3D oxygen vacancy order and defect-property relations in multiferroic (LuFeO$_3$)$_9$/(LuFe$_2$O$_4$)$_1$ superlattices

Oxide heterostructures exhibit a vast variety of unique physical properties. Examples are unconventional superconductivity in layered nickelates and topological polar order in (PbTiO$_3$)$_n$/(SrTiO$_3$)$_n$ superlattices. Although it is clear that variations in oxygen content are crucial for the electronic correlation phenomena in oxides, it remains a major challenge to quantify their impact. Here, we measure the chemical composition in multiferroic (LuFeO$_3$)$_9$/(LuFe$_2$O$_4$)$_1$ superlattices, revealing a one-to-one correlation between the distribution of oxygen vacancies and the electric and magnetic properties. Using atom probe tomography, we observe oxygen vacancies arranging in a layered three-dimensional structure with a local density on the order of 10$^{14}$ cm$^{-2}$, congruent with the formula-unit-thick ferrimagnetic LuFe$_2$O$_4$ layers. The vacancy order is promoted by the locally reduced formation energy and plays a key role in stabilizing the ferroelectric domains and ferrimagnetism in the LuFeO$_3$ and LuFe$_2$O$_4$ layers, respectively. The results demonstrate the importance of oxygen vacancies for the room-temperature multiferroicity in this system and establish an approach for quantifying the oxygen defects with atomic-scale precision in 3D, giving new opportunities for deterministic defect-enabled property control in oxide heterostructures.