Competing Interactions in the $S$=3/2 Kagome Staircase Co$_{3}$V$_{2}$O$_{8}$: Evolution of the Commensurate and Incommensurate Phases in a Magnetic Field

Single crystal neutron diffraction studies have been performed on the $S$~=~3/2 kagome staircase compound Co$_{3}$V$_{2}$O$_{8}$ with a magnetic field applied along the magnetization easy-axis ($\vec{H}$~$||$~$\vec{a}$). Previous zero-field measurements reported incommensurate, transversely polarized spin density wave (SDW) phases [with a temperature dependent propagation vector of $\vec{k}$~=~(0~$\delta$~0)] interspersed with multiple commensurate lock-in transitions at temperatures above the ferromagnetic ground state. For small applied fields along $\vec{a}$, $\mu_{0}H$~$\approx$~0.05~T, the commensurate lock-in phases are destabilized in favor of the incommensurate SDW, while slightly larger applied fields restore the commensurate lock-in phase with $\delta$~=~1/2 and yield a new commensurate phase with $\delta$~=~2/5. For measurements in an applied field, higher-order scattering is observed that corresponds to the second-harmonic.