Machine Learning for Electron-phonon Interactions From Finite Difference

TL;DR

A machine learning electron-phonon interaction (MLEPI) pipeline that predicts force constants and electronic Hamiltonians for modeling EPIs from finite difference calculations, improving efficiency by orders of magnitude without compromising accuracy.

Abstract

First-principles investigations of electron-phonon interactions (EPIs) play a crucial role in understanding a wide range of phenomena in physics and materials science. Among various approaches, the finite difference method offers a direct route to capture higher-order EPIs and is compatible with diverse electronic structure solvers. However, its considerable computational cost limits its broader application. To overcome this bottleneck, we present a machine learning electron-phonon interaction (MLEPI) pipeline that predicts force constants and electronic Hamiltonians for modeling EPIs from finite difference calculations, improving efficiency by orders of magnitude without compromising accuracy. The performance of MLEPI is validated by studying the temperature dependence of the electronic band properties in bilayer graphene, where both first- and second order EPIs are treated on an equal footing. Using a heterogeneous edge network, the pipeline integrates both interlayer and intralayer interactions, making it particularly suitable for studying multilayer materials. With its inherent adaptability and ease of transfer to other applications, our methodology provides a robust tool with a very favorable accuracy/efficiency balance for investigating EPIs in large-scale material systems.

Figures (5)

• Figure 1: MLEPI pipeline for ML-based EPI calculations in materials. For an arbitrary material, (a) DFT is employed to generate phonon training sets that contain structural information, energies and atomic forces, as well as electronic training sets that include structural details and Hamiltonians. These datasets are used to train (b) a phonon neural network and (d) an electronic neural network, respectively. The trained phonon neural network predicts the interatomic forces for perturbed structures, facilitating the calculation of phonon-eigenmode determined probability distributions $dP_{\mu}(\lambda, T)$ required for (c) MC sampling. The structures sampled via MC are then input into the trained electronic neural network to infer corresponding Hamiltonians, which are instrumental for subsequent calculations of EPI-related physical quantities. $\Delta \sigma$ schematically represents the perturbation of the atomic structure around the equilibrium configuration.
• Figure 2: A schematic illustrates how a multi-layered material system can be constructed as a heterogeneous graph $\mathcal{G} = (\mathcal{V}, \mathcal{R}, \mathcal{E})$. Without loss of generality, we consider a bilayer material and arbitrarily select a central atom $i$ from Layer 1. Its neighboring atoms $j \in \mathcal{N}_i$ are identified based on a specified cutoff radius. In the graph, the superscript of $j$, either intra or inter, indicates whether it is in the same layer or a different layer compared to atom $i$. The resulting heterogeneous graph encompasses two types of relations: $\mathcal{R} = \{\text{rel}_{\text{intra}}, \text{rel}_{\text{inter}}\}$. The connecting edge $e_{ij}^{\text{intra}}$ (represented by a solid gray line) signifies an intra-layer relationship $\text{rel}_{\text{intra}}$, while the connecting edge $e_{ij}^{\text{inter}}$ (shown as an orange dashed line) signifies an inter-layer relationship $\text{rel}_{\text{inter}}$. For the central atom $i$, edges corresponding to the intra-layer relationship $\text{rel}_{\text{intra}}$ are input into the neural network module $f_{\theta}^{\text{intra}}$ (where $\theta$ denotes learnable parameters), while edges corresponding to the inter-layer relationship $\text{rel}_{\text{inter}}$ are input into $f_{\theta}^{\text{inter}}$. This process yields the node embedding $h_i$ for the central atom $i$.
• Figure 3: The detailed architecture of (a) HedgeNet-Phonon and (b) HedgeNet-Electron. In this work, MACE batatia2022mace and DeepH-E3 gong2023general are utilized as the backbones for the phonon network (HedgeNet-Phonon) and electronic network (HedgeNet-Electron), respectively. The architectural details of the vanilla MACE and DeepH-E3 models can be found in Sections B and C in SM, respectively.
• Figure 4: (a) Schematic structure of bilayer graphene. (b) Comparison of the phonon spectrum of a 6$\times$6$\times$1 bilayer graphene supercell predicted by HedgeNet-Phonon model, trained on a 4$\times$4$\times$1 bilayer graphene dataset, with DFT results. (c) Comparison of the band structure inference results of twisted bilayer graphene (magic angle $\theta=$1.08$^\circ$, 11,164 atoms) from HedgeNet-Electron, trained on the bilayer graphene dataset provided by DeepH li2022deep and DeepH-E3 gong2023general, with DFT and continuum model results (The DFT and continuum model results are extracted from Ref. lucignano2019crucialli2022deep, and gong2023general). (d) Comparison of the band structure results of a randomly selected MC sampling configuration from HedgeNet-Electron and DFT. (e) Iteration error of the Fermi velocity of the bilayer graphene modulated by EPI effects at 0 K, calculated according to the Equation \ref{['eq:obs']}, as a function of the number of sampling points. The converge trendency is similiar for other temperatures. (f) Variation of the Fermi velocity of the bilayer graphene modulated by EPI effects with temperature, calculated with 500 sampling points. The dashed red line indicates Fermi velocity calculated without accounting for EPIs.
• Figure 5: Computation time for (a) HedgeNet-Phonon inference and (b) HedgeNet-Electron inference applied to bilayer graphene as a function of system size, with DFT calculation time included for reference.