Exploring the High-Temperature Stabilization of Cubic Zirconia from Anharmonic Lattice Dynamics

Finite-temperature stability of crystals is of continuous relevance in solid-state chemistry with many important properties only emerging in high-temperature polymorphs. Currently, the discovery of new phases is largely serendipitous due to a lack of computational methods to predict crystal stability with temperature. Conventional methods use harmonic phonon theory, but this breaks down when imaginary phonon modes are present. Anharmonic phonon methods are required to describe dynamically stabilized phases. We investigate the high-temperature tetragonal-to-cubic phase transition of ZrO2 based on first-principles anharmonic lattice dynamics and molecular dynamics simulations as an archetypical example of a phase transition involving a soft phonon mode. Anharmonic lattice dynamics calculations and free energy analysis suggest that the stability of cubic zirconia cannot be attributed solely to anharmonic stabilization and is thus absent for the pristine crystal. Instead, an additional entropic stabilization is suggested to arise from spontaneous defect formation, which is also responsible for superionic conductivity at elevated temperatures.

where U n is the n'th order term in the expansion. These are determined from the force constants, Φ, at n'th order U n = 1 n! Φ µ1...µn (l 1 κ 1 ; . . . ; l n κ n )u µ1 (l 1 κ 1 ) · · · u µn (l n κ n ), where u µ (lκ) is the displacement of atom κ in cell l along the cartesian direction µ.
In the harmonic approximation, only the U 2 contribution is retained, and the solution is a set of eigenfrequencies and eigenvector as a function of the reciprocal vector, and band index, j.
Within SCPH theory, anharmonic phonon frequencies can be obtained through the Green's function, G qjj , by solving the Dyson equation given in matrix form as where Σ q (ω) is the anharmonic self-energy. Solving the Dyson equation leads to the following equation, which needs to be solved self-consistently for the anharmonic phonon frequencies, ω The SCPH equations can be solved to various levels of theory by systematically including higher order diagrams to the phonon self-energy. The simplest level of theory, here termed SC1 theory, includes only the contributions from the loop-diagram, which is based on fourth-order force constants. At this level of theory, and neglecting off-diagonal components, the SCPH equations can be written in diagonal form as [1] where n is is the Bose-Einstein distribution giving the population of the phonon mode at the relevant temperature, which is how temperature is introduced into the SCPH equations. This leads to a set of renormalised phonon frequencies Ω, which are all real, even if imaginary harmonic frequencies are present. As all frequencies are real by construction of equations, a phase transition should be predicted by following the change in frequency above the phase transition temperature and extrapolating to zero frequency. The next level of anharmonicity to include originates from the bubble diagram, which is based on third order force constants [3]. In the current implementation, the fully anharmonic Dyson equation, eq. 3, is approximated as where Σ B q is the bubble self-energy, and the self-consistency from the Dyson equation is removed, since the Green's function from the SC1 theory, G S q , is close to the full Green's function. This leads to the following self-consistent equations where Ω S q are the solutions to the SC1 equations in eq. 6. Solving this non-linear (NL) equation results in phonon quasi-particle (QP) frequencies termed QP-NL. Further approximation can be made by setting ω in the self-energy equal to 0 (QP[0] theory) or to Ω S q (QP[S] theory) as used in Fig. 2 in the main text. Finally, anharmonic free energies can be calculated within the SCPH framework [4]. The phonon frequencies from SC1 theory are used and the free energy is calculated as a summation over these in a similar way as what is done in the harmonic approximation, but with an additional term to satisfy the thermodynamic relation between entropy and free energy. To improve accuracy, the contribution from the bubble diagram is added as an additional term through the bubble self-energy as described above.

SUPPORTING NOTE II: ENTROPY FROM FRENKEL DEFECT PAIRS
A simple model for the configurational entropy of Frenkel defects in fluorite type structures has been derived by Voronin [5]. In this model, all interstitial sites are considered accessible and the configurational entropy from random occupation of interstitials and vacancies is determined. The configurational entropy (per atom) is given as where x is the fraction of occupied interstitials. The entropy as a function of defect concentration is plotted in Fig. S8 showing that a defect concentration of around 3% would result in a similar entropic stabilisation as the energetic difference between cubic and tetragonal phases of zirconia.   [6] to obtain only real frequency for calculation of the vibrational free energy. The frequency at 1000 K is used since this is near the flat region of the freqency with temperature and therefore gives a reasonable overall effective frequency for a qualitative comparison of free energies.    [5] given in Eq.
(2) in the main text. The interstitial concentration is given as the fraction of occupied interstitials.