Metadynamics Simulations of the High-Pressure Phases of Silicon Employing a High-Dimensional Neural Network Potential

We study in a systematic way the complex sequence of the high-pressure phases of silicon obtained upon compression by combining an accurate high-dimensional neural network representation of the density-functional theory potential-energy surface with the metadynamics scheme. Starting from the thermodynamically stable diamond structure at ambient conditions we are able to identify all structural phase transitions up to the highest-pressure fcc phase at about 100 GPa. The results are in excellent agreement with experiment. The method developed promises to be of great value in the study of inorganic solids, including those having metallic phases.

Figure 1

Enthalpies of the optimized crystal structures obtained in a metadynamics simulation at 300 K starting from the cubic diamond structure at an external pressure of 12 GPa.

Figure 2

Enthalpies of the optimized crystal structures obtained in a metadynamics simulation at 300 K starting from the $\beta$-tin structure at an external pressure of 16 GPa. Note the much smaller enthalpy differences compared to Fig. 1.

Figure 3

Enthalpies of the optimized crystal structures obtained in a metadynamics simulation at 800 K starting from the simple hexagonal (sh) structure at an external pressure of 45 GPa.

Figure 4

Enthalpies of the relaxed structures obtained in a metadynamics simulation at 300 K starting from the Cmca structure at an external pressure of 50 GPa.

Figure 5

Enthalpies of the relaxed structures obtained in a metadynamics simulation at 300 K starting from the hcp structure at an external pressure of 90 GPa.