Depolarizing Field and ``Real'' Hysteresis Loops in Nanometer Scale Ferroelectric Films

We give detailed analysis of the effect of depolarizing field in nanometer-size ferroelectric capacitors studied by Kim \textit{et al.} [Phys. Rev. Lett. \textbf{95}, 237602 (2005)]. We calculate a critical thickness of the homogeneous state and its stability with respect to domain formation for strained thin films of BaTiO$_3$ on SrRuO$_3$/SrTiO$_3$ substrate within the Landau theory. While the former (2.5nm) is the same as given by ab- initio calculations, the actual critical thickness is set by the domains at 1.6nm [1]. There is a large Merz's activation field for polarization relaxation. Remarkably, the results show a \emph {negative} slope of the ``actual'' hysteresis loops, a hallmark of the domain structures in ideal thin films with imperfect screening[2]. \newline [1] A.M. Bratkovsky and A.P. Levanyuk, Appl. Phys. Lett. (in print), cond-mat/0608283. \newline [2] A.M. Bratkovsky and A.P. Levanyuk, Phys. Rev. B {\bf 63}, 132103 (2001).