  We study the electrocaloric (EC) effect in bulk BaTiO$_3$ (BTO) using
molecular dynamics simulations of a first principles-based effective
Hamiltonian, combined with direct measurements of the adiabatic EC temperature
change in BTO single crystals. We examine in particular the dependence of the
EC effect on the direction of the applied electric field at all three
ferroelectric transitions, and we show that the EC response is strongly
anisotropic. Most strikingly, an inverse caloric effect, i.e., a temperature
increase under field removal, can be observed at both
ferroelectric-ferroelectric transitions for certain orientations of the applied
field. Using the generalized Clausius-Clapeyron equation, we show that the
inverse effect occurs exactly for those cases where the field orientation
favors the higher temperature/higher entropy phase. Our simulations show that
temperature changes of around 1 K can in principle be obtained at the
tetragonal-orthorhombic transition close to room temperature, even for small
applied fields, provided that the applied field is strong enough to drive the
system across the first order transition line. Our direct EC measurements for
BTO single crystals at the cubic-tetragonal and at the tetragonal-orthorhombic
transitions are in good qualitative agreement with our theoretical predictions,
and in particular confirm the occurrence of an inverse EC effect at the
tetragonal-orthorhombic transition for electric fields applied along the [001]
pseudo-cubic direction.
