  Improper ferroelectrics are described by two order parameters: a primary one,
driving a transition to long-range distortive, magnetic or otherwise
non-electric order, and the electric polarization, which is induced by the
primary order parameter as a secondary, complementary effect. Using
low-temperature scanning probe microscopy, we show that improper ferroelectric
domains in YMnO$_3$ can be locally switched by electric field poling. However,
subsequent temperature changes restore the as-grown domain structure as
determined by the primary lattice distortion. The backswitching is explained by
uncompensated bound charges occuring at the newly written domain walls due to
the lack of mobile screening charges at low temperature. Thus, the polarization
of improper ferroelectrics is in many ways subject to the same electrostatics
as in their proper counterparts, yet complemented by additional functionalities
arising from the primary order parameter. Tailoring the complex interplay
between primary order parameter, polarization, and electrostatics is therefore
likely to result in novel functionalities specific to improper ferroelectrics.
