Order-disorder transition in the prototypical antiferroelectric ${\mathrm{PbZrO}}_3$

The prototypical antiferroelectric ${\mathrm{PbZrO}}_3$ has several unsettled questions, such as the nature of the antiferroelectric transition, a possible intermediate phase, and the microscopic origin of the $Pbam$ ground state. Using first-principles calculations, we show that no phonon becomes truly soft at the cubic-to-$Pbam$ transition temperature, and the order-disorder character of this transition is clearly demonstrated based on molecular dynamics simulations and potential energy surfaces. The out-of-phase octahedral tilting is an important degree of freedom, which can collaborate with other phonon distortions and form a complex energy landscape with multiple minima. Candidates of the possible intermediate phase are suggested based on the calculated kinetic barriers between energy minima, and the development of a first-principles-based effective Hamiltonian. The use of this latter scheme further reveals that specific bilinear interactions between local dipoles and octahedral tiltings play a major role in the formation of the $Pbam$ ground state, which contrasts with most of the previous explanations.

Figure 1

Calculated temperature dependence of phonon frequencies in the cubic phase. (a) Phonon dispersion spectra at various temperatures. For clarity, only branches with $\omega <400{\mathrm{cm}}^{-1}$ are shown. For comparison, the DFPT phonon is also displayed (dashed lines). (b) Temperature dependence of the frequency of the ferroelectric mode $\left(\mathrm{\Gamma }\right)$, in-phase tilting mode $\left(M\right)$, out-of-phase tilting mode $\left(R\right)$, and antipolar mode $\left(\mathrm{\Sigma }\right)$, in comparison with available experimental data [13].

Figure 2

Pair correlation function as an indication of structural distortions at 600 K. (a) Ab initio MD sampling. (b) Stochastic sampling. Black arrows point to the differences.

Figure 3

Potential energy landscapes in PZO. (a) 0-K potential energy as a function of the amplitude of unstable high-symmetry phonon modes, i.e., $\mathrm{\Gamma }$-point ferroelectric distortion, $M$-point in-phase tilting, and $R$-point out-of-phase tilting. (b) 0-K PES as functions of $a^{-}{a}^{-}{c}^0$ and ${\mathrm{\Sigma }}_2+S_2$ (left panel, $Pbam$ as minima), $a^{-}{a}^{-}{c}^0$ and $a^0{a}^0{c}^{-}+{\mathrm{\Gamma }}_4^{-}$ (middle panel, $R3c$ as minima), and $a^{-}{a}^{-}{c}^0$ and $a^0{a}^0{c}^{+}+X_5^{-}$ (right panel, $Pnma$ as minima). (c) Simplified picture of the dynamic average structures. The circles in each square $(Pbam,R3c$, or $Pnma$ phase) denote the equivalent energy minima. The solid circles represent the minima being dynamically sampled at finite temperature. Note that the four degeneracies chosen in this schematization are for drawing convenience and do not correspond to the real degeneracy of each phase.

Figure 4

The effect of different types of dipole-tilting couplings on the total energies of the $Pbam,Pnma,R3c$, and long-modulated (LM) phases of ${\mathrm{PbZrO}}_3$ based on Monte Carlo simulations at 1 K. The simulated LM phase corresponds to ${\mathit{q}}_{\text{LM}}=(2\pi /a)(3/16,3/16,0)$. (a) Biquadratic coupling. (b) Trilinear coupling. (c) Bilinear coupling. The vertical dashed lines denote the coupling constants obtained by DFT calculations.