Deep Neural Network for Phonon-Assisted Optical Spectra in Semiconductors

TL;DR

This work tackles the computational bottleneck of simulating phonon-assisted optical spectra at finite temperature by integrating DeePMD-based molecular dynamics with a DeePTB tight-binding framework, enabling ab initio-fidelity calculations in large supercells using high-accuracy functionals like HSE and SCAN. The approach leverages Williams-Lax theory to compute the temperature-dependent imaginary dielectric function $oldsymbol{ abla}$, enabling accurate captures of phonon-induced bandgap renormalization and indirect/direct absorption in Si and GaAs from 100–400 K. Key findings include convergence of Si spectra for supercells with linear size $L \ge 4$, and successful reproduction of GaAs phonon-assisted features below the direct gap, validating the method for indirect/direct transitions and complex electron-phonon coupling. The framework promises high-throughput capabilities for studying temperature-dependent phenomena in complex materials and is reinforced by publicly available DeePTB code, facilitating broader adoption and extension to alloys and nanostructures.

Abstract

Ab initio based accurate simulation of phonon-assisted optical spectra of semiconductors at finite temperatures remains a formidable challenge, as it requires large supercells for phonon sampling and computationally expensive high-accuracy exchange-correlation (XC) functionals. In this work, we present an efficient approach that combines deep learning tight-binding and potential models to address this challenge with ab initio fidelity. By leveraging molecular dynamics for atomic configuration sampling and deep learning-enabled rapid Hamiltonian evaluation, our approach enables large-scale simulations of temperature-dependent optical properties using advanced XC functionals (HSE, SCAN). Demonstrated on silicon and gallium arsenide across temperature 100-400 K, the method accurately captures phonon-induced bandgap renormalization and indirect/direct absorption processes which are in excellent agreement with experimental findings over five orders of magnitude. This work establishes a pathway for high-throughput investigation of electron-phonon coupled phenomena in complex materials, overcoming traditional computational limitations arising from large supercell used with computationally expensive XC-functionals.

Figures (5)

• Figure 1: Workflow for calculating phonon-assisted optical spectra using DeePMD and DeePTB models. DeePMD generates atomic structures through molecular dynamics simulations. Small-cell structures are then labeled by DFT eigenvalues to train the DeePTB model. Large-cell structures from DeePMD are then used with the trained DeePTB model to predict TB Hamiltonians $H(R)$ and hence the transition matrix for optical spectra calculations.
• Figure 2: Absorption coefficient of silicon (Si) at 300 K. Blue dashed lines represent calculations with atoms clamped at their equilibrium positions. Red solid lines denote calculations with atom configurations sampled in MD simulations at 300K. Experimental data greenOptical1995 for Si are shown as grey discs. Calculations were performed using $8 \times 8 \times 8$ conventional supercells and $2\times 2 \times 2$${\bm{k}}$-grid for Brillouin zone (BZ) sampling with a Gaussian broadening of 30 meV.
• Figure 3: Convergence of supercell size and number of snapshots (a) The optical absorption spectra of Si were calculated from ensemble averages with different supercell sizes. Calculated ensemble averaged spectra considering five snapshots for (b) conventional, (c) 2 $\times$ 2 $\times$ 2, (d) 4 $\times$ 4 $\times$ 4, and (e) 8 $\times$ 8 $\times$ 8 cell size. The shaded region represents the deviation of spectra from the ensemble-averaged one. As the cell size increases, the deviation in spectra gets substantially reduced.
• Figure 4: Temperature-renormalized indirect band gaps of Si. (a) Tauc plot for determining the indirect band gap as a function of temperature. The markers, in different colors, represent $(\omega^2\epsilon_2)^{1/2}$ at various temperatures, with solid lines of the same color showing the corresponding linear fits. The indirect band gap is obtained from the intercept with the horizontal axis. (b) The temperature dependence of the indirect band gap from the present theory (red discs) and the experimental data alexTemperature1996 (black squares). The solid and dashed lines serve as a guide to the eye.
• Figure 5: Absorption coefficient of GaAs at 300 K. Blue dashed lines represent calculations with atoms clamped at their equilibrium positions. Red solid lines denote calculations with atom configurations sampled in MD simulations at 300K. Experimental data for GaAs are shown as grey sturgeOptical1962 and black aspnesDielectric1983 discs. Calculations were performed using $8 \times 8 \times 8$ conventional supercells and $5\times 5 \times 5$${\bm{k}}$-grid for Brillouin zone (BZ) sampling with a Gaussian broadening of 50 meV.