${\mathrm{BaZrO}}_3$ stability under pressure: The role of nonlocal exchange and correlation

The ground-state structure of ${\mathrm{BaZrO}}_3$ is experimentally known to be cubic down to absolute zero. However, there exist several measured properties and experimental characterizations that earlier computational works have failed to accurately describe and explain within this cubic symmetry. Among these properties and observations are the dielectric constant and the parallel mean-squared relative displacement value that tracks the fluctuations in distance for Ba-O atom pairs. Previous density-functional theory (DFT) studies have resolved the issue by assuming that ${\mathrm{BaZrO}}_3$ undergoes a phase transition from cubic to tetragonal $I4/mcm$ symmetry, possibly while forming a glasslike state that reflects cubic symmetry on average. In this paper, we show that the set of experimental results can indeed be satisfactorily explained by DFT entirely within the cubic symmetry. We find that past theory limitations arose from the choice of exchange-correlation-functional approximations and that the inclusion of Fock exchange in hybrids significantly improves the DFT performance. We also find that the inclusion of nonlocal correlation effects is beneficial. We conclude by making a prediction for the phase-transition pressure for the transition from cubic to tetragonal symmetry at zero kelvin.

Figure 1

${\mathrm{BaZrO}}_3$ structure and illustration of the Glazer angle rotation $\varphi$, taking the cubic $Pm\overline{3}m$ structure into the tetragonal $I4/mcm$ structure. The Ba-O distance $u_{\parallel }$ is also marked. Note that the barium and oxygen ions are in different planes.

Figure 2

Variation of the $R_{25}$-mode frequency squared by lattice constants obtained in a quasiharmonic approximation. Adapted from Fig. 5 of Ref. [11], showing zero-point-energy effects, moving the raw DFT lattice constants $a_0$ (squares) to the ZPE-corrected values $a_{\text{ZPE}}$ (circles) for CX0p, HSE, CX, and PBE.

Figure 3

Thermal expansion $\alpha$ as a function of temperature computed using the QHA. The experimental values are taken from Ref. [4].

Figure 4

Energy as a function of volume (left) and enthalpy as a function of pressure (right) for the three different phases attainable from an $R$-mode instability computed using the PBE functional. Here the energies and volumes are given relative to the reference values $E_0$ and $V_0$ defined as the energy minimum and corresponding volume of the cubic phase.

Figure 5

Illustration of the functional square used to analyze the role of nonlocal exchange and nonlocal correlation and our description of ${\mathrm{BaZrO}}_3$.

Figure 6

Color coding of the relative difference $\mathrm{\Delta }{\mathrm{\Phi }}_{ij}/{\mathrm{\Phi }}_{ij}$ of the elements in the force constant matrices between PBE and HSE (left) and between HSE and CX0p (right). Positive values in the left (right) panel indicate that larger force constants are found in HSE (CX0p) than in PBE (HSE).

Figure 7

Energy as a function of volume (left) and enthalpy as a function of pressure (right) for the three different phases attainable from an $R$-mode instability computed using the PBE functional.

Figure 8

Determination of the critical pressure for (a) PBE and (b) HSE using fitting as described in the text, with (blue) the Glazer rotation angle squared (in degrees) and (red) the tetragonal strain as discriminator.

Figure 9

Determination of the critical pressure using a straight-line fit of the $R_{25}$-mode frequency as a function of lattice constant.

Figure 10

Graphical illustration summarizing the prediction from the functionals HSE (top left), CX0p (top right), PBE (bottom left), and CX (bottom right). The relative error in percent compared with experiments for the lattice constant $a$ (magnified ten times), the $R_{25}$-mode frequency ${\omega }_R$, the dielectric constant $\epsilon$, the oxygen-barium MSRD at 300 K ${\langle \mathrm{\Delta }{u}_{\parallel }^2\rangle }_{\text{BaO}}$, and the root mean square for the thermal expansion. The last column (in each box), marked $\sigma$, shows a root-mean-square error summary of the previous columns.

Figure 11

The temperature dependence of (a) the dielectric constant, (b) the Last-mode frequency (TO1) computed using the quasiharmonic approximation. Values for CX are only given at temperatures above 125 K (see Sec. 2a). Experimental values are taken from Refs. [6, 34].

Figure 12

The dielectric constant of ${\mathrm{BaZrO}}_3$ at different lattice constants computed with PBE. The vertical dashed lines mark the phase transitions between tetragonal and cubic and between cubic and ferroelectric orthorhombic phases.