Nano-ARPES observation of a flat band at the \bar{M} point in 2.4{\deg} twisted bilayer graphene

Technical Summary: Nano-ARPES Observation of a Flat Band at the $\bar{M}$ Point in 2.4° Twisted Bilayer Graphene

Problem Statement
Twisted bilayer graphene (TBG) has emerged as a primary platform for investigating correlated electron phases, such as superconductivity and Mott insulation, which arise when the twist angle approaches the "magic angle" (~1.1°). At this angle, moiré superlattices induce flat bands near the Fermi level, significantly enhancing electron-electron interactions. However, the electronic structure of TBG at twist angles larger than the magic angle remains incompletely understood, particularly regarding regions of the Brillouin zone (BZ) away from the high-symmetry $\bar{K}$ point. While previous studies have focused on low-energy Dirac bands near $\bar{K}$, the potential for moiré coupling, structural relaxation, or interfacial interactions to generate localized states at other momentum points (such as $\bar{M}$) and at higher binding energies has not been systematically explored via angle-resolved photoemission spectroscopy (ARPES).

Methodology
The study utilized a combination of advanced sample fabrication, structural characterization, and spectroscopic measurements:

• Sample Fabrication: Monolayer graphene and hexagonal boron nitride (h-BN) flakes were mechanically exfoliated and assembled using a "tear and stack" dry transfer technique. The graphene layers were rotated by a target angle of 2° relative to each other. A thick h-BN flake was used as a substrate to ensure an atomically flat surface, with a large twist angle between graphene and h-BN chosen to suppress additional moiré patterns.
• Structural Characterization: Conductive atomic force microscopy (c-AFM) was employed to visualize the moiré superlattice. Fast Fourier transform (FFT) analysis of the c-AFM images confirmed a moiré wavelength of 5.8 ± 0.2 nm, corresponding to a twist angle of 2.4 ± 0.2°.
• Nano-ARPES Measurements: Electronic band structures were measured along the $\bar{\Gamma}-\bar{M}$ and $\bar{\Gamma}-\bar{K}$ high-symmetry directions at the ANTARES beamline of the SOLEIL synchrotron. Measurements were conducted at 70 K with a beam spot size of 600 nm, photon energy of 100 eV, and energy/angular resolutions of 70 meV and 0.5°, respectively.
• Theoretical Modeling: The electronic structure was simulated using two approaches:
  1. Tight-Binding (TB): An atomistic $\pi$-orbital model using a supercell of 6,892 atoms (2.4° twist) with nearest-neighbor intralayer hopping and distance-dependent interlayer hopping.
  2. Density Functional Theory (DFT): Calculations using the Quantum ESPRESSO package with PBE functionals and van der Waals corrections, performed on a smaller supercell (390 atoms, 2.31° twist) to manage computational costs.

Key Results

• Absence of Flat Bands at $\bar{K}$: Along the $\bar{\Gamma}-\bar{K}$ direction, the nano-ARPES spectra revealed two highly dispersive bands corresponding to the top and bottom graphene layers, with the valence band maximum at the $\bar{K}$ point. No weakly dispersive or flat features were observed near the Fermi level or the $\bar{K}$ point.
• Discovery of a Flat Band at $\bar{M}$: In contrast, measurements along the $\bar{\Gamma}-\bar{M}$ direction revealed a pronounced, weakly dispersive feature near the $\bar{M}$ point at a binding energy of approximately 2.6 eV.
• Curvature Analysis: A 2D curvature analysis of the ARPES data confirmed the existence of a distinct flat band at the $\bar{M}$ point with a bandwidth of ~0.2 eV. The Fermi velocity at this point was calculated to be $6.82 \times 10^4$ m/s, nearly two orders of magnitude lower than the velocity at the $\bar{K}$ point ($1.14 \times 10^6$ m/s).
• Theoretical Discrepancy: Both TB and DFT calculations successfully reproduced the principal dispersive bands observed in the experiment. However, neither model predicted the additional flat band at the $\bar{M}$ point. The TB model, which assumes a rigid lattice, and the DFT model, limited by a smaller supercell size, failed to capture this feature.

Significance and Claims
The authors claim this work represents the first observation of flat bands in twisted bilayer graphene away from the $\bar{K}$ point and at a twist angle (2.4°) significantly larger than the magic angle.

The study highlights that:

1. Momentum Dependence: The electronic structure of TBG exhibits strong momentum dependence, with localized states emerging at the $\bar{M}$ point rather than the traditionally studied $\bar{K}$ point.
2. Limitations of Ideal Models: The inability of standard rigid-lattice TB and limited supercell DFT models to reproduce the flat band suggests that the feature does not originate from the ideal periodic electronic structure of TBG.
3. Role of Structural Perturbations: The authors propose that the flat band likely arises from structural or interfacial effects not captured by current models, such as local strain, structural reconstruction, domain walls, or residual interactions with the h-BN substrate.
4. Broader Exploration: The findings underscore the necessity of probing the entire Brillouin zone, rather than focusing solely on the low-energy Dirac bands near $\bar{K}$, to fully capture the electronic states in twisted van der Waals structures.

The paper concludes that while the microscopic origin of the $\bar{M}$-point flat band remains to be definitively identified, its existence points to a richer electronic landscape in TBG involving localized states at higher binding energies and larger twist angles, motivating further investigation into structural relaxation and inhomogeneity.