First-principles accurate total energy surfaces for polar structural distortions of ${\text{BaTiO}}_3$, ${\text{PbTiO}}_3$, and ${\text{SrTiO}}_3$: Consequences for structural transition temperatures

Specific forms of the exchange-correlation energy functionals in first-principles density-functional-theory-based calculations, such as the local-density approximation (LDA) and generalized-gradient approximations (GGA), give rise to structural lattice parameters with typical errors of $-2\mathrm{\%}$ and 2%. Due to a strong coupling between structure and polarization, the order parameter of ferroelectric transitions, they result in large errors in estimation of temperature-dependent ferroelectric structural transition properties. Here, we employ a recently developed GGA functional of Wu and Cohen [Phys. Rev. B 73, 235116 (2006)] and determine total energy surfaces for zone-center distortions of ${\text{BaTiO}}_3$, ${\text{PbTiO}}_3$, and ${\text{SrTiO}}_3$ and compare them with the ones obtained with calculations based on standard LDA and GGA. Confirming that the Wu and Cohen functional allows better estimation of structural properties at 0 K, we determine a set of parameters defining the effective Hamiltonian for ferroelectric transition in ${\text{BaTiO}}_3$. Using the new set of parameters, we perform molecular-dynamics simulations under effective pressures $p=0.0\text{GPa}$, $p=-2.0\text{GPa}$, and $p=-0.005T\text{GPa}$. The simulations under $p=-0.005T\text{GPa}$, which is for simulating thermal expansion, show a clear improvement in the cubic to tetragonal transition temperature and $c/a$ parameter of its ferroelectric tetragonal phase, while the description of transitions at lower temperatures to orthorhombic and rhombohedral phases is marginally improved. Our findings augur well for use of Wu-Cohen functional in studies of ferroelectrics at nanoscale, particularly in the form of epitaxial films where the properties depend crucially on the lattice mismatch.

Figure 1

(Color online) Total energy surfaces for zone-center distortions of a ferroelectric $AB{\text{O}}_3$ perovskite on a two-dimensional subspace $\left(v_z^A,v_z^B\right)$ of the atomic-displacement space $\left(v_z^A,v_z^B,v_z^{{\text{O}}_{\text{I}}},v_z^{{\text{O}}_{\text{II}}},v_z^{{\text{O}}_{\text{III}}}\right)$. (a) Schematic contour plot for atomic displacements from the centrosymmetric cubic structure (b), $e_{zz}=0$, is compared to (c) that from the tetragonal structure (d), $e_{zz}>0$. Thick solid lines are the valley lines for fixed $e_{zz}$’s. Dashed lines show the direction of the ${\Gamma }_{15}$ soft-mode eigenvector $\mathbf{\xi }$ at zero strain. Note that the direction is tangential to the valley line at $v_z^{\tau }=0$ for $e_{zz}=0$, but this is not the case for $e_{zz}\ne 0$.

Figure 2

(Color online) [(a)–(d)] Total energy surfaces for zone-center distortions of ${\text{BaTiO}}_3$. (e) $u$ dependence of strain along [110] distortion. (f) Direction $\mathbf{\xi }\left(\mathit{u}\right)$ of atomic displacements along [001] distortion.

Figure 3

(Color online) (A) Half of eigenvalues of the $3\times 3$ long-range dipole-dipole interaction matrix $\tilde{\Phi }\left(\mathit{k}\right)$ [Fourier transform of Eq. (10) in Ref. 5] are plotted along symmetric axes in the first Brillouin zone of the simple-cubic lattice. Special points and $\mathit{k}/\left(2\pi /a\right)=\left(\frac{1}{4},\frac{1}{4},0\right)$ (the center of the $\Sigma$ axis) are indicated with vertical dotted lines. Labels (a)–(g) corresponds to Eqs. (15a, 15b, 15c, 15d, 15e, 15f, 15g), respectively. Tics in the unit of $\frac{Z^{\ast 2}}{{ϵ}_{\infty }{a}_0^3}$ is placed in left side. Tics in the unit of electron volt, in the case of the parameter set of Table , is placed in right side. (B) Half of eigenvalues of the total $\left(\text{long-range}+\text{short-range}\right)$ interaction matrix ${\tilde{\Phi }}^{\text{quad}}\left(\mathit{k}\right)$ [Eq. (13) in Ref. 5].

Figure 4

(Color online) Calculated dipole moment per unit cell as a function of $u$ for atomic displacements along [001] distortion (solid line). $Z^{\ast }u$ is also plotted for comparison (dashed line).

Figure 5

(Color online) Simulated temperature dependence of lattice parameters under (a) $p=-0.0\text{GPa}$, (b) $p=-2.0\text{GPa}$, and (c) $p=-0.005T\text{GPa}$.

Figure 6

(Color online) [(a)–(d)] Total energy surfaces for zone-center distortions of ${\text{PbTiO}}_3$. (e) $u$ dependence of strain along [110] distortion. (f) Direction $\mathbf{\xi }\left(\mathit{u}\right)$ of atomic displacements along [001] distortion.

Figure 7

(Color online) [(a)–(d)] Total energy surfaces for zone-center distortions of ${\text{SrTiO}}_3$. (e) $u$ dependence of strain along [110] distortion. (f) Direction $\mathbf{\xi }\left(\mathit{u}\right)$ of atomic displacements along [001] distortion.