Unconventional magnetic order in the frustrated diamond-lattice antiferromagnet ${\mathrm{CoAl}}_2{\mathrm{O}}_4$ studied by neutron diffraction and classical Monte Carlo simulation

${\mathrm{CoAl}}_2{\mathrm{O}}_4$ spinel with magnetic ${\mathrm{Co}}^{2+}$ ions on the diamond lattice is known to be magnetically frustrated. We compare neutron single-crystal diffraction patterns measured in zero and applied magnetic fields with the ones obtained from classical Monte Carlo models. In simulations we test the influence of various parameters on diffraction patterns: the ratio of nearest-, $J_1$, and next-nearest-, $J_2$, neighbor interactions, magnetic field applied along the principal crystallographic directions, and random disorder on the $A\left({\mathrm{Co}}^{2+}\right)$ and $B\left({\mathrm{Al}}^{3+}\right)$ sites. We conclude that the models considered so far explain the broadening of magnetic Bragg peaks in zero magnetic field and their anisotropic response to applied magnetic field only partly. As bulk properties of our single crystal are isotropic, we suggest that its microstructure, specifically $\langle 111\rangle$-twin boundaries, could be a reason for the nonconventional magnetic order in ${\mathrm{CoAl}}_2{\mathrm{O}}_4$.

Figure 1

Elastic neutron-scattering $H\text{−}HL$ maps calculated for the ideal model with $J_2/J_1=0.10$ (left) and 0.17 (right). Middle: the $00L$ cuts of the $H\text{−}HL$ maps with the $J_2/J_1$ ratio 0.10 (red), 0.15 (green), and 0.17 (blue). Note the logarithmic scale for the intensity.

Figure 2

$00L$ cuts as a function of $A$ (left) and $B$ (right) defect concentration for the model with $J_2/J_1=0.10$: 0% (black), 4% (green), 8% (blue), and 12% (red). Note the logarithmic scale for the intensity.

Figure 3

$00L$ cuts for zero-field (black), 1.5 T (blue), 4 T (green), and 5 T (red) calculated for the ideal $J_2/J_1=0.10$ model with magnetic fields applied along $\left[110\right]$ (left) and $\left[100\right]$ (middle) and for the model with 8% defects for 4 T with field along $\left[100\right]$ (red) and $\left[110\right]$ (green) (right).

Figure 4

Elastic neutron-scattering $0KL$ (left) and $HK\text{−}K$ (right) maps calculated for the ideal model with $J_2/J_1=0.10$ with a magnetic field applied along $\left[100\right]$.

Figure 5

Left: temperature dependence of magnetic susceptibility $M/H$ measured with a dc field $H=0.005$ T. Solid symbols represent field cooled and open symbols zero-field cooled data. Green: magnetic field applied along the $\left[111\right]$ direction; blue: $\left[110\right]$; red $\left[100\right]$. Inset: inverse susceptibility for three principal directions. Middle: temperature variation of ac magnetic susceptibility ${\chi }^{\prime }$ for different frequencies with a dc field $H=0.1$ T along $\left[111\right]$. Right: magnetization at 2 K (open symbols) and 30 K (solid symbols). Insets: (a) change of $M/H$ with $H\parallel \left[111\right]$; (b) variation of $M/H$ with temperature at different strength of magnetic field along $\left[111\right]$.

Figure 6

Left: temperature variation of heat capacity $C_P$ measured in zero and 6 T magnetic field applied along three principal directions: red $\left[100\right]$, blue $\left[110\right]$, and green $\left[111\right]$. Heat capacity of ${\mathrm{ZnAl}}_2{\mathrm{O}}_4$ (nonmagnetic analog) taken from Ref. [7] is also plotted. Inset: difference of $C_P$ measured in 6 T and without field. Right: magnetic entropy $▵S_{\mathrm{mag}}$ calculated from these data. The dashed horizontal line indicates the high temperature limit of $▵S_{\mathrm{mag}}=11.56$ J/mol K for $S=3/2$. Inset: magnetic contribution to the heat capacity $C_{\mathrm{mag}}$ fitted to a power law with $\alpha =2.1$.

Figure 7

$\langle 200\rangle$ reflections of ${\mathrm{CoAl}}_2{\mathrm{O}}_4$ crystal measured at 1.8 K without (0 T, black) and with (5 T, red) magnetic field applied along $\left[110\right]$ (left), $\left[111\right]$ (middle), and $\left[010\right]$ (right).

Figure 8

$\left[1\text{−}10\right]$ projection of the ${AB}_2{\mathrm{O}}_4$ spinel structure with (a) ideal arrangement of $\mathrm{Co}{}^{+2} A$ tetrahedra (violete) and $\mathrm{Al}{}^{+3} B$ octahedra (green) and $\langle 111\rangle$-twin boundaries preserving the cation composition (b) or resulting in the $B$-rich (c) or $A$-rich (d) regions.