Accurate Deep Potential Model of Temperature-Dependent Elastic Constants for Phosphorus-Doped Silicon

Accurate predictions of elastic properties under varying doping concentrations and temperatures are critical for designing reliable silicon-based micro-/nano-electro-mechanical systems (MEMS/NEMS). Empirical potentials typically lack accuracy for elastic predictions, whereas density functional theory (DFT) calculations are precise but computationally expensive. In this study, we developed a highly accurate and efficient machine learning-based Deep Potential (DP) model to predict the elastic constants of phosphorus-doped silicon (Si_64-xP_x, x = 0, 1, 2, 3, 4) within a temperature range of 0-500 K. The DP model was rigorously validated against benchmark DFT results. At 0 K, the elastic constants predicted by our DP model exhibited excellent agreement with experimental data, achieving a mean absolute percentage error (MAPE) of only 2.88%. We investigated the effects of doping on elastic constants in single-crystal silicon and determined their second-order temperature coefficients. The calculations demonstrated distinct doping-induced variations, showing pronounced decreases in C₁₁ and C₄₄ and a moderate increase in C₁₂. Finite-element analyses using the fitted temperature coefficients indicated improved thermal stability of silicon resonators through phosphorus doping. Our study explores the integration of machine learning-based atomic-scale simulations with MEMS/NEMS design, providing practical guidance for optimal dopant selection to enhance silicon resonator thermal stability.