  We use density functional theory to investigate the possibility of polar and
multiferroic states in free-standing, perovskite-based nanodots at their limit
of miniaturization: single unit cell with termination to allow centrosymmetry.
We consider both A-O and B-O2 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, regardless of the presence of ferroelectricity in the bulk. These
dots all carry a net magnetic moment, except Ca8TiO6. On the other hand, all
B-O2 terminated nanodots considered in this study relax to a non-cubic ground
state. Rather surprisingly, a subset of these structures (BaTi8O12, BaHf8O12,
BaZr8O12, SrTi8O12, CaTi8O12 and KNb8O12) exhibit polar ground states. We
propose a new structural parameter, the cluster tolerance factor (CTF), to
determine whether a particular combination of A and B will yield a polar ground
state in the nanodot geometry, analogous to the Goldschmidt factor in bulk
ferroelectrics. We also find that in all polar B-O2 terminated nanodots are
also magnetic. The multiferroic state 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.
