Figure 1

(a) The crystal structure of $R$${}_2$V${}_2$O${}_7$. The global coordinate system $\mathrm{XYZ}$ is indicated. (b) The spin exchange paths between ${\mathrm{V}}^{4+}$ ions. (c) The two local coordinate systems used for the ideal VO${}_6$ octahedron. The $\mathit{xyz}$ coordinate system is defined for the ideal VO${}_6$ octahedron with the $x$, $y$, and $z$ axes taken along the V-O bonds. In the $x^{\text{'}}{y}^{\text{'}}{z}^{\text{'}}$ coordinate system, the $z^{\text{'}}$ axis is taken along one threefold rotational axis of the ideal VO${}_6$ octahedron. In $R$${}_2$V${}_2$O${}_7$ each VO${}_6$ octahedron is axially compressed slightly with only one threefold rotational axis. (d) The DM vectors ${\mathbf{D}}_{\mathit{ij}}$ with $i<j$ of the V tetrahedron, where $i$ and $j$ denote the V site labels. (e) The two spin configurations used to extract the DM parameter. (f) The two spin configurations used to extract the single-ion anisotropy parameter.Reuse & Permissions

Figure 2

(a) The band structure of the FM state from the $\text{DFT}+U$ calculation. (b) The band structure of the AFM state (with two up and two down spins in each ${\mathrm{V}}_4$ tetrahedron) from the $\text{DFT}+U$ calculation. Solid lines and dashed lines represent up-spin and down-spin bands, respectively.Reuse & Permissions

Figure 3

(a) The total and partial density of states of the FM state from the $\text{DFT}+U$ calculation. The inset shows the density distribution associated with the occupied V 3$d^1$ states. (b) A schematic illustration of electron configuration of the ${\mathrm{V}}^{4+}$ ion. (c) The interactions between the 3$d$ states of two adjacent ${\mathrm{V}}^{4+}$ ions when their spins have the FM and AFM arrangements. Only the hopping processes which lead to the overall energy lowering of the up-spin $a_{1g}$ state of spin site $1$ are shown. (d) The interactions between the 3$d$ states of two adjacent ${\mathrm{V}}^{4+}$ ions leading to the CBM positions when their spins have the FM and AFM arrangements.Reuse & Permissions

Figure 4

The spin configuration of the magnetic ground state of $R$${}_2$V${}_2$O${}_7$ from the Monte Carlo simulated annealing method.Reuse & Permissions