  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.
