First-principles study of high-field piezoelectricity in tetragonal ${\text{PbTiO}}_3$

We calculate from first principles the nonlinear piezoelectric response of ferroelectric ${\text{PbTiO}}_3$ for the case of a polarization-enhancing electric field applied along the tetragonal axis. We focus mainly on the case of fixed in-plane lattice constants, corresponding to epitaxially constrained thin films. We find that the dependence of the $c/a$ ratio on electric field is almost linear in the range up to 500 MV/m with little saturation. This result contrasts with expectations from Landau-Devonshire approaches based on experimental results obtained at lower fields but is in qualitative agreement with a recent experiment in which higher fields were attained using pulsed-field methods. We also study cases in which the in-plane epitaxy constraint is removed, or an artificial negative pressure is applied, or both. These calculations demonstrate that ${\text{PbTiO}}_3$ can show a strikingly nonlinear piezoelectric response under modified elastic boundary conditions.

Figure 1

(Color online) Out-of-plane strain response to applied electric field, relative to zero-field ferroelectric state, under in-plane epitaxial constraint. Experiment: data of Grigoriev et al. (Ref. 4) for several PZT 20/80 capacitors of varying thickness. Theoretical points with fitted lines: computed results for pure ${\text{PbTiO}}_3$ using two computer codes. Inset shows the ABINIT results over a more extended range of fields.

Figure 2

(Color online) Variation in properties of tetragonal ${\text{PbTiO}}_3$ with electric field under in-plane epitaxial constraint. (a) Piezoelectric coefficient $d_{33}$ (calculated with both codes). (b) Out-of-plane displacements of ions from zero-field positions (calculated with ABINIT).

Figure 3

(Color online) Variation in properties of tetragonal ${\text{PbTiO}}_3$ with electric field under in-plane epitaxial constraint, calculated with ABINIT. (a) Born effective charges. (b) Polarization.

Figure 4

(Color online) Field-induced strain, relative to zero-field zero-pressure state, for tetragonal ${\text{PbTiO}}_3$ under in-plane epitaxial constraint, calculated with LAUTREC.

Figure 5

(Color online) Field-induced strain, relative to out-of-plane lattice constant of zero-field zero-pressure state, for tetragonal ${\text{PbTiO}}_3$ under free-stress (0 GPa) and negative-pressure ($-1$ and $-2\text{GPa}$) boundary conditions, calculated with LAUTREC. Top curves: out-of-plane strain. Bottom curves: in-plane strain.

Figure 6

(Color online) Variation in properties of tetragonal ${\text{PbTiO}}_3$ with electric field under free-stress boundary conditions, calculated with LAUTREC. (a) Piezoelectric coefficient $d_{33}$. (b) Out-of-plane displacements of ions from zero-field positions.