  Going down to the limit of ultrathin films holds promise for a new generation
of devices such as ferroelectric tunnel junctions or resistive memories.
However, these length scales also make the devices sensitive to parasitic
effects related to miniaturization, and a better understanding of what happens
as size is reduced is of practical importance for the future development of
these devices.
  As the experimental advances in materials preparation and characterization
have come together with great progress in theoretical modeling of
ferroelectrics, both theorists and experimentalists can finally probe the same
length and time scales. This allows realtime feedback between theory and
experiment, with new discoveries now routinely made both in the laboratory and
on the computer. Throughout this chapter, we will highlight the recent advances
in density functional theory based modeling and the role it played in our
understanding of ultrathin ferroelectrics. We will begin with a brief
introduction to ferroelectricity and ferroelectric oxides, followed by an
overview of the major theoretical developments. We will then discuss some of
the subtleties of ferroelectricity in perovskite oxides, before turning our
attention to the main subject of the chapter -- ferroelectricity in ultrathin
films. We will discuss in detail the influence of the mechanical, electrical
and chemical boundary conditions on the stability of the polar state in a
parallel plate capacitor geometry, introducing the notion of depolarization
fields that tend to destabilize ferroelectricity. We will look at other ways in
which a thin ferroelectric can preserve its polar state, focusing on
ferroelectric domains and domain walls. Finally, we will briefly discuss
artificially layered ferroelectrics and the potential they hold as tailor-made
materials for electronic applications.
