We use density functional theory to investigate the possibility of polar and multiferroic states in free-standing, perovskite-based nanodots at the atomic limit of miniaturization: single unit cells with terminations which allow centro-symmetry. We consider both A-O and B-O-2 terminations for three families of nanodots: (i) A = Ba with B = Ti, Zr, and Hf; (ii) A = Ca and Sr with B = Ti; and (iii) A = Na and K with B = Nb. We find all A-O terminated dots to be non-polar and to exhibit cubic symmetry (except for K8NbO6), regardless of the presence of ferroelectricity in the bulk. In contrast, all the B-O-2 terminated nanodots considered relax to a non-cubic ground state. Rather surprisingly, all of these structures exhibit polar ground states (except NaNb8O12). We propose a new structural parameter, the cluster tolerance factor (CTF), to determine whether a particular chemistry will result in a polar ground state nanodot, analogous to the Goldschmidt factor for bulk ferroelectrics. In addition, we find that all A-O terminated (except Ca8TiO6) and all polar B-O-2 terminated nanodots are magnetic, where none show magnetism in the bulk. As with bulk systems, multiferroicity in the B-O-2 terminated dots originates from separation between spin density in peripheral B atoms and polarity primarily caused by the off-center central A atom. Our findings stress that surface termination plays a crucial role in determining whether ferroelectricity is completely suppressed in perovskite-based materials at their limit of miniaturization.
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