Locally resolved electronic textures of reconstruction domains in marginally twisted monolayer-bilayer graphene

Technical Summary: Locally Resolved Electronic Textures of Reconstruction Domains in Marginally Twisted Monolayer–Bilayer Graphene

Problem Statement
While the "magic-angle" regime of twisted bilayer graphene (TBG) and complex multilayer systems has been extensively studied for its flat bands and correlated states, the physics of the "marginally twisted" regime (twist angles $\theta \approx 0^\circ$) remains qualitatively distinct and underexplored. In this limit, the system is not governed by a uniform moiré potential but rather undergoes structural reconstruction into a network of domains with uniform stacking orders separated by domain walls. Previous transport experiments in twisted monolayer-bilayer graphene (1L/2L) at larger angles ($0.9^\circ$–$1.6^\circ$) have revealed tunable correlated states, but the electronic properties of the reconstruction domains in the marginal twist limit, where the system relaxes into distinct stacking configurations, have not been directly resolved. The interplay between atomic-scale structural relaxation, out-of-plane deformations, and the resulting local electronic textures in 1L/2L graphene at near-zero twist angles constitutes the central problem addressed in this work.

Methodology
The authors investigated marginally twisted 1L/2L graphene heterostructures fabricated on hexagonal boron nitride (hBN) substrates using a tear-and-stack technique. The twist angle was determined to be $\theta = 0.30^\circ \pm 0.01^\circ$ based on the moiré superlattice period ($\lambda \approx 47$ nm).

The study employed a combined experimental and theoretical approach:

1. Scanning Tunneling Microscopy and Spectroscopy (STM/STS): Experiments were conducted at room temperature (300 K) in ultra-high vacuum. The authors acquired topographic maps and spatially resolved spectroscopic maps ($dI/dV$) to probe the local density of states (LDOS) across different stacking domains.
2. Theoretical Modeling:
   • Electronic Structure: A hybrid $k \cdot p$ tight-binding model for trilayer graphene was used to compute band dispersions and projected density of states (DOS). The model included self-consistent Hartree corrections and proximity-induced potentials ($\Delta_{hBN}$ and $\Delta_{vac}$) to account for the substrate and vacuum interfaces. Thermal broadening (82 meV) was applied to simulate room-temperature conditions.
   • Lattice Relaxation: Mesoscale lattice relaxation calculations were performed using a multiscale approach combining interpolated adhesion energy densities (derived from DFT) and continuum elasticity theory. This modeled in-plane ($\mathbf{u}$) and out-of-plane ($\zeta$) displacement fields to minimize the total energy of the moiré unit cell, accounting for bending rigidity and interlayer interactions.

Key Contributions and Results

• Identification of Three Distinct Stacking Domains: The system relaxes into a triangular network of three high-symmetry stacking configurations: Bernal (ABA), Rhombohedral (ABC), and the energetically unfavorable AAB stacking. STM topography reveals that AAB nodes appear as bright features with greater apparent height ($\sim 330$ pm relative to other domains) compared to the ABA and ABC triangles.
• Domain-Specific Electronic Textures: STS measurements reveal unique electronic fingerprints for each stacking domain:
  • ABA (Bernal): Exhibits a characteristic V-shaped DOS profile linked to Dirac-type bands.
  • ABC (Rhombohedral): Shows a prominent conduction band peak corresponding to flat surface bands, though the experimental peak is $\sim 50$ meV deeper than theoretical predictions.
  • AAB: Displays a V-shaped spectrum with a characteristic peak at the Fermi level.
• Bias-Dependent Hierarchy Switching: The relative tunneling intensity between Bernal (ABA) and Rhombohedral (ABC) domains inverts as a function of bias voltage. At low bias in the conduction band, ABC domains exhibit higher $dI/dV$ intensity than ABA, but this trend reverses at higher energies. This inversion is captured in spectroscopic maps and reproduced by normalized LDOS calculations.
• Domain Wall "Twirling" and Out-of-Plane Deformation: Theoretical calculations predict that to minimize energy, the three graphene layers bend in the same direction at the AAB stacking nodes. This out-of-plane deformation induces a "twirling" of the domain wall network around these nodes. This structural feature is experimentally observed in the $dI/dV$ maps, where domain walls appear to connect around the AAB nodes at specific biases (e.g., 100 mV), matching the theoretical prediction of lattice relaxation.
• Domain Wall Width: The width of the domain walls separating the stacking domains was estimated to be $(8 \pm 3)$ nm from experimental data, consistent with theoretical calculations of approximately 7 nm.

Significance
The paper establishes that in the marginally twisted regime, the electronic landscape of 1L/2L graphene is governed not by a periodic moiré potential, but by atomic-scale structural reconstruction into distinct stacking domains. The work demonstrates that:

1. Structural relaxation creates a network of domains with unique, spatially resolved electronic textures.
2. Out-of-plane deformations play a critical role in the formation of the domain wall network, specifically driving the "twirling" texture around AAB nodes.
3. The electronic hierarchy between different stacking orders is tunable via bias voltage.

These findings highlight structural relaxation as a primary determinant of electronic behavior in twisted van der Waals systems. The authors conclude that the marginal twist regime offers a largely untapped space for engineering electronic properties through the control of stacking order and structural degrees of freedom in van der Waals heterostructures, providing fundamental insights into structure-property relationships in moiré-driven phenomena.