The magnetic structure of the $zigzag$ chain family Na${}_x$Ca${}_{1-x}$V${}_2$O${}_4$ determined by muon-spin rotation

We present muon-spin rotation measurements on polycrystalline samples of the complete family of the antiferromagnetic (AF) $zigzag$ chain compounds, Na${}_x$Ca${}_{1-x}$V${}_2$O${}_4$. In this family, we explore the magnetic properties from the metallic NaV${}_2$O${}_4$ to the insulating CaV${}_2$O${}_4$. We find a critical $x_c(\sim 0.833)$ which separates the low and high Na-concentration dependent transition temperature and its magnetic ground state. In the $x<x_c$ compounds, the magnetic ordered phase is characterized by a single homogenous phase and the formation of incommensurate spin-density-wave order. Whereas in the $x>x_c$ compounds, multiple sub-phases appear with temperature and $x$. Based on the muon data obtained in zero external magnetic field, a careful dipolar field simulation was able to reproduce the muon behavior and indicates a modulated helical incommensurate spin structure of the metallic AF phase. The incommensurate modulation period obtained by the simulation agrees with that determined by neutron diffraction.