Mapping the energy surface of PbTiO${}_3$ in multidimensional electric-displacement space

In recent years, methods have been developed that allow first-principles electronic-structure calculations to be carried out under conditions of fixed electric field. For some purposes, however, it is more convenient to work at fixed electric displacement field. Initial implementations of the fixed-displacement-field approach have been limited to constraining the field along one spatial dimension only. Here, we generalize this approach to treat the full three-dimensional displacement field as a constraint and compute the internal-energy landscape as a function of this multidimensional displacement-field vector. Using PbTiO${}_3$ as a prototypical system, we identify stable or metastable tetragonal, orthorhombic, and rhombohedral structures as the displacement field evolves along the [001], [110], and [111] directions, respectively. The energy minimum along [001] is found to be deeper than that along [110] or [111], as expected for a system having a tetragonal ground state. The barriers connecting these minima are found to be quite small, consistent with the current understanding that the large piezoelectric effects in PbTiO${}_3$ arise from the easy rotation of the polarization vector.

Figure 1

(a) Reduced electric field $\overline{\varepsilon }$ and (b) internal energy $U$ as a function of reduced electric displacement field $d$ along the [001] direction. The lautrec results are reproduced from Ref. 8. The solid curve in (a) is a cubic-spline fit to the abinit results; the solid curve in (b) is the numerical integral of the curve in (a).

Figure 2

Internal energy surface $U\left(\mathbf{D}\right)$ of PbTiO${}_3$ plotted for $\mathbf{D}$ lying in the plane spanned by the [100] and [010] directions. (a) Perspective plot. (b) Contour plot, with 3.0 meV separating contour levels. The minimum at T and saddle point at O represent spontaneously polarized tetragonal and orthorhombic states, respectively.

Figure 3

Internal energy surface $U\left(\mathbf{D}\right)$ of PbTiO${}_3$ plotted for $\mathbf{D}$ lying in the plane spanned by the [110] and [001] directions. (a) Perspective plot. (b) Contour plot, with 3.0 meV separating contour levels. The minima at T and O and the saddle point at R represent spontaneously polarized tetragonal, orthorhombic, and rhombohedral states, respectively.

Figure 4

Dark blue: isosurfaces of $U\left(\mathbf{D}\right)$ for PbTiO${}_3$ plotted at $U=-44$ meV. Contoured plane: alternative view of data of Fig. 2b.

Figure 5

Electric equations of state of the form $\mathcal{E}\left(D\right)$ (a), $D\left(P\right)$ (b), and $P\left(\mathcal{E}\right)$ (c), plotted for fields constrained to lie along the [001], [110], or [111] directions. All units are a.u. Numbered dots on the [001] curves indicate special states as described in the text.

Figure 6

Energy functionals vs their respective natural variables along [001], [110], and [111] directions: $U\left(D\right)$ (a), $E_{\mathrm{KS}}\left(P\right)$ (b), and $\mathcal{F}\left(\mathcal{E}\right)$ (c). All field variables are in units of a.u. Dots labeled 1–5 indicate the same special states as in Fig. 5.