[Paper Review] Atomistic substrate relaxation effects in the band gaps of graphene on hexagonal boron nitride

Jiaqi An, Nicolas Leconte+3 more|arXiv (Cornell University)|Feb 21, 2026

Graphene research and applications0 citations

TL;DR

The paper analyzes how atomistic substrate lattice relaxation schemes affect the primary and secondary band gaps of graphene on h-BN across twist angles, using a hybrid tight-binding model with TAPW and atomistic relaxations.

ABSTRACT

We assess the impact of atomistic substrate lattice relaxation schemes in the primary band gap at charge neutrality and the secondary valence band gap of graphene on hexagonal boron nitride (G/h-BN) as a function of twist angle. For zero twist angle, the primary gap decreases from $\sim 30$~meV in fully relaxed suspended G/h-BN bilayers, to $\sim 9$~meV when the remote h-BN substrate layer is kept rigid, and down to $\sim 3$~meV in completely rigid structures. In the presence of relaxations, the primary gap shows a maximum near $\sim 0.6^{\circ}$ coinciding with energetic stabilization due to alignment between the moiré pattern and the graphene lattice vectors, while the secondary valence band gap drops from $\sim 12$~meV down to zero beyond twist angles of $\sim 1^{\circ}$. A small but finite primary gap on the order of $\sim 1$~meV, with a mass sign favoring electronic occupation of carbon atop boron, persists across twist angles from $0^{\circ}$ to $30^{\circ}$ for all sliding configurations, and switches sign for twist angles between $30^{\circ}$ and $60^{\circ}$.

Motivation & Objective

• Assess how different substrate lattice relaxation schemes influence the primary (Dirac) and secondary band gaps in graphene on h-BN as a function of twist angle.
• Quantify the role of substrate relaxation in gap magnitudes and their angular evolution, including energy minimization effects.
• Identify any commensuration- or reconstruction-induced enhancements of the primary gap and the angle ranges where gaps persist or close.

Proposed method

• Construct commensurate moiré supercells for G/h-BN using four integers (p, q, p', q').
• Develop a hybrid tight-binding (HTC) model with intra-layer F2G2 terms and inter-layer two-center tunneling based on distance vectors and moiré displacement d.
• Compute the average mass term with the truncated atomic plane wave (TAPW) method to extract the primary gap at charge neutrality.
• Relax structures with LAMMPS using DRIP (reparametrized) for interlayer and ExTeP/REBO2 for intralayer interactions and perform energy minimization.
• Map interlayer moiré effects onto TB parameters with distance-dependent corrections and a global strain adjustment.
• Utilize TAPW to connect atomistic TB to an effective low-energy description and obtain the average mass term ΔA−ΔB to relate to the primary gap.
• Consider multiple relaxed vs rigid configurations (fully relaxed suspended, rigid substrate layer, remote rigid substrate layer, and fully rigid) to isolate substrate effects.

Figure 1: (Color online) Number of atoms in the commensurate supercells constructed using the four indices defined in Eq. 4 . A lattice tolerance of $\pm 0.01$ Å is allowed with respect to the equilibrium h-BN lattice constant $a_{h-BN}^{0}=2.504$ Å, increasing the density of accessible twist angles

Experimental results

Research questions

• RQ1 How do substrate lattice relaxation schemes modify the primary and secondary band gaps in G/h-BN across twist angles?
• RQ2 Is there a twist-angle-dependent enhancement of the primary gap tied to moiré-graphene alignment or lattice reconstruction?
• RQ3 What is the relationship between the average mass term and the observed band gaps under different relaxation constraints?
• RQ4 How does the secondary gap evolve with twist angle under various substrate relaxations (e.g., does it close at a particular angle)?

Key findings

• For zero twist, the primary gap varies with relaxation: ~30 meV in fully relaxed suspended G/h-BN, ~9 meV with a rigid remote h-BN layer, and ~3 meV in completely rigid structures.
• A maximum in the primary gap occurs near twist angle ~0.6°, coinciding with energetic stabilization due to moiré–graphene lattice alignment.
• The secondary valence band gap decreases from ~12 meV to zero beyond twist angles of ~1°, effectively closing at larger angles.
• A small but finite primary gap (~1 meV) persists across twist angles 0° to 30° for all sliding configurations and changes sign between 30° and 60°.
• Relaxation effects generally increase the primary gap and decrease the secondary gap compared to the rigid case, with the largest gap predictions when all layers relax and damping when a substrate is present.
• The non-monotonic behavior around ~0.5°-0.6° is linked to lattice reconstruction and stacking distributions driven by relaxation; the average mass term tracks the band gap evolution and accounts for most of its magnitude (over 70%).

Figure 2: (color online) Top view of the aligned G/h-BN moiré pattern generated with indices $i=55,j=0,i^{\prime}=54,j^{\prime}=0$ . In the AA stacking region, the carbon atoms are on top of boron and nitrogen atoms, respectively. In the AB region, the carbon atom is on top of the nitrogen atom and