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.Comment: 32 pages + 230 references, 8 figures; Chapter 12 in Oxide Ultrathin Films: Science and Technology (2012) - Edited by Gianfranco Pacchioni and Sergei Valeri - WILEY-VCH ISBN 978-3-527-33016-
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