  The sensitivity of materials properties, particularly those of perovskite
oxides, to epitaxial strain has been exploited to great advantage to create
materials with new or enhanced properties. Although it has certainly been
recognized that mismatch in the thermal expansion coefficients of the bulk and
substrate material will contribute to the misfit strain, the significance of
this contribution for ferroelectric perovskite thin-films has not been
systematically explored. We use first-principles density functional theory and
the example of ferroelectric PbTiO$_3$ thin-films on various substrates to show
that ignoring the thermal expansion of the substrate (that is, assuming that
the in-plane lattice parameter of the film remains roughly constant as a
function of temperature) results in ferroelectric transition temperatures and
structural trends that are completely qualitatively different from calculations
in which thermal expansion mismatch is properly taken into account. Our work
suggests that the concept of a misfit strain defined as a single number is
particularly ill-defined for PbTiO$_3$ and invites further study of the
interplay between thermal expansion mismatch and structural and functional
properties in other thin-film materials.
