Noncollinear magnetic structure and magnetoelectric coupling in buckled honeycomb Co$_4$Nb$_2$O$_9$:A single crystal neutron diffraction study

Through an analysis of single-crystal neutron diffraction data, we present the magnetic structure and magnetoelectric properties of ${\mathrm{Co}}_{4}{\mathrm{Nb}}_{2}{\mathrm{O}}_{9}$ under various magnetic fields. In zero field, neutron diffraction experiments below ${T}_{N}=27\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ reveal that the ${\mathrm{Co}}^{2+}$ moments order in the (ab) plane without any spin canting along the $c$ axis, manifested by the magnetic symmetry $C2/{c}^{\ensuremath{'}}$. Along each Co chain parallel to the $c$ axis, the moments of nearest-neighbor Co atoms order ferromagnetically with a small cant away from the next-nearest-neighbor Co moments. Under the applied magnetic field $H\ensuremath{\parallel}a$, three magnetic domains were aligned with their major magnetic moments perpendicular to the magnetic field with no other observable magnetic transitions. The influences of magnetic fields on the magnetic structures associated with the observed magnetoelectric coupling are discussed.