First-principles calculations of the ferroelastic transition between rutile-type and ${\text{CaCl}}_2$-type ${\text{SiO}}_2$ at high pressures

The tetragonal to orthorhombic ferroelastic phase transition between rutile- and ${\text{CaCl}}_2$-type ${\text{SiO}}_2$ at high pressures is studied using first-principles calculations and the Landau free-energy expansion. The phase transition is systematically investigated in terms of characteristic phonon modes with ${\text{B}}_{1g}$ and ${\text{A}}_g$ symmetries, shear moduli, transverse-acoustic mode, rotation angle of the ${\text{SiO}}_6$ octahedra, spontaneous symmetry-breaking and volume strains, and enthalpy. The results show that these physical behaviors at the transition are well described using the Landau free-energy expansion parametrized by the first-principles calculations.

Figure 1

(Color online) (a) Unit cell of the rutile-type structure. The dark gray and light gray (red) denote silicon and oxygen atoms, respectively. (b) Polyhedral representation of the ${\text{B}}_{1g}$ $\left({\text{A}}_g\right)$ mode of the tetragonal (orthorhombic) structure projected onto the $a\text{−}b$ plane. The directions of the octahedral rotations are depicted by arrows. The square with dashed lines represents the unit cell. (c) Schematic representation of the tetragonal shear strain. The dashed and solid lines represent the tetragonal and orthorhombic lattices, respectively. The arrows show the shear direction.

Figure 2

(Color online) Squared frequencies of the ${\text{B}}_{1g}$ and ${\text{A}}_g$ modes as a function of pressure. The filled and open circles depict the values for the tetragonal and orthorhombic structures, respectively. The solid and dashed lines are the fits to respective values.

Figure 3

(Color online) (a) Internal energies per unit volume as a function of the rotation angle $\varphi$ relative to those at $\varphi =0$. The circles, triangles, and squares denote the values at 20, 60, and 100 GPa, respectively. The curves through the symbols are guides for the eyes. (b) Second- and fourth-order coefficients $A$ and $B$ of Eq. (29) per unit volume as a function of pressure. The circles and squares denote $A$ and $B$, respectively. (c) Squared rotation angles ${\varphi }^2$ and ${\varphi }^{\prime 2}$ at equilibrium as a function of pressure. The circles and triangles depict $\varphi$ and ${\varphi }^{\prime }$, respectively. The solid and dashed curves are the fits to $\varphi$ and ${\varphi }^{\prime }$ above the transition pressure, respectively.

Figure 4

(Color online) (a) $\left(c_{11}^0-c_{12}^0\right)/2$ as a function of pressure. The circles, squares, and triangles depict those calculated with strains of 0.001, 0.005, and 0.01, respectively. (b) $\left(c_{11}-c_{12}\right)/2$ and $\left(c_{11}+c_{22}-2c_{12}\right)/4$ as a function of pressure. Filled and open circles depict the values calculated using the tetragonal and orthorhombic structures, respectively, and the solid and dashed curves are the fits to the respective values.

Figure 5

(Color online) Redefined unit cell and original unit cell projected onto $a\text{−}b$ plane, which are depicted by the solid and dashed lines, respectively. The number of atoms in the redefined unit cell is twice that in the original unit cell.

Figure 6

(Color online) (a) TA-mode phonon branches using the elongated supercells (solid curves) and the $2\times 2\times 3$ supercells (dashed curves). The circles on the solid curves represent the exact $k$ points represented by the elongated supercells. The vertical dashed-dotted line depicts $\mathbf{k}=\left[1/16,1/16,0\right]$. Imaginary frequencies are represented as negative values. (b) Squared TA-mode frequencies of the elongated supercell at $\mathbf{k}=\left[1/16,1/16,0\right]$ as a function of pressure. The filled and open circles show the values for the tetragonal and orthorhombic structures, respectively, and the solid and dashed curves are the fits to the respective values.

Figure 7

(Color online) Lattice parameters as a function of pressure. The filled and open circles depict those of the tetragonal and orthorhombic structures, respectively. The crosses denote $\sqrt{a_o\times b_o}$.

Figure 8

(Color online) (a) Squared symmetry-breaking strain ${\left(e_1-e_2\right)}^2$ as a function of pressure. (b) Volume strain as a function of pressure. The curves are the fits to the values above the transition pressure.

Figure 9

(Color online) Enthalpy of the orthorhombic structure relative to that of the tetragonal structure as a function of pressure normalized by the unit volume of the tetragonal structure. The curve depicts the fit to the values above 54 GPa.