VacHopPy: A Python package for vacancy hopping analysis based on molecular dynamics simulations

TL;DR

The paper addresses the challenge of embedding atomistic vacancy diffusion information into continuum models by introducing VacHopPy, an open-source Python package that extracts a single, effective set of hopping parameters from MD data. The method combines TS-based vacancy trajectory determination with a formal construction of an effective lattice, yielding parameters ($\bar{a}$, $\bar{E}_a$, $\bar{z}$, $\bar{\nu}$, $\bar{D}_{\mathrm{rand}}$, $\bar{\tau}$, $f$) that reproduce multi-path diffusion across temperatures. Validation on fcc Al, rutile TiO$_2$, and monoclinic HfO$_2$ shows good agreement with experiments and reveals how path competition, anisotropic vibrations, and phase transitions shape diffusion, while providing a practical route to plug these parameters into continuum models. The framework thus bridges atomistic and continuum scales, enabling more reliable multiscale simulations for diffusion-controlled processes in complex crystals.

Abstract

Multiscale modeling, which integrates material properties from ab initio calculations into continuum-scale simulations, is a promising strategy for optimizing semiconductor devices. However, a key challenge remains: while ab initio methods provide diffusion parameters specific to individual migration paths, continuum equations require a single effective set of parameters that captures the overall diffusion behavior. To address this issue, we present VacHopPy, an open-source Python package for vacancy hopping analysis based on molecular dynamics (MD). VacHopPy extracts an effective set of hopping parameters, including hopping distance, hopping barrier, number of effective paths, correlation factor, and attempt frequency, by statistically integrating energetic, kinetic, and geometric contributions across all paths. It also includes tools for tracking vacancy trajectories and for detecting phase transitions during MD simulations. The applicability of VacHopPy is demonstrated in three representative materials: face-centered cubic Al, rutile TiO2, and monoclinic HfO2. The extracted effective parameters reproduce temperature-dependent diffusion behavior and are in good agreement with previous experimental data. Provided in a simplified form, these parameters are well suited for continuum-scale models and remain valid over a wide temperature range spanning several hundred kelvins. Furthermore, VacHopPy inherently accounts for anisotropy in thermal vibrations, a factor often overlooked, making it suitable for simulating diffusion in complex crystals. Overall, VacHopPy establishes a robust bridge between atomic- and continuum-scale models, enabling more reliable multiscale simulation

Figures (9)

• Figure 1: (a) PES diagram at $T=0~K$. $\mathbf{F}$ acting on the atom before passing the TS (orange) is directed toward the initial site ($\cos \theta_i > \cos \theta_f$), where $\theta_i$ and $\theta_f$ denote the angles between the $\mathbf{F}$ and the directions toward the initial and final sites, respectively. In contrast, $\mathbf{F}$ acting on the atom after crossing the TS (blue) is directed toward the final site ($\cos \theta_i < \cos \theta_f$). (b) PES diagram at $T>0~K$. The PES is perturbed by random thermal fluctuations, making it challenging to determine the relative position of an atom to the TS based on instantaneous $\mathbf{F}$.
• Figure 2: Multiscale modeling strategy using effective hopping parameters. Solid and dashed circles represent atoms and vacancies, respectively. At the atomic scale, the hopping parameters $p_{(i,j)}$ are path-dependent, resulting in multiple distinct sets within a given system. VacHopPy consolidates these into a single set of effective hopping parameters $\bar{p}$, expressed in a material-independent form. This abstraction resolves the ambiguity in parameter selection and enables the use of universal continuum-scale models whose governing equations are formulated in terms of $\bar{p}$ (e.g., $F(\bar{p})=0$ and $G(\bar{p})=0$).
• Figure 3: Representative workflow of the VacHopPy package. For clarity, only the main API components and CLI commands are shown. Detailed descriptions of all functionalities and usage tutorials are provided in the VacHopPy documentation (https://vachoppy.readthedocs.io).
• Figure 4: Atomic structure and a single vacancy hopping path in fcc Al. Light and dark grey circles represent Al atoms in different layers. The characteristics of the A1 path are summarized in the accompanying table.
• Figure 5: Atomic structure and three vacancy hopping paths in rutile TiO2. The blue and red circles represent Ti and O atoms, respectively. The characteristics of each path are summarized in the accompanying table.