Structural phase transitions in Si and SiO${}_2$ crystals via the random phase approximation

We have assessed the performance of the non-self-consistent random phase approximation (RPA) on two pressure-induced structural phase transitions, diamond to β-Sn Si in Si and α-quartz to stishovite in SiO${}_2$. The calculated equilibrium lattice properties of the four structures are in better agreement with experimental results than are those from several semilocal functionals. The energy differences between the high- and low-pressure phases are found to be 0.37 eV/Si and 0.39 eV/SiO${}_2$, respectively. The transition pressure obtained from our RPA calculations for diamond to β-Sn in Si is 12.2 GPa, in excellent agreement with the experimental value 11.3–12.6 GPa. However, the α-quartz to stishovite phase-transition pressure in SiO${}_2$ is found to be 5.6 GPa, lower than the experimental 7.46 GPa; the Perdew-Burke-Ernzerhof (PBE) semilocal functional gives the transition pressure closest to experiment in this case. We conclude that the non-self-consistent, nonlocal RPA accurately describes the insulator-to-metal transition in Si, where semilocal density functionals tend to fail. But the RPA error cancellation that is nearly perfect in many solids, including Si, may be less perfect in solid SiO${}_2$, as it is in many molecules.

Figure 1

The energy versus volume curves of the Si and SiO${}_2$ systems. The dotted line is the common tangent line obtained from the Birch-Murnaghan third-order EOS, and the solid lines in the graphs are fitted to the data points by the same EOS: (a) Si and (b) SiO${}_2$.

Figure 2

Calculated RPA energy landscapes of SiO${}_2$ phases, with all geometry parameters other than the lattice constants $c$ and $a$ from PBE. The energy minimum is in the darkest blue area. The experimental lattice constants are indicated by gray circles. The gray squares show the lattice constants found by minimizing the RPA energy with respect to $a$ only, at fixed PBE $c$/$a$. The unit for lattice constants is angstroms (Å). The local minimum is chosen to be the zero reference energy: (a) α-quartz and (b) stishovite. (Unlike SiO${}_2$, Si has no internal geometry parameter.)