Structural stability and lattice dynamics of ${\text{SiO}}_2$ cristobalite

Among the phases of ${\text{SiO}}_2$ are $\alpha$ and $\beta$ cristobalites, which have a long and somewhat controversial history of proposed structural assignments and phase-transition mechanisms. Recently, Zhang and Scott found new indications that the higher-temperature $\beta$ phase has space group $I\overline{4}2d$ and, by assuming a group-subgroup relationship between phases, they argued that the lower-temperature $\alpha$ phase should have lower symmetry than that of the widely accepted $P{4}_1{2}_{1}2$ space group. With this motivation, we use first-principles calculations to investigate the energy, structure, and local stability of $P{4}_1{2}_{1}2$ and $I\overline{4}2d$ structures. We also compute the frequencies of the zone-center phonon modes in both structures, as well as certain zone-boundary modes in the $I\overline{4}2d$ structure, and compare with experiment. We then argue that the various $P{4}_1{2}_{1}2$ and $I\overline{4}2d$ enantiomorphs can be grouped into three clusters, each of which is identified with a three-dimensional manifold of structures of $P{2}_1{2}_1{2}_1$ symmetry in which the $P{4}_1{2}_{1}2$ and $I\overline{4}2d$ appear as higher-symmetry special cases. We find that there are relatively high energy barriers between manifolds, but low barriers within a manifold. Exploring the energy landscape within one of these manifolds, we find a minimal-energy path connecting $P{4}_1{2}_{1}2$ and $I\overline{4}2d$ structures with a surprisingly low barrier of $\sim 5\text{meV}$ per formula unit. Possible implications for the phase-transition mechanism are discussed.

Figure 1

(Color online) Projection on $x\text{−}y$ plane of the (a) $\tilde{\alpha }$ and (b) $\tilde{\beta }$ structures, proposed as candidates for $\alpha$ and $\beta$ cristobalite phases, respectively. Darker shading is used to represent more distant tetrahedra so that the spiral structure of the connected tetrahedrons becomes evident; double vertical lines indicate that the adjoining tetrahedra are actually disconnected because they are separated in the $z$ direction.

Figure 2

Ground-state energy per formula unit (eV) vs volume per formula unit $\left({\text{Å}}^3\right)$ for $\tilde{\alpha }$, $\tilde{\beta }$, and cubic cristobalite structures of ${\text{SiO}}_2$.

Figure 3

(Color online) Five linearized RUMs in ideal cristobalite $Z=4$ structure.

Figure 4

Energy in plane defined by the $\tilde{\alpha }$ structure (filled circle), $\tilde{\beta }$ structure (filled square), and saddle point (cross). Coordinates are chosen such that these structures occur at (0,0), (1,0), and (0.5,1), respectively. The energy difference separating contours is 3 meV per ${\text{SiO}}_2$ formula unit.

Figure 5

Energy as a function of rotation angles ${\varphi }_{{\alpha }_1}$ and ${\varphi }_{{\beta }_1}$, corresponding to rotations ${\tilde{\alpha }}_1$ and ${\tilde{\beta }}_1$ shown in Fig. 3. The origin corresponds to “ideal” cristobalite. Filled squares at top and bottom denote $\tilde{\beta }$ minima, filled circles at left and right denote ${\alpha }_1$ minima, and crosses denote saddle points, as in Fig. 4. The energy difference separating contours is 3 meV per ${\text{SiO}}_2$ formula unit.

Figure 6

Energy as a function of rotation angles ${\varphi }_{{\alpha }_1}$ and ${\varphi }_{{\alpha }_1^{\prime }}$, corresponding to rotations ${\tilde{\alpha }}_1$ and ${\tilde{\alpha }}_1^{\prime }$ shown in Fig. 3. The origin corresponds to “ideal” cristobalite. The four minima (filled circles) correspond to the ${\tilde{\alpha }}_1^{\prime }$ structure (top and bottom) and to the ${\tilde{\alpha }}_1$ structure (left and right). The energy difference separating contours is 3 meV per ${\text{SiO}}_2$ formula unit.

Figure 7

Sketch of important states in one of the three-dimensional rigid-unit subspaces discussed in the text. Local energy minima associated with $\tilde{\alpha }$ $\left(D_4\right)$ and $\tilde{\beta }$ $\left(D_{2d}\right)$ structures are indicated by filled circles and squares, respectively. Remainder of space, including saddle points (crosses), has $D_2$ symmetry.

Figure 8

Structural parameters of the $\tilde{\alpha }$ structure vs volume per formula unit. Top panel: $c/a$ ratio. Bottom panel: absolute values of deviations of internal parameters $u\left(\text{Si}\right)$ (squares), $x\left(\text{O}\right)$ (triangles), $y\left(\text{O}\right)$ (crosses), and $z\left(\text{O}\right)$ (circles) from ideal-cubic values. Symbols represent first-principles calculations; lines are fits to an ideal rigid-unit geometry.

Figure 9

Structural parameters of the $\tilde{\beta }$ structure vs volume per formula unit. Top panel: $c/a$ ratio. Bottom panel: $x\left(\text{O}\right)$. Symbols represent first-principles calculations; lines are fits to an ideal rigid-unit geometry.