Buckling and flat bands in twisted bilayer graphene

Technical Summary: Buckling and Flat Bands in Twisted Bilayer Graphene

Problem Statement
Twisted bilayer graphene (TBG) at "magic" twist angles ($\theta \approx 1.1^\circ$) exhibits flat electronic bands with a high density of states (DOS), serving as a platform for correlated electronic phases such as superconductivity and insulating states. While periodic buckling in monolayer graphene has been proposed as a tunable mechanism to generate flat bands via strain-induced pseudomagnetic fields (PMFs), the interplay between intrinsic moiré physics and externally controlled periodic buckling in bilayer systems remains largely unexplored. Specifically, it is unclear how combining these two mechanisms affects band flattening, symmetry properties, and the density of states across different twist angles, particularly whether they act additively or competitively.

Methodology
The authors employ an atom-resolved tight-binding model to investigate TBG under periodic buckling.

• Hamiltonian: The system is described using a Slater-Koster parametrization for distance-dependent hopping strengths, including both in-plane ($t_0$) and out-of-plane ($t_\perp$) couplings up to sixth-order nearest neighbors to ensure accurate reproduction of the magic-angle band structure.
• Buckling Model: Periodic buckling is modeled by introducing a spatially varying pseudomagnetic field (PMF), $B(\mathbf{r})$, derived from empirical relations in monolayer graphene. This PMF is generated by modulating the in-plane hopping strengths ($\delta t_i$) according to the strain-induced gauge field formalism. The buckling profile is defined as a superlattice centered either on AA-stacked or AB-stacked regions of the moiré unit cell.
• Parameters: The study covers a wide range of twist angles, from large angles ($\theta \approx 3.15^\circ$) down to the magic angle ($\theta \approx 1.20^\circ$). The buckling strength is tuned via the PMF amplitude ($B_0$), scaled relative to the moiré length ($L_m$) to maintain consistent physical deformation across different twist angles.
• Analysis: The electronic structure is analyzed via sparse diagonalization of the Hamiltonian. Key metrics include the band dispersion, the opening of gaps at Dirac points, sublattice-resolved charge densities, and the integrated density of states (IDOS) near zero energy ($\Delta E_U \approx 27$ meV) as a proxy for the propensity of correlated phenomena.

Key Contributions and Results

1. Large-Angle TBG: Enhanced Flattening and Symmetry Breaking

   • Enhanced Flattening: In large-angle TBG, periodic buckling initially enhances band flattening more effectively than in monolayer graphene for comparable buckling strengths. This is attributed to interlayer coupling, which allows electrons to tunnel between layers, effectively reducing in-plane kinetic energy and enhancing localization in high-PMF regions.
   • Non-Monotonic Behavior: Unlike monolayer graphene, where flattening is monotonic with PMF strength, TBG exhibits non-monotonic behavior. At sufficiently strong buckling, the bands become more dispersive.
   • Gap Opening: A critical finding is that the combination of twisting and buckling breaks the $C_{2z}$ inversion symmetry (which protects Dirac point degeneracy in untwisted bilayers and monolayers). This symmetry breaking opens a gap at the Dirac points ($K, K'$). For large twist angles, where the intrinsic bandwidth is large, this gap opening contributes to an overall flattening of the spectrum near zero energy.
   • High-Energy Bands: Buckling transforms the continuum of high-energy bands into isolated flat bands, which are significantly flatter in TBG than in monolayer graphene due to interlayer coupling.

2. Magic-Angle TBG: Competing Mechanisms

   • Sublattice Polarization vs. Symmetry: At the magic angle, pristine TBG features flat bands with sublattice-symmetric charge distributions. Buckling, however, induces a sublattice polarization (breaking sublattice symmetry) via valley-dependent gauge fields.
   • Competition: The two flattening mechanisms—twist-induced moiré localization and buckling-induced PMF localization—are fundamentally incompatible. Twist-induced flat bands prefer sublattice symmetry, while buckling forces sublattice polarization. Consequently, they compete rather than add.
   • Dispersive Outcome: At the magic angle, the intrinsic bandwidth is already narrow. The gap opening at the Dirac points (caused by symmetry breaking) and the sublattice polarization actually increase the overall bandwidth, making the bands more dispersive compared to pristine magic-angle TBG.
   • Location Dependence: Centering the buckle on AA-stacked regions (where charge density is high) yields flatter isolated bands at higher energies compared to centering it on AB-stacked regions, though both configurations result in increased dispersion of the low-energy moiré bands.

3. Intermediate Angles and Robustness

   • IDOS Enhancement: For twist angles slightly larger than the magic angle (e.g., $1.3^\circ - 2.0^\circ$), periodic buckling can significantly enhance the IDOS, more than doubling that of pristine TBG for specific buckling strengths.
   • Robustness: While buckling does not produce flatter bands at the magic angle, it allows buckled TBG to rival pristine magic-angle TBG in flatness over a wide range of non-magic angles. This suggests that buckled TBG is robust against twist-angle disorder, as the buckling strength can be tuned to maintain high DOS even if the twist angle deviates from the magic value.

Significance and Claims
The paper claims that active buckling of TBG, achievable through substrate engineering, provides a versatile and robust platform for studying strongly correlated electronic phenomena.

• Mechanism Insight: The work elucidates that while buckling and twisting can both flatten bands, their underlying symmetry requirements (sublattice symmetry preservation vs. breaking) lead to competition at the magic angle, preventing additive effects.
• Tunability: The study demonstrates that buckling strength serves as a tuning parameter to enhance flatness and IDOS in large-angle TBG and to extend the range of twist angles hosting flat-band physics.
• Experimental Feasibility: The authors assert that the required effects are experimentally accessible, citing prior observations of similar periodic buckling in monolayer graphene and strain-induced PMFs reaching hundreds of Tesla, which exceed the strengths required in this theoretical study.
• Conclusion: Buckled TBG is presented not as a replacement for magic-angle TBG, but as a complementary system that can stabilize correlated phases over a broader range of twist angles and potentially enhance the zero-energy DOS through the hybridization of buckling-induced flat bands with moiré bands.