Stability and lattice dynamics of SiO$_2$ cristobalite

Among the phases of SiO$_2$ are alpha and beta cristobalite. Despite early indications that the higher-temperature beta phase might be cubic (Fd$\overline{3}$m), it is now accepted that it is in fact tetragonal (I$\overline{4}$2d), and that the experiments suggesting a cubic structure were averaging spatially or dynamically over tetragonal domains. Recently, Zhang and Scott (J.\ Phys.\ Cond.Matt.\ {\bf 19}, 275201) suggested that the lower-temperature alpha phase, widely accepted to be tetragonal (P$4_12_12$), might be an artifact in a similar way. With this motivation we investigate the energy landscape in the vicinity of cristobalite phases using first-principles calculations. We use the ABINIT implementation of density-functional theory in a plane-wave pseudopotential framework. We find that both the P$4_12_12$ alpha and I$\overline{4}$2d beta phases are local minima, thus reinforcing that the identification of the alpha phase as belonging to the P$4_12_12$ structure. We compute the frequencies of phonon modes at high-symmetry k-points in both structures and compare with experiment. We also identify a minimum-energy path connecting the alpha and beta phases through an intermediate orthorhombic phase (P$2_12_12_1$), and find a surprisingly low barrier of $\sim$5\,meV per formula unit. We note that a simple rigid-unit mode picture gives a good rough description of these energetics, and we map out the minimum-energy path in the space of rigid unit rotations in a physically insightful way.