Twistraintronics in Square Moire Superlattices of Stacked Graphene Layers

Technical Summary: Twistraintronics in Square Moiré Superlattices of Stacked Graphene Layers

Problem and Context
The discovery of unconventional superconductivity and strongly correlated phases in twisted bilayer graphene (TBG) has established moiré heterostructures as a primary platform for exploring quantum matter. These phenomena rely critically on the electronic modulation induced by the moiré potential, particularly near the "magic angle" where flat bands enhance electronic correlations. While standard twist-only configurations yield trigonal moiré patterns, the addition of strain offers a pathway to diverse moiré geometries. Theoretical work has predicted that specific combinations of twist and strain could generate square moiré patterns in stacked hexagonal lattices, yet experimental realization of such patterns, particularly those induced by externally applied strains, had not been reported. This study addresses the gap in experimentally controlling and observing square moiré superlattices and their associated electronic properties.

Methodology
The researchers utilized graphene samples synthesized via thermal decomposition of 6H-SiC(000-1), a method known to produce high-quality, large-area graphene domains with rotational disorder. The experimental approach centered on "twistraintronics"—the active tuning of electronic properties via the interplay of twist and strain.

• Strain Manipulation: Instead of applying global strain, the team manipulated local strain at the nanoscale by laterally displacing native graphene wrinkles using the tip of a Scanning Tunneling Microscope (STM). These wrinkles, formed during sample preparation due to thermal expansion mismatch, were pushed over distances exceeding 100 nm to alter the local strain field.
• Characterization: The team employed STM for topographic imaging and Scanning Tunneling Spectroscopy (STS) to measure differential conductance ($dI/dV$), which is proportional to the Local Density of States (LDOS). Measurements were conducted at base temperatures of 3–4 K.
• Theoretical Modeling: A continuum model of TBG was extended to include strain effects. This model accounted for:
  1. Modifications to moiré vectors and interlayer coupling.
  2. Strain-induced scalar and gauge fields shifting Dirac points.
  3. Electrostatic interactions via a self-consistent Hartree potential to account for charge inhomogeneities.
     The model sought the specific combination of twist and shear strain that minimizes elastic energy to reproduce experimental observations.

Key Contributions and Results

1. First Observation of Controlled Square Moiré Patterns: The paper reports the first experimental observation of reversible, strain-induced square moiré patterns in stacked graphene. By displacing a wrinkle, the authors drove a transition from the native trigonal moiré order to a square order and back, demonstrating active control over the superlattice geometry.
2. Structural Characterization: STM imaging revealed that the square patterns consist of elliptically shaped AA-stacking domains. The transition from trigonal to square geometry was shown to be reversible, confirming that the strain field could be dynamically tuned.
3. Electronic Correlations and Spectroscopy: STS measurements along different paths of the square unit cell revealed narrow electronic bands with two prominent Van Hove singularities (VHs) near the Fermi energy ($E_F$). Crucially, these singularities exhibited a distinct splitting into two components. The data showed a peak at $E_F$ and another approximately 20–30 meV below it, both split by the applied strain.
4. Theoretical Validation: The continuum model, specifically under the condition of minimum elastic energy (achieved via shear strain), successfully reproduced the experimental features. The model predicted that the shear strain configuration leads to the observed elliptical AA domains and the specific splitting of the Van Hove singularities. The best fit was obtained for a filling of $\nu \approx +1$ electron per moiré unit cell.
5. Strain-Induced Symmetry Breaking: The study demonstrated that while the geometric appearance of square patterns can arise from a continuous family of twist-strain configurations, the electronic properties (specifically the VH splitting) are uniquely determined by the minimum energy shear strain configuration. This allows the geometry and electronic state to be uniquely identified through combined STM and STS analysis.

Significance
The authors claim this work demonstrates that "twistraintronics" enables the realization of highly correlated electronic states in moiré heterostructures with geometries previously inaccessible. By combining twist and strain, the system achieves a square-symmetric moiré lattice with strong electronic correlations, narrow bands, and a high density of states near the Fermi level. The presence of these features, particularly the proximity of the Fermi level to Van Hove singularities and the strong Hartree contribution, suggests that this system is a promising platform for investigating anisotropic superconductivity in square-symmetric lattices. The work establishes a method to actively tune the properties of moiré superlattices within a single sample, providing a unique tool for investigating strain effects on correlated electronic states.