Primitive-cell-resolved Crystallography for Moiré Bilayers from Imaging

Abstract

Accurate geometric decoding of moiré bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moiré transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moiré crystallography framework that treats the beating-to-moiré relation in full generality and introduces a complete descriptor set $\{θ_r,\boldsymbol{\varepsilon},(T_{Mt},T_{Mb}),N_B\}$, where the integer moiré--layer matrices $(T_{Mt},T_{Mb})$ and the beating number $N_B$ determine the commensurate unit cell. A hybrid analytical--numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain $(T_{Mt},T_{Mb})$ and $N_B$, and extracts $(θ_r,\varepsilon_b,θ_u,\varepsilon_u)$ with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a $N_B=3$ primitive cell rather than a $N_B=9$ aligned supercell, reducing the atomistic basis threefold and correcting the moiré Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.

Figures (4)

• Figure 1: (a) Schematic of the moiré (M) and beating (B) lattices for a bilayer with twist angle $6.836^{\circ}$ and zero strain. The primitive vectors are $\{\mathbf{a}_{M1},\mathbf{a}_{M2}\}$ and $\{\mathbf{a}_{B1},\mathbf{a}_{B2}\}$. The quantities $(M_1,M_2,M_3)=(|\mathbf{a}_{M1}|,|\mathbf{a}_{M2}|,|\mathbf{a}_{M1}+\mathbf{a}_{M2}|)$ and $(B_1,B_2,B_3)=(|\mathbf{a}_{B1}|,|\mathbf{a}_{B2}|,|\mathbf{a}_{B1}+\mathbf{a}_{B2}|)$ are defined accordingly. The three panels on the right enlarge the circled regions, showing registries at AA, BB, and hollow-hollow sites; green dots mark beating centers. (b) Reciprocal-space diffraction schematics. The top (bottom) panel illustrates cases where bottom-layer information is detectable (undetectable). Red, blue, and green circles indicate diffraction peaks from the top layer, bottom layer, and beating, respectively. Insets in the lower right corners indicate a thin top layer (top panel) and a thick top layer (bottom panel). (c) The moiré primitive vectors $\{\mathbf{a}_{M1},\mathbf{a}_{M2}\}$ are related to the top-layer lattice vectors $\{\mathbf{a}_{t1},\mathbf{a}_{t2}\}$ via an integer transformation matrix $T_{Mt}$ (or to the bottom-layer vectors $\{\mathbf{a}_{b1},\mathbf{a}_{b2}\}$ via $T_{Mb}$). (d) Key geometric control parameters for generating a moiré superlattice: twist angle $(\theta_r)$ and strain (biaxial $(\epsilon_b)$; uniaxial magnitude $(\epsilon_u)$ with direction $(\theta_u)$). The material's elastic response is characterized by its Poisson's ratio $(\delta)$.
• Figure 2: Reciprocal-space diffraction patterns for a bilayer specified by the integer transformation matrices $(i,j,k,l)=(5,-6,6,9)$ and $(m,n,q,r)=(6,-4,5,9)$, yielding a beating number of $N_B=2$. (a) Full detection: diffraction peaks from the top layer, bottom layer, and the beating pattern. Arrows indicate the corresponding primitive reciprocal-lattice vectors $\{\mathbf{k}_{t1},\mathbf{k}_{t2}\}$ (blue), $\{\mathbf{k}_{b1},\mathbf{k}_{b2}\}$ (red), and $\{\mathbf{k}_{B1},\mathbf{k}_{B2}\}$ (magenta). (b) Bottom layer undetectable: only top-layer and beating peaks are observed. Red hollow circles mark a candidate reconstruction of the missing bottom-layer peaks obtained under the assumption that two observed beating peaks (e.g., $\mathbf{k}_2$ and $\mathbf{k}_3$) are taken to form the primitive beating basis $\{\mathbf{k}_{B1},\mathbf{k}_{B2}\}=\{\mathbf{k}_{2},\mathbf{k}_{3}\}$, together with Eq. \ref{['Eq_Beating_def_fundamental']}.
• Figure 3: Schematic workflow for determining the integer transformation matrices from diffraction data, contrasting the special case $N_B=1$ (a–c) with the general case $N_B\neq1$ (d–f). Simulations use $(i,j,k,l,m,n,q,r)=(3,-4,4,7,\,4,-3,3,7)$ for $N_B=1$ and $(3,-4,5,7,\,4,-3,3,7)$ for $N_B=2$. (a,d) A reciprocal beating grid (black lines) is constructed from the primitive beating vectors $\{\mathbf{k}_{B1},\mathbf{k}_{B2}\}$ (magenta arrows). (b) For $N_B=1$, the top-layer ($\{\mathbf{k}_{t1},\mathbf{k}_{t2}\}$, blue arrows) and bottom-layer ($\{\mathbf{k}_{b1},\mathbf{k}_{b2}\}$, red arrows) peaks lie exactly on the beating-grid vertices, i.e., have integer coordinates. (c) The same peaks coincide with the vertices of the moiré reciprocal lattice (green hollow circles) defined by $\{\mathbf{k}_{M1},\mathbf{k}_{M2}\}$ (black arrows); in this special case the beating and moiré grids coincide. (e) For $N_B=2$, the layer peaks do not fall on beating-grid vertices; their fractional coordinates within each cell are obtained by subdividing the grid (green lines), yielding the coprime integers $(\alpha_{ij},\beta_{ij},N_{tb})$ of Eq. \ref{['Eq_ktkb_kB_STM']}. (f) In contrast, the layer peaks always land exactly on the moiré lattice vertices (green hollow circles), confirming that the moiré lattice is the Bravais lattice of the bilayer. These coordinates feed Eq. \ref{['Eq_step4']} to solve $(i,j,k,l,m,n,q,r)$.
• Figure 4: Visualizing the moiré crystallography framework: a non-aligned case study. (a) Simulated diffraction pattern of the bilayer graphene system based on imaging data from Ref. Ref_strain_PRL_2018. The reciprocal beating grid (black lines) is defined by the observable beating vectors $\mathbf{k}_{B1}$ and $\mathbf{k}_{B2}$ (magenta arrows, shown magnified in (b)). (b-e) Magnified views of the regions marked by black, red, and blue boxes in (a). (b) Direct comparison between the beating basis $\{\mathbf{k}_{B1}, \mathbf{k}_{B2}\}$ (magenta) and the primitive moiré basis $\{\mathbf{k}_{M1}, \mathbf{k}_{M2}\}$ (green), showing their distinct orientations and magnitudes. (c) Integer occupancy on the beating grid. A specific direction where the layer diffraction peaks $\mathbf{k}_{t2}$ and $\mathbf{k}_{b2}$ (blue, red) coincide with the vertices of the beating grid, satisfying $\mathbf{k}_{t2} - \mathbf{k}_{b2} = \mathbf{k}_{B2}$. (d) Fractional occupancy and the moiré grid in the aligned ($\mathbf{k}_{B\alpha} \parallel \mathbf{k}_{M\alpha}$) configuration Ref_strain_PRL_2018. In contrast to (c), the layer peaks $\mathbf{k}_{t1}$ and $\mathbf{k}_{b1}$ are located at $1/3$ fractional positions relative to the beating grid. The orange sub-grid represents the moiré lattice derived by assuming the beating and moiré vectors are aligned Ref_strain_PRL_2018, which results in a redundant $N_B=9$ expansion. (e) Primitive moiré grid via general matrix decoupling. The green grid represents the result of our general matrix decoupling method, which accounts for non-aligned configurations in the general case. This primitive moiré grid identifies the $N_B=3$ Bravais lattice and coincides with all observed layer diffraction peaks $\{\mathbf{k}_{t1}, \mathbf{k}_{b1}, \mathbf{k}_{t2}, \mathbf{k}_{b2}\}$ without redundant expansion.