Deep-learning atomistic semi-empirical pseudopotential model for nanomaterials

TL;DR

DeepPseudopot presents a transferable, machine-learned atomistic semi-empirical pseudopotential that reproduces DFT+GW quasiparticle energies and deformation potentials with high accuracy at a fraction of the computational cost. It combines a neural-network local pseudopotential $v_{\mathrm{loc}}^{\alpha}(G)$ with analytically parameterized non-local $v_{\mathrm{nl}}^{\alpha}$ and spin-orbit $v_{\mathrm{soc}}^{\alpha}$ terms to form a Hamiltonian $\hat{H}=\hat{T}+\hat{V}_{\mathrm{loc}}+\hat{V}_{\mathrm{nl}}+\hat{V}_{\mathrm{soc}}$, trained on bulk GW data and deformation potentials across Si and group-III–V semiconductors. The model achieves GW-level accuracy for band structures and edge properties, transfers well to Si allotropes and III–V alloys, and enables efficient quasiparticle calculations for large nanocrystals, alloyed systems, and defect contexts, including exciton–phonon coupling via subsequent BSE and vibronic analyses. This framework offers a scalable path for data-driven design of optoelectronic nanomaterials with practical GW/BSE-level predictive power at reduced cost. Its data-efficient training and physically motivated Hamiltonian design position it as a versatile tool for high-throughput discovery and characterization of complex nanomaterials.

Abstract

The semi-empirical pseudopotential method (SEPM) has been widely applied to provide computational insights into the electronic structure, photophysics, and charge carrier dynamics of nanoscale materials. We present "DeepPseudopot", a machine-learned atomistic pseudopotential model that extends the SEPM framework by combining a flexible neural network representation of the local pseudopotential with parameterized non-local and spin-orbit coupling terms. Trained on bulk quasiparticle band structures and deformation potentials from GW calculations, the model captures many-body and relativistic effects with very high accuracy across diverse semiconducting materials, as illustrated for silicon and group III-V semiconductors. DeepPseudopot's accuracy, efficiency, and transferability make it well-suited for data-driven in silico design and discovery of novel optoelectronic nanomaterials.

Figures (5)

• Figure 1: Workflow for developing the DeepPseudopot model. (a) Reference data generation. Quasiparticle band structures and hydrostatic deformation potentials are computed using DFT+GW for multiple crystal structures. (b) Model setup. The atomistic machine learning model is initialized, with the local pseudopotential represented by a neural network and the non-local and spin-orbit coupling terms modeled by parameterized functional forms. (c) Hamiltonian construction and model training. The DeepPseudopot Hamiltonian is constructed from structure factors, wavevector data, and the model pseudopotentials, then diagonalized to obtain the predicted quasiparticle band structures and deformation potentials. The model is trained by minimizing the loss function based on these properties.
• Figure 2: Band properties of silicon from training the DeepPseudopot model. The reference DFT+GW data (red), the DeepPseudopot model predictions (blue), and the simple functional form pseudopotential fitted using Monte Carlo sampling (yellow) are consistently color-coded across all panels. (a) Band structure of cubic diamond silicon. (b) Accuracy matrix for interband transition energies between high-symmetry points in the Brillouin zone. Grid colors and blue text show the absolute energy errors between reference values and the DeepPseudopot prediction. Yellow text shows the corresponding errors from the simple functional form pseudopotential for comparison. (c) Local pseudopotentials plotted in reciprocal space and real space. The simple functional form pseudopotential fitted using gradient descent is shown as grey dashed lines. (d) Effective masses (top) and deformation potentials (bottom). Insets show zoomed-in comparisons of effective masses. (e) Training loss evolution starting from a random initialization.
• Figure 3: DeepPseudopot model predictions for the hexagonal diamond (lonsdaleite) and body-centered tetragonal (bct) structures of silicon. Consistent with Figure 2, reference DFT+GW data are shown in red, DeepPseudopot model predictions in blue, and simple functional form pseudopotentials in yellow. (a) The interband transition energies between the valence band maximum and various $\mathbf{k}$-points of the conduction band edge in the lonsdaleite structure. The fundamental band gaps are highlighted. (b) The reference and predicted band structures of the lonsdaleite structure. (c, d) Same as panels (a) and (b), but for the bct structure.
• Figure 4: Band properties of group III-V semiconductors from training the DeepPseudopot model. Reference DFT+GW data are shown in red throughout; predictions from the trained DeepPseudopot model are shown in blue. (a) Band structures of InAs, InP, GaAs, GaP, from the trained DeepPseudopot model (blue lines) compared to GW reference (red dots). Insets show zoomed-in views around the VBM, highlighting the spin-orbit splitting energy. (b) Real-space local pseudopotentials for each element. (c) Predicted band structure of alloyed $\mathrm{In}_{0.5}\mathrm{Ga}_{0.5}\mathrm{P}$ around the $\Gamma$ point. The inset shows the 8-atom zincblende conventional cell with cation substitutions used in the calculation. Atom colors: Pink-In, Green-Ga, Purple-P. (d) Fundamental band gaps of bulk $\mathrm{In}_{1-x}\mathrm{Ga}_{x}\mathrm{P}$ alloy supercells.
• Figure 5: Optoelectronic properties and electron-phonon coupling of group III-V NCs calculated using the trained DeepPseudopot model. (a) Size-dependent optical gaps of InAs, InP, GaAs, and GaP NCs compared with experiments (hollow squares). micic_synthesis_1994micic_highly_1996guzelian_colloidal_1996ondry_reductive_2024 (b) Calculated (orange solid line) and experimental (black dash-dot) absorption spectra for a $4.0$ nm InP NC. Exciton energies (bars below axis) and oscillator strengths (bars above axis) are shown. (c) Exciton-phonon coupling spectral density (green) and phonon density of states (black dashed line) of a $6.0$ nm GaAs NC. (d) Calculated (solid dots) and experimental (hollow squares) Stokes shifts for GaAs NCs as a function of size. (e) Fundamental band gaps of $\mathrm{In}_{1-x}\mathrm{Ga}_{x}\mathrm{P}$ and $\mathrm{Ga}\mathrm{P}_{1-x}\mathrm{As}_{x}$ ternary alloys. Bulk alloy gaps at the $\Gamma$, $X$, $L$ valleys (dotted lines) were interpolated using a simple quadratic form with experimental bowing parameters vurgaftman_band_2001, with the lowest-energy branch at each composition highlighted as the solid line. Bulk direct-to-indirect crossover compositions are annotated in purple. The NC gaps are shown as dots colored by the dominant GaP valley character (inset). The grey line shows average NC gaps across three random alloy configurations per composition.