  We calculate from first principles the nonlinear piezoelectric response of
ferroelectric PbTiO3 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 PbTiO3 can show a strikingly non-linear
piezoelectric response under modified elastic boundary conditions.
