Robust high-fidelity DFT study of the lithium-graphite phase diagram

Lithium intercalation in graphite is an important process in the context of anodes for lithium ion batteries. With increasing demands on lithium ion batteries to operate at lower temperatures and higher currents, it is crucial to understand lithium intercalation in graphite due to issues associated with lithium plating. Lithium intercalation into graphite has been extensively studied theoretically using density functional theory (DFT) calculations, complemented by experimental studies through x-ray diffraction, spectroscopy, optical imaging, and other techniques. In this work, we present a first-principles-based model using DFT calculations, employing the Bayesian error estimation functional van der Waals (BEEF-vdW) as the exchange correlation functional, and Ising model to determine the phase transformations and, subsequently, the thermodynamic intercalation potential diagram. We explore a configurational phase space of about $1\ifmmode\times\else\texttimes\fi{}{10}^{9}$ structures by accurately determining the important interactions for the Ising model. The BEEF-vdW exchange correlation functional employed accurately captures a range of interactions including vdW, covalent, and ionic interactions. We incorporate phonon contributions at finite temperatures and configurational entropy to get high accuracy in free energy and potentials. We utilize the built-in error estimation capabilities of the BEEF-vdW exchange correlation functional and develop a methodological framework for determining the uncertainty associated with DFT calculated phase diagrams and intercalation potentials. The framework also determines the confidence of each predicted stable phase. The confidence value of a phase can help us to identify regions of solid solutions and phase transformations accurately. Given the subtle differences in energy between lithium intercalation into graphite and lithium plating $(<0.1\phantom{\rule{0.16em}{0ex}}\mathrm{eV})$, we believe such an error estimation framework is crucial to know the reliability of DFT predictions.