Density functional investigation of the spin frustration and the field-driven long-range ordering in the honeycomb lattice system Bi$_{3}$Mn$_{4}$O$_{12}$(NO$_{3})$

Bi$_{3}$Mn$_{4}$O$_{12}$(NO$_{3})$, consisting of the honeycomb lattices of Mn$^{4+}$ (d$^{3})$ ions, has dominant antiferromagnetic interactions ($\theta _{CW}=-$257 K) but its spins do not order down to 0.4 K. However, applied magnetic fields induce a long-range magnetic order, which is believed to arise from the spin canting due to the Dzyaloshinskii-Moriya interaction. To explain these observations, we examined the spin exchanges between the Mn$^{4+}$ ions (J$_{1}$, J$_{2}$, J$_{c})$ by DFT+U calculations and the preferred orientation of their spins by DFT+U+SOC calculations. The spin frustration is reproduced by U close to zero with J$_{2}$/J$_{1}$ $\approx $ 1/2. The cause for the field-induced long-range magnetic ordering was explored on the basis of DFT+U+SOC calculations.