GEARS H: Accurate machine-learned Hamiltonians for next-generation device-scale modeling

TL;DR

GEARS H presents a compact, E(3)-equivariant, LCAO-based Hamiltonian learning framework that enables device-scale electronic structure simulations with DFT-like accuracy. By decomposing the Hamiltonian into atom-centered and bond-centered descriptors and employing scale-shift readouts, it achieves sub-meV Hamiltonian-element MAEs across diverse materials, including molecules, 2D materials, alloys, and amorphous solids. The authors demonstrate a large-scale, Ni-doped a-HfO$_2$/WSe$_2$ case study, performing a Bayesian analysis to quantify how Ni doping, density, and Se vacancies influence hole concentration, and show inference times sufficient to explore thousands of structures in hours. The work provides a production-ready pathway for predictive, large-scale device modeling and offers broad applicability across complex chemical environments, supported by public code, datasets, and pretrained models.

Abstract

We introduce GEARS H, a state-of-the-art machine-learning Hamiltonian framework for large-scale electronic structure simulations. Using GEARS H, we present a statistical analysis of the hole concentration induced in defective $\mathrm{WSe}_2$ interfaced with Ni-doped amorphous $\mathrm{HfO}_2$ as a function of the Ni doping rate, system density, and Se vacancy rate in 72 systems ranging from 3326 to 4160 atoms-a quantity and scale of interface electronic structure calculation beyond the reach of conventional density functional theory codes and other machine-learning-based methods. We further demonstrate the versatility of our architecture by training models for a molecular system, 2D materials with and without defects, solid solution crystals, and bulk amorphous systems with covalent and ionic bonds. The mean absolute error of the inferred Hamiltonian matrix elements from the validation set is below 2.4 meV for all of these models. GEARS H outperforms other proposed machine-learning Hamiltonian frameworks, and our results indicate that machine-learning Hamiltonian methods, starting with GEARS H, are now production-ready techniques for DFT-accuracy device-scale simulation.

Figures (13)

• Figure 1: (a) GEARS H architecture overview. (b) Validation set eigenvalue errors relative to the reference eigenvalues. (c) Validation set Hamiltonian matrix element errors relative to the reference matrix elements with an asinh scale to help resolve bins that have smaller counts. (d) Cumulative distribution function of the eigenvalue errors larger than 0.1m. (e) Cumulative distribution function of the Hamiltonian matrix element errors larger than 0.1m. (f) Sample structure from the training dataset. Hf are gold, Ni are turquoise, O are red, W are gray, and Se are green. The MAEs shown in (b-c) are averaged across training set structures, whereas the mean values in (d-e) are taken across all errors. GEARS H performs very well on this highly complex system.
• Figure 2: (a) Ni doping rates, Se vacancy rates, and total system density of the large scale systems used in the statistical analysis. Diagonal figures are histograms with 20 bins of the distribution of each individual quantity, while the off-diagonal figures show pair plots with the corresponding hole concentrations. (b) Results of the Bayesian analysis showing interactions between the posterior likelihood of the parameters of the model. Diagonal histograms show the distribution of each parameter and the residual $\sigma$. Off-diagonal subplots show marginal joint distributions of the parameters with iso-likelihood contours. Most interactions are weak with the exception of $c_{\rm Ni}$ and $c_{V_{\rm{Se}}}$, indicating that both the $p$-doping due to Ni and the $n$-doping due to Se-vacancies can be strong or weak together. (c) Selected system used in the statistical study. Hf, O, W, and Se atoms have been hidden to highlight the Ni dopants spread through the a-HfO2.
• Figure 3: Sample structure from the training set and 2D histograms of eigenvalue and Hamiltonian matrix element errors of GEARS H models trained on a wide variety of materials chosen for their varied local environments. These include 1) Li-TFSI, a molecular system with 6 atomic species, 2) $\ce{WSe_{2-x}}$, a 2D TMD, 3) a collection of 9 different 2D TMDs with 8 atomic species, 4) AgAu, a metallic alloy, 5) a-SiO2, a covalent solid, and 6) a-HfO2, a mixed ionic-covalent solid. Errors shown were calculated on the validation sets, and $H_{ij}$ errors are shown with an asinh scale. Note that error scales differ between figures, due to the variation in the range of errors. GEARS H performs extremely well across all classes of materials.
• Figure S1: GEARS H architecture overview. The blocks discussed in detail below share colors with the blocks in the overview.
• Figure S2: Species aware radial basis block diagram. This layer outputs the 2-body descriptor that is the starting point for the atom-centered block.