Comparative first-principles studies of prototypical ferroelectric materials by LDA, GGA, and SCAN meta-GGA

Originating from a broken spatial inversion symmetry, ferroelectricity is a functionality of materials with an electric dipole that can be switched by external electric fields. Spontaneous polarization is a crucial ferroelectric property, and its amplitude is determined by the strength of polar structural distortions. Density functional theory (DFT) is one of the most widely used theoretical methods to study ferroelectric properties, yet it is limited by the levels of approximations in electron exchange-correlation. On the one hand, the local density approximation (LDA) is considered to be more accurate for the conventional perovskite ferroelectrics such as $\mathrm{BaTi}{\mathrm{O}}_3$ and $\mathrm{PbTi}{\mathrm{O}}_3$ than the generalized gradient approximation (GGA), which suffers from the so-called super-tetragonality error. On the other hand, GGA is more suitable for hydrogen-bonded ferroelectrics than LDA, which largely overestimates the strength of hydrogen bonding in general. We show here that the recently developed general-purpose strongly constrained and appropriately normed (SCAN) meta-GGA functional significantly improves over the traditional LDA/GGA for structural, electric, and energetic properties of diversely bonded ferroelectric materials with a comparable computational effort and thus enhances largely the predictive power of DFT in studies of ferroelectric materials. We also address the observed system-dependent performances of LDA and GGA for ferroelectrics from a chemical bonding point of view.

Figure 1

The low-temperature ferroelectric phase of several prototypical ferroelectric materials. (a) Tetragonal $\mathrm{BaTi}{\mathrm{O}}_3$ and $\mathrm{PbTi}{\mathrm{O}}_3$ with space group $P4mm$, (b) perovskite-like $\mathrm{LiNb}{\mathrm{O}}_3$ and $\mathrm{BiFe}{\mathrm{O}}_3$ with space group $R3c$, (c) orthorhombic $\mathrm{K}{\mathrm{H}}_2\mathrm{P}{\mathrm{O}}_4$ with space group $Fdd2$, (d) orthorhombic 2-phenylmalondialdehyde (${\mathrm{C}}_9{\mathrm{H}}_8{\mathrm{O}}_2$, known as PhMDA) with space group $Pna{2}_1$, and (e) hexagonal $\mathrm{YMn}{\mathrm{O}}_3$ with space group $R3c$. The black arrows show the spontaneous polarization directions with respect to lattices. Note that the O−H···O bonds in $\mathrm{K}{\mathrm{H}}_2\mathrm{P}{\mathrm{O}}_4$ are almost within the $ab$-plane (i.e., the basal plane), but the bonds in PhMDA have both the in-plane and out-of-plane components.

Figure 2

Plots of the properties reported in Table 1 for $\mathrm{BaTi}{\mathrm{O}}_3$ (left panel), $\mathrm{PbTi}{\mathrm{O}}_3$ (middle panel), and $\mathrm{LiNb}{\mathrm{O}}_3$ (right panel). The PP and AE are pseudopotential and all-electron potential, respectively. The annotated values are relative errors in comparison with the experimental values. In subplot (b), an error bar is plotted for the experimental values (see text for details); in subplot (n), the dotted bar indicates that the experimental value of $\mathrm{PbTi}{\mathrm{O}}_3$ spontaneous polarization is quite scattered $(50\sim 100\mu \mathrm{C}/\mathrm{c}{\mathrm{m}}^2)$, and the upper limit of $100\mu {\mathrm{C}/\mathrm{cm}}^2$ is used here (see discussions in the main text).

Figure 3

Total energy as a function of the amplitude of the polar distortion for (a) $\mathrm{BaTi}{\mathrm{O}}_3$ and (b) $\mathrm{PbTi}{\mathrm{O}}_3$ within the LDA, PBE, SCAN, and HSE functionals. In the calculations, the lattice constants are fixed to the relaxed values of the ferroelectric phase, and the internal ionic positions are interpolated along the direction of the polar mode. The double-well depth here is quantitatively slightly different from the energetic difference in Table 1, for which all the structural parameters are fully relaxed.

Figure 4

Ratios of the theoretically calculated and experimentally measured lattice constants from Table 2.

Figure 5

Phonon dispersion relations of cubic $\mathrm{BaTi}{\mathrm{O}}_3, \mathrm{PbTi}{\mathrm{O}}_3$, and $\mathrm{SrTi}{\mathrm{O}}_3$ calculated by LDA, PBE, and SCAN. In the left panel, the crystal structures are fully relaxed by each functional (see Table 2), and in the right panel the experimental lattice parameters of $a\left(\mathrm{BaTi}{\mathrm{O}}_3\right)=4.001\AA$ [87], $a\left(\mathrm{PbTi}{\mathrm{O}}_3\right)=3.93\AA$ [61], and $a\left(\mathrm{SrTi}{\mathrm{O}}_3\right)=3.905\AA$ [87] are used. The splitting between longitudinal optical (LO) and transverse optical (TO) phonons (i.e., the LO-TO splitting) is considered in the calculation.

Figure 6

Total energy as a function of the amplitude of the polar distortion between centric and polar configurations for (a) $\mathrm{K}{\mathrm{H}}_2\mathrm{P}{\mathrm{O}}_4$ and (b) PhMDA within the LDA, PBE, and SCAN functionals. In the calculations, the lattice constants are fixed to the relaxed values of the low-temperature structure, and the internal ion positions are interpolated. The double-well depth here is quantitatively slightly different from the energetic difference in Table 4, for which all the structural parameters are fully relaxed.

Figure 7

Plots of the lattice parameter $a$, lattice distortion $c/a$, and unit cell volume $\mathrm{\Omega }$ reported in Table 5. The annotated values are the errors of the calculated results relative to the experimental values.