Predicting Defect Stability and Annealing Kinetics in Two-Dimensional PtSe$_2$ Using Steepest Entropy Ascent Quantum Thermodynamics
The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework was used to calculate the stability of a collection of point defects in 2D PtSe$_2$ and predict the kinetics with which defects rearrange during thermal annealing. The framework provides a non-equilibrium, ensemble-based framework with a self-consistent link between mechanics (both quantum and classical) and thermodynamics. It employs an equation of motion derived from the principle of steepest entropy ascent (maximum entropy production) to predict the time evolution of a set of occupation probabilities that define the states of a system undergoing a non-equilibrium process. The system is described by a degenerate energy landscape of eigenvalues, and the entropy is found from the occupation probabilities and the eigenlevel degeneracies. Scanning tunneling microscopy was used to identify the structure and distribution of point defects observed experimentally in a 2D PtSe$_2$ film. A catalog of observed defects included six unique point defects (vacancies and anti-site defects on Pt and Se sublattices) and twenty combinations of multiple point defects in close proximity. The defect energies were estimated with density functional theory (DFT), while the degeneracies, or density of states, for the 2D film with all possible combinations or arrangements of cataloged defects was constructed using a non-Markovian Monte-Carlo approach (i.e., the Replica-Exchange-Wang-Landau algorithm) with a q-state Potts model. The energy landscape and associated degeneracies were determined for a 2D PtSe$_2$ film two molecules thick and $30 \times 30$ unit cells in area (total of 5400 atoms). The SEAQT equation of motion was applied to the energy landscape to determine how an arbitrary density and arrangement of the six defect types evolve during annealing.