Engineering multiferroism through localized hole doping in a flat-band ferroelectric HfO₂

Flat phonon bands in ferroelectric HfO₂give rise to a unique structural pattern of alternating polar and spacer oxygen layers at the sub-nanometer scale. Here, we demonstrate that exploiting these localized polar layers through substitutional nitrogen (N) doping can induce electrically tunable magnetism, enabling multiferroism in this simple binary oxide. First-principles density functional theory calculations reveal that N substitution for oxygen preferentially occurs at the electrically switchable oxygen sites within the polar layer, rather than the oxygen sites in the spacer layer. This site-selective N substitution is driven by the preference of N for sp²bonding and results in a localized hole at the N site. The hole carries a magnetic moment of approximately0.7μB, leading to A-type antiferromagnetic ordering between N dopants. Remarkably, we found that a162∘rotation of the single-ion anisotropy easy-axis occurs with the reversal of ferroelectric polarization. This is because the spin's easy-axis, induced by hole, is tied to the local lattice distortion. This magnetoelectric coupling is achieved without any transition metal ions, relying solely on hole doping. Furthermore, the substitutional N-doped HfO₂retains robust ferroelectricity ((Pr≈45.5μCcm-2)and an insulating state even at substitutional doping levels up to 12.5%. Our work unveils a design strategy for electrically-controlled magnetism in HfO₂, harnessing flat-band ferroelectricity to localize dopant-induced holes in switchable polar layers, thereby coupling ferroelectric and magnetic orders in a silicon-compatible oxide.