Machine learning approach for vibronically renormalized electronic band structures

TL;DR

This work addresses the computational bottleneck of finite-temperature electron-phonon renormalization by pairing a stochastic frozen-phonon sampling scheme with a symmetry-aware neural network. The method uses a Td-point-group–invariant descriptor to map frozen phonon configurations to observables, enabling accurate prediction of the temperature dependence of the silicon band gap $E_g(T)$ with far fewer $E_g$ evaluations. Key contributions include a group-theoretic descriptor that preserves crystal symmetry, a four-layer neural network trained on a modest ab initio dataset, and transfer learning capability to interpolate across intermediate temperatures, reducing computational cost while maintaining fidelity. The approach provides a general, first-principles–driven pathway to incorporate vibronic effects into electronic structure calculations for materials design.

Abstract

We present a machine learning (ML) method for efficient computation of vibrational thermal expectation values of physical properties from first principles. Our approach is based on the non-perturbative frozen phonon formulation in which stochastic Monte Carlo algorithm is employed to sample configurations of nuclei in a supercell at finite temperatures based on a first-principles phonon model. A deep-learning neural network is trained to accurately predict physical properties associated with sampled phonon configurations, thus bypassing the time-consuming {\em ab initio} calculations. To incorporate the point-group symmetry of the electronic system into the ML model, group-theoretical methods are used to develop a symmetry-invariant descriptor for phonon configurations in the supercell. We apply our ML approach to compute the temperature dependent electronic energy gap of silicon based on density functional theory (DFT). We show that, with less than a hundred DFT calculations for training the neural network model, an order of magnitude larger number of sampling can be achieved for the computation of the vibrational thermal expectation values. Our work highlights the promising potential of ML techniques for finite temperature first-principles electronic structure methods.

Figures (7)

• Figure 1: Machine learning model for band gap prediction for a given atomic displacement field $\mathbf u_i$ within the a box of linear size $\ell$. The ML model is composed of two central components: the descriptor and the neural network. The input of the ML model is a cubic block of displacement vectors $\mathbf u_i$. The descriptor corresponds to a representation or feature variables $\bm G = (G_1, G_2, G_3, \cdots)$ of the displacement vectors that is invariant under symmetry operations of the point group of the lattice, which in the case of Si is the $T_d$ group. The complex dependence of the band gap on the nuclei configurations is encoded in the NN which takes the symmetry-invariant feature variables $\bm G$ as well as the temperature $T$ as the input, and the predicted gap energy at the output.
• Figure 2: Group of atoms sharing same distance to the center of the box form an invariant block of the representation matrix of the symmetry group. Panels (a), (b), and (c) show such invariant groups of 4, 6, and 12 atoms, respectively, in a diamond lattice.
• Figure 3: Ground state (a) electron and (b) phonon dispersion of Si crystal along high-symmetry directions of the lattice.
• Figure 4: Histogram showing distribution of phonon-induced band gap correction at (a) $T = 0$K and (b) $T = 200$ K using 100 data points for Si-supercell of size 6 $\times$ 6 $\times$ 6.
• Figure 5: Scatter plot showing phonon-induced band gap correction (in meV units) predicted by ML model versus DFT calculation for $6\times 6 \times 6$ Si-supercell at four different temperature values.