Dynamical stabilization of cubic ${\text{ZrO}}_2$ by phonon-phonon interactions: Ab initio calculations

Cubic zirconia exhibits a soft phonon mode $\left(X_2^{-}\right)$ which becomes dynamically unstable at low temperatures. Previous ab initio investigations into the temperature-induced stabilization of the soft mode treated it as an independent anharmonic oscillator. Calculations presented here, using the self-consistent ab initio lattice-dynamical method to evaluate the phonons at 2570 K, show that the soft mode should not be treated independently of other phonon modes. Phonon-phonon interactions stabilize the $X_2^{-}$ mode. Furthermore, the effective potential experienced by the mode takes on a quadratic form.

Figure 1

(Color online) Schematic potential-energy surfaces for systems with two vibrational modes. The solid black curve illustrates the potential seen by the second mode for a fixed finite amplitude of the first mode. In (a), both modes are independent and stable. In (b), the second mode is unstable and independent of the first mode, i.e., always sees the same potential. In (c), the second mode is not independent of the first mode; for small amplitudes of the first mode, the second mode is unstable; for a large-enough amplitude of the first mode, the second mode becomes stable.

Figure 2

The phonon dispersion of cubic ${\text{ZrO}}_2$ calculated with a 24-atom supercell (top row), an 81-atom supercell (middle row), and a 192-atom supercell (bottom row). The solid (and dashed) lines represent the first-principles phonon calculations. The left column (a) shows the results from the direct force method calculations (with the negative axis convention used for imaginary frequencies); the right column (b) shows the finite-temperature SCAILD calculations. The dashed lines on the left-hand side of the figure indicate imaginary phonon frequencies obtained in the direct force method calculations. At 2570 K these modes become stable, i.e., real, by including interaction between the phonons.

Figure 3

(Color online) Schematic figure of the atomic movements corresponding to the soft longitudinal $X_2^{-}$-point phonon mode of cubic ${\text{ZrO}}_2$. Here two unit cells of ${\text{ZrO}}_2$ are depicted with large red spheres representing oxygen atoms and small white spheres representing zirconium atoms. The green arrows, here pointing parallel to the [100] direction of the crystal, indicate the directions of the displacement of the oxygen atoms corresponding to the $X_2^{-}$ phonon mode. (The Zr atoms are not displaced by this mode.)

Figure 4

(Color online) Phonon density of states for cubic ${\text{ZrO}}_2$ calculated with the SCAILD method. The dashed blue curve corresponds to the 24-atom supercell calculation, the thin red curve to the 81-atom supercell calculation, and the thick black curve to the 192-atom supercell calculation.

Figure 5

(Color online) Calculated potential experienced by cubic zirconia’s $X_2^{-}$ phonon mode as a function of oxygen atom displacement in the independent phonon approximation and at 2570 K. (a) The calculated potential-energy curve has a local maximum at ${\mathcal{A}}_{\mathbf{k}s}^{\sigma }/\sqrt{N}=0$, reflecting cubic zirconia’s lack of stability at low temperatures. (b) The calculated projected force in 24-atom supercells acting on the $X_2^{-}$ phonon mode at 2570 K reflects cubic zirconia’s stability at high temperatures. Empty symbols represent results evaluated for amplitudes either determined by Eq. (1) (empty circles) or fixed at values evident from the figure (empty diamonds, triangles, and squares). Filled red circles denote the geometrical mean values of the projected forces from each of the five sets of calculations, and the width of the error bars are two times the square root of the mean-square deviation of the projected forces relative to the geometrical mean values. The red dashed line plots a linear fit $F_p\left({\mathcal{A}}_{\mathbf{k}s}^{\sigma }/\sqrt{N}\right)$ of the mean values of the projected forces with $\alpha =-\frac{1}{2}\frac{d{F}_p\left(x\right)}{dx}=17.7{\text{THz}}^2$. (c) Based on the linear fit, the phonon mode experiences an effective potential-energy curve $\Delta E=M_{\sigma }\alpha {\left({\mathcal{A}}_{\mathbf{k}s}^{\sigma }/\sqrt{N}\right)}^2$ at 2570 K that is quadratic.