphene ($a = 2.456$Å) to MoTe$_2$ ($a = 3.521$Å), taken from the ICSD ICSD. Colors distinguish materials for ease of comparison. A wider palette of materials is illustrated in the Supplemental Material in Sect. VII.