  An effective Hamiltonian scheme is developed to investigate structural and
magnetic properties of BiFeO3 nanodots under short-circuit-like electrical
boundary conditions. Various striking effects are discovered. Examples include
(a) scaling laws involving the inverse of the dots' size for the magnetic and
electric transition temperatures; (b) the washing out of some structural phases
present in the bulk via size effects; (c) the possibility of tailoring the
difference between the Neel and Curie temperatures, by playing with the size
and electrical boundary conditions; and (d) an universal critical thickness of
the order of 1.6 nm below which the dots do not possess any long-range ordering
for the electrical and magnetic dipoles, as well as, for the oxygen octahedral
tiltings.
