Multiferroics in nanoscale dimensions are promising for novel functional device paradigms, such as magnetoelectric memories, due to an intriguing cross-coupling between coexisting ferroelectric and (anti)ferromagnetic order parameters. However, the ferroic order is inevitably destroyed below the critical dimension of several nanometers. Here, we demonstrate a new path towards atomic-size multiferroics while resolving the controversial origin of dilute ferromagnetism that unexpectedly emerges in nanoparticles of nonmagnetic ferroelectric ${\mathrm{PbTiO}}_3$. Systematic exploration using predictive quantum-mechanical calculations demonstrates that oxygen vacancies formed at surfaces induce ferromagnetism due to local nonstoichiometry and orbital symmetry breaking. The localized character of the emerged magnetization allows an individual oxygen vacancy to act as an atomic-scale multiferroic element with a nonlinear magnetoelectric effect that involves rich ferromagnetic-antiferromagnetic-nonmagnetic phase transitions in response to switching of the spontaneous polarization.

• Received 30 January 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.107202