Machine-learning-enabled methodology for the ab-initio simulations of sub-$μ$m-wide nanoribbons

TL;DR

This work tackles the challenge of simulating large, mesoscopic nanostructures with first-principles reliability. It introduces a gauge-independent Wannier tight-binding (GI-WTB) framework and a machine-learning-enabled extension (ML-GI-WTB) that uses geometric variables to interpolate TB parameters across widths and edge configurations. Applied to MoS$_{2}$ armchair nanoribbons, the approach reproduces DFT band structures, enables parameter fitting across widths up to sub-$\mu$m scales, and provides access to wavefunctions for identifying bulk, edge, and bulk–edge states. The method offers a scalable route to predict electronic properties in realistic nanostructures beyond conventional DFT, with potential impact on nanoscale device design.

Abstract

Simulation of mesoscopic nanostructures is a central challenge in condensed matter physics and device applications. First-principles methods provide accurate electronic structures but are computationally prohibitive for large systems, while empirical band theories are efficient yet limited by parameter fitting that neglects wavefunction information and often yields non-transferable parameters. We propose a methodology that bridges these approaches, achieving first-principles-level reliability with computational efficiency through a machine-learning-enabled tight-binding framework. Our approach starts with Wannier tight-binding (WTB) parameters from small nanostructures, which serve as training data for machine learning (ML). To remove the gauge freedom of Wannier functions that obscures size- and geometry-dependent parameter trends, we construct gauge-independent (GI) bases and transform the WTB model into a gauge-independent WTB (GI-WTB) model. This enables robust parameter fitting and ML prediction of parameter variations, yielding the machine-learning GI-WTB (ML-GI-WTB) model. Applied to MoS2 armchair-edge nanoribbons, the ML-GI-WTB model shows excellent agreement with first-principles results and enables reliable simulations of sub-$μ$m-wide nanoribbons. This framework provides a scalable tool for predicting electronic properties of realistic nanostructures beyond the reach of conventional first-principles methods.

Figures (6)

• Figure 1: (a) Top-down view of the structure-relaxed monolayer MoS$_{2}$ armchair-edge nanoribbon (A-NR), where the lattice translational symmetry is defined by $\boldsymbol{a} _{2} = a _{2} \hat{\boldsymbol{y}}$. Mo and S atoms are depicted in purple and orange, respectively. The integer $N _{a}$ denotes the total number of atomic chains, which characterizes the ribbon width, while the $L$-index is a geometric factor indicating the position of each atomic chain. The $L$-index equals zero at the edge and is positive or negative for chains on the left or right, respectively. Here, we refer to this ribbon as $N _{a}$-A-NR. (b) The first Brillouin zone (BZ) of the $N _{a}$-A-NR, where the gray points indicate the mesh used for both the DFT calculations and Wannierization. (c) Band structure of the monolayer MoS$_{2}$ 11-A-NR with a width of 1.60 nm, obtained from DFT (gray) and the Wannier tight-binding (WTB) model (cyan). (d) Band structure of the same nanoribbon obtained from the gauge-independent Wannier tight-binding (GI-WTB) model (pink). In all cases, the bands are aligned by shifting the valence band maximum to zero.
• Figure 2: (a) $N _{a}$ dependence of the on-site energy for the $d _{z ^{2}}$-orbital at the atom indicated in the top schematic (illustrated using 11-A-NR). (b) $N _{a}$ dependence of the hopping parameter between the $d _{z ^{2}}$- and $d _{x ^{2} - y ^{2}}$-orbitals, as denoted by the arrow in the top schematic (again shown using 11-A-NR). The cyan and pink data points are obtained from the WTB and GI-WTB models, respectively. The blue dashed line for the machine-learning WTB (ML-WTB) model is fitted to the cyan data points, while the red dashed line for the machine-learning GI-WTB (ML-GI-WTB) model is fitted to the pink data points. (c) Band structure of monolayer MoS$_{2}$ 31-A-NR from the ML-WTB model (blue) compared with DFT (gray). (d) Band structure of monolayer MoS$_{2}$ 31-A-NR from the ML-GI-WTB model (red) compared with DFT (gray). Double-headed arrows indicate the energy band gap, $E _{g}$.
• Figure 3: (a) Dependence of the energy gap $E _{g}$ on $N _{a}$, ranging from $N _{a} = 11$ to $N _{a} = 1261$. The upper axis shows the corresponding widths of the monolayer MoS$_{2}$$N _{a}$-A-NR. Data points are color-coded according to the model, with gray for DFT, cyan for WTB, pink for GI-WTB, blue for ML-WTB, and red for ML-GI-WTB. (b) Band structure of the sub-$\mu$m-wide monolayer MoS$_{2}$ 631-A-NR computed using the ML-GI-WTB model.
• Figure 4: (a) Schematic illustration of the relative average position $\bar{x} _{n \boldsymbol{k}}$ for the Bloch state $| \psi _{n , \boldsymbol{k}} \rangle$ (see definition in Eq. \ref{['x_bar']}) in monolayer MoS$_{2}$$N_{a}$-A-NR. The deep blue line marks $\bar{x} _{n \boldsymbol{k}} = 0$ at the ribbon center. The deep red line marks $\bar{x} _{n \boldsymbol{k}} = 1$ at the ribbon edge. The light green line marks $\bar{x} _{n \boldsymbol{k}} = 1/2$ at the midpoint between the ribbon center and the edge. Other values of $\bar{x} _{n \boldsymbol{k}}$ between $0$ and $1$ are marked by colored lines as indicated. Mo atoms are shown in black and S atoms are shown in gray to avoid confusion from overuse of colors. (b) Energy levels at the $\Gamma$-point for MoS$_{2}$$N_{a}$-A-NRs of different widths, where the $\bar{x} _{n \boldsymbol{k}}$ of each Bloch state is color-coded according to the scale on the right and demonstrated in (a). For the 50.08 nm wide NR, several states are selected as representative examples of the probability distributions (indicated by color-coded arrows). The radius of the magenta circles represents the probability at each atomic site. Numbers at the upper right of each probability plot indicate the applied scaling factors.
• Figure 5: Band structure comparison between DFT (gray) and models. (a) WTB band structure (cyan) of 2D-bulk monolayer MoS$_{2}$. (b) Band structure of monolayer MoS$_{2}$$N_a$-A-NRs from the TB model (green) constructed using 2D-bulk WTB parameters in (a).