Graph Neural Network for Unified Electronic and Interatomic Potentials: Strain-tunable Electronic Structures in 2D Materials

Abstract

We introduce UEIPNet, an equivariant graph neural network designed to predict both interatomic potentials and tight-binding (TB) Hamiltonians for an atomic structure. The UEIPNet is trained using density functional theory calculations followed by Wannier projection to predict energies and forces as node-level targets and Wannier-projected TB matrices as edge-level targets. This enables physically consistent modeling of coupled mechanical electronic responses with near-DFT accuracy. Trained on bilayer graphene and monolayer MoS2 DFT data, UEIPNet captures key deformation-electronic effects: in twisted bilayer graphene, it reveals how interlayer spacing, in-plane strain, and out-of-plane corrugation drive isolated flat-band formation, and further shows that modulating substrate interaction strength can generate flat bands even away from the magic angle. For monolayer MoS2, the UEIPNet accurately reproduces phonon dispersions, strain-dependent band-gap evolution, and local density of states modulations under non-uniform strain. The UEIPNet offers a generalized, efficient, and scalable framework for studying deformation-electronic coupling in large-scale atomistic systems, bridging classical atomistic simulations and electronic-structure calculations.

Figures (6)

• Figure 1: Overview of the training and inference workflow of UEIPNet. (a) Data generation pipeline for training UEIPNet. For each atomic structure, DFT calculations provide total energy, atomic forces, and the Kohn-Sham Hamiltonian ($\hat{\mathbf{H}}$). Then $\hat{\mathbf{H}}$ is projected onto a localized Wannier basis to generate TB matrix elements for each pair of two orbitals. These energy and forces (node target), and TB matrix (edge target) are jointly used to train UEIPNet. (b) Inference stage using the trained UEIPNet model. Given an input atomic structure, the model predicts both the interatomic potential (red arrows for predicted forces) from node features, and the local TB Hamiltonians $\mathbf{H}_{ij}^{\rm TB}$ from edge features. As an example, we present $p$-orbital systems, where each $\mathbf{H}_{ij}^{\rm TB}$ is a $3 \times 3$ matrix. The color in the matrix visualization represents the magnitude and sign of each element.
• Figure 2: The architecture of UEIPNet. (a) A graph is constructed from an atomic structure, with initial features based on atomic species and geometric information. (b) Simplified overview of the UEIPNet workflow, consist of four modules: Node update (pink), Edge update (green), IP module (yellow), and TB module (blue). The workflow has two parallel pathways (denoted as IP mode and TB mode). The IP module predicts energy and forces and the TB module predicts TB Hamiltonians ($\mathbf{H}^{\rm TB}$). (c) Schematics of detailed process within UEIPNet. See details on Section \ref{['sec:SI:gnn']} in SM.
• Figure 3: Validation of the UEIPNet model for IP and TB predictions in bilayer graphene. a) Three representative stacking configurations: AA, AB, and SP stackings. Red and blue spheres represent atoms in the top and bottom graphene layers, respectively. b) Energy evolution as a function of interlayer spacing for AA and AB stacking. DFT results (solid dots) are compared with UEIPNet predictions (dotted lines). For both, the minimum energy of AB-stacking is set to zero for relative comparison. c) Generalized stacking fault energy (GSFE) for interlayer shear in bilayer graphene, obtained from DFT and UEIPNet. Lateral displacements $\Delta a_1$ and $\Delta a_2$ are expressed in units of the lattice vectors $\mathbf{a}_1 = \sqrt{3}a,\hat{\mathbf{x}}$ and $\mathbf{a}_2 = a(\hat{\mathbf{x}} + \frac{3}{2}\hat{\mathbf{y}})$, where $a$ is the carbon–carbon bond length. The energy difference per atom between AA and AB stackings ($\Delta E_{\rm AA-AB}$) is also shown at the bottom. d) Electronic band structures of bilayer graphene under various configurations: AB stacking, AB stacking with atomic perturbations, and TBG at $\theta = 9.43\degree$ (148 atoms). Results from DFT (black), UEIPNet (red), and empirical tight-binding model (LETB letb)(blue) are shown.
• Figure 4: The effect of structural deformation on electronic band structure of TBG. (a) Energy maps of the unrelaxed and relaxed TBG at $1.05\degree$, with stacking domains indicated. (b) Electronic band structures of unrelaxed (black) and relaxed (blue) TBG predicted by the UEIPNet, with the electronic gap, hole gap, and bandwidth highlighted. (c) Deformation modes in TBG: interlayer distance (orange), in-plane strain (green), and out-of-plane corrugation (red). The middle row represent each mode, with corresponding coefficients labeled as $d$, $A_{1,2,3}$, and $B_{1,2,3}$. The bottom row illustrates the isolated impact of each mode on the band structure of TBG: (i) $d$ only, (ii) $A_{1,2,3}$ only, and (iii) $B_{1,2,3}$ only. Deformation coefficients are taken from all-atom simulations of the $1.05\degree$ TBG shown in (a) and (b). (d) Evolution of band structure features (electronic gap, hole gap, and bandwidth) and deformation coefficients ($d$, $A_{1,2,3}$, and $B_{1,2,3}$) as a function of twist angle for unrelaxed, relaxed, and relaxed on substrate cases. Experimental deformation data are shown for comparison; Experiment$^*$cao2018; Experiment$^{**}$sung2022choi2024. (e) Bandwidth map versus twist angle and substrate strength ($\epsilon^*$), demonstrating that substrate pulling induce isolated flat bands even at non-magic angles. The red dotted line marks the minimal-bandwidth trajectory. Insets show band structures at $\theta = 1.05\degree$ for $\epsilon^* = 0.9$ (red) and $\epsilon^* = 0.7$ (blue), showing substrate-driven modulation of band flatness.
• Figure 5: Validation of UEIPNet for IP and TB models of monolayer MoS$2$. (a) Phonon dispersion relations from UEIPNet and DFT along the high-symmetry path $\Gamma$–M–K–$\Gamma$. (b) Electronic band structures from UEIPNet and DFT for unstrained and uniformly strained MoS$2$ ($\varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{xy} = 0.015$), with energies referenced to the Fermi level ($E_\mathrm{F} = 0$). (c) Band gap evolution under uniform 2D strain in $\varepsilon_{xx}$ and $\varepsilon_{yy}$ for various fixed values of $\varepsilon_{xy}$. Circles and crosses indicate direct and indirect gaps, respectively.