Imaging supermoiré relaxation in helical trilayer graphene

Technical Summary: Imaging Supermoiré Relaxation in Helical Trilayer Graphene

Problem and Motivation
In twisted van der Waals materials, local atomic relaxation driven by competing stacking energetics and elastic strain can significantly alter electronic structures. While lattice reconstruction in bilayers is known to generate ferroelectric domains and topological solitons, the behavior in multilayers is more complex due to interference between multiple moiré lattices. This interference creates a secondary, larger length scale known as the "supermoiré." In helical trilayer graphene (HTG), where three graphene layers are consecutively twisted by equal angles $\theta$, theory predicts that the system relaxes into large triangular domains of uniform moiré periodicity, separated by domain walls and meeting at AAA stacking sites. These domains are predicted to host topological boundary modes. However, direct experimental visualization of this supermoiré relaxation, its susceptibility to strain, and the specific electronic character of the domain walls has remained unaddressed.

Methodology
The authors employed scanning single-electron transistor (SET) microscopy to image the local electronic compressibility ($d\mu/dn$) and chemical potential ($\mu$) of an HTG device with a twist angle of approximately $1.45^\circ$. The device was encapsulated in hexagonal boron nitride (hBN) with a graphite bottom gate.

• Measurement Modalities: The SET operated in both direct current (d.c.) and alternating current (a.c.) modes. The d.c. mode tracks the chemical potential step ($\Delta\mu$) across incompressible states, while the a.c. mode measures inverse compressibility. The authors utilized the difference in response between these two modalities to qualitatively probe local conductivity, as high resistance leads to RC time constant effects that enhance a.c. signals.
• Strain Engineering: The study involved thermal cycling and device handling (including a tip crash and subsequent tip replacement) to induce changes in the strain state of the sample, allowing for the observation of supermoiré domain reshaping.
• Theoretical Modeling: The authors performed band structure calculations using a generalized Bistritzer-MacDonald continuum model, incorporating momentum-dependent tunneling terms, remote interlayer hopping, and strain effects. They also simulated the finite spatial resolution of the SET tip to compare theoretical predictions with experimental data.

Key Results

1. Imaging Supermoiré Domains: The SET measurements revealed periodic spatial modulations in the electronic compressibility at a length scale of hundreds of nanometers, significantly larger than the local moiré wavelength ($\lambda_M \approx 10$ nm). These modulations form a triangular lattice of local minima (identified as AAA stacking sites) surrounded by a honeycomb structure of local maxima (identified as the centers of $h$ and $\bar{h}$ domains). This confirms that HTG relaxes into large, moiré-periodic domains separated by domain walls.
2. Strain-Induced Domain Reshaping: The study demonstrated that the size of the supermoiré domains is highly sensitive to global heterostrain. After thermal cycling and device changes, the supermoiré domain sizes increased by up to four times the theoretical prediction for an unstrained sample, and the domains became more isotropic. Crucially, this reshaping occurred without altering the local twist angle or the electronic band structure within the domains, establishing a decoupling between the moiré and supermoiré length scales.
3. Enhanced Domain Wall Conductance: By comparing a.c. and d.c. measurements, the authors observed that the degree of "a.c. enhancement" (indicative of higher resistance) was lowest at the domain walls and AAA sites, and highest in the domain centers. This implies that the domain walls are more conductive than the domain interiors. This observation is consistent with theoretical predictions that the domain walls host counter-propagating topological edge modes (helical edge states) when the domains are tuned to incompressible fillings (e.g., $\nu = \pm 4$).
4. Magnetic Field Response: Landau fan measurements showed Hofstadter spectra consistent with a single moiré periodicity and broken $C_{2z}$ symmetry, further supporting the relaxation model. The absence of signatures corresponding to two distinct moiré periods ruled out the possibility of a large twist angle mismatch preventing relaxation.

Significance and Claims
The paper claims to provide the first real-space visualization of moiré-periodic domains in HTG and the direct observation of strain-engineered supermoiré relaxation. The work establishes that:

• Lattice relaxation in HTG creates a network of topological boundary modes at domain walls, which exhibit enhanced conductivity.
• The supermoiré length scale can be independently tuned via strain without perturbing the local moiré physics, offering a route to design reconfigurable topological networks.
• The separation of length scales allows for the potential design of fractional Chern mosaics and single-domain devices.

The authors emphasize that their findings clarify the interplay between lattice reconstruction and strain, providing a framework for interpreting transport measurements in mesoscopic moiré devices. They suggest that similar strain-engineering effects could be applicable to other twisted multilayer systems, such as transition metal dichalcogenides and 2D magnets, for engineering intertwined topology and correlations at the supermoiré scale.