Surrogate models for vibrational entropy based on a spatial decomposition

Abstract

The temperature-dependent behavior of defect densities within a crystalline structure is intricately linked to the phenomenon of vibrational entropy. Traditional methods for evaluating vibrational entropy are computationally intensive, limiting their practical utility. We show that total entropy can be decomposed into atomic site contributions and rigorously estimate the locality of site entropy. This analysis suggests that vibrational entropy can be effectively predicted using a surrogate model for site entropy. We employ machine learning to develop such a surrogate models employing the Atomic Cluster Expansion model. We supplement our rigorous analysis with an empirical convergence study. In addition we demonstrate the performance of our method for predicting vibrational formation entropy and attempt frequency of the transition rates, on point defects such as vacancies and interstitials.

Figures (11)

• Figure 1: Depiction of the spectrum of $\mathbf{T}$, where $\sigma(\mathbf{T}) \cap (0, \infty) \subset [m,M]$ along with the appropriate contour $\mathcal{C}$.
• Figure 2: Derivatives of site entropy for the 1 dimensional toy model including 800 atoms. The dark green dots represent the maximum $\frac{\partial S}{\partial u_n}$ values within specified logarithmic bins along the $x$ axis.
• Figure 3: Derivatives of site entropy for the 2 dimensional toy model including 2025 atoms. The dark green dots represent the maximum $\frac{\partial S}{\partial u_n}$ values within specified logarithmic bins along the x axis.
• Figure 4: Absolute difference in truncated site entropy $\widetilde{\mathcal{S}}_{\ell}$ and non-truncated site entropy $\mathcal{S}_{\ell}$ for the 1D (left) and 2D (right) toy models with different $\delta$ values. The 1D model included 800 atoms, while the 2D model included 4225 atoms, with the site entropies evaluated near the center of the domains.
• Figure 5: Total entropy trained on 70 configurations including 31 Silicon atoms and tested on 50 rattled configurations of bulk Silicon with a $2a_0 \times 2a_0 \times a_0$ supercell and and 50 rattled configurations of Silicon including a vacancy.

Theorems & Definitions (3)

• proof
• proof : Proof of Theorem \ref{['thm:local']}
• proof : Proof of Theorem \ref{['thm:err_local']}