Construction of Long-Range Magnetic Sequences on Different Surfaces of BaTiO₃

Using the first-principles spin-density-functional theory calculations, we studied the origin of ferromagnetism from non-magnetic ferroelectric barium titanate (BaTiO₃ ) and found out vacancies in different surface can successfully contribute to the origin of ferromagnetism. Accurately, our findings demonstrate that both O and Ti vacancies induce ferromagnetism on the (001) and (010) surfaces of BaTiO₃ , and the optimal Ti-O bond length can control the vacancy-induced spin density that is delocalized or concentrated in the real space outside the vacancy, and it helps to enhance our understanding on the long-range magnetic order induced by the vacancy. In addition, intrinsic magnetism is shown on the defect-free (110) surface, and the structure is found to be a near-ideal two-dimensional Ising ferromagnet with large magnetocrystalline anisotropy, and it supplies the platform for studying basic spin behavior of BaTiO₃ and more according materials.