Origin of magnetic frustration in ${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$

${\mathrm{Bi}}_3{\mathrm{Mn}}_4{\mathrm{O}}_{12}\left({\mathrm{NO}}_3\right)$ (BMNO) is a honeycomb bilayers antiferromagnet, not showing any ordering down to very low temperatures despite having a relatively large Curie-Weiss temperature. Using ab initio density functional theory, we extract an effective spin Hamiltonian for this compound. The proposed spin Hamiltonian consists of antiferrimagnetic Heisenberg terms with coupling constants ranging up to third intralayer and fourth interlayer neighbors. Performing Monte Carlo simulation, we obtain the temperature dependence of magnetic susceptibility and so the Curie-Weiss temperature and find the coupling constants which best match with the experimental value. We discover that depending on the strength of the interlayer exchange couplings, two collinear spin configurations compete with each other in this system. Both states have in plane Néel character, however, at small interlayer coupling spin directions in the two layers are antiparallel (${\mathrm{N}}_1$ state) and discontinuously transform to parallel (${\mathrm{N}}_2$ state) by enlarging the interlayer couplings at a first order transition point. Classical Monte Carlo simulation and density matrix renormalization group calculations confirm that exchange couplings obtained for BMNO are in such a way that put this material at the phase boundary of a first order phase transition, where the trading between these two collinear spin states prevents it from setting in a magnetically ordered state.

Figure 1

The $2\times 2$ supercell of BMNO. The thicker green lines show the primitive cell. Two (yellow transparent) planes show honeycomb lattices which are made of Mn atoms.

Figure 2

$J_1, J_2$, and $J_3$ indicate the Heisenberg exchange coupling constants between first, second, and third nearest neighbor in-plane ${\mathrm{Mn}}^{+4}$ ions, respectively. $J_{1c}, J_{2c}, J_{3c}$, and $J_{4c}$ indicate the Heisenberg exchange coupling constants between interplane first, second, and third nearest neighbor ${\mathrm{Mn}}^{+4}$ ions, respectively.

Figure 3

(top): The ab initio energy landscape (per Mn atom) of 54 magnetic configurations obtained by DFT+$U$ with $U=1.5$ eV. The energies of magnetic configurations are respect to fully ferromagnetic state whose energy is set to 0. (bottom): Three magnetic configurations ${\mathrm{N}}_1, {\mathrm{N}}_2$, and ${\mathrm{N}}_3$.

Figure 4

Temperature dependence of the inverse normalized DC susceptibility ($1/\chi$) of BMNO obtained in experiment and MC simulations for a different set of ab initio exchange couplings derived by using $U=1.5\mathrm{eV}$ (filled circles), $U=2.0\mathrm{eV}$ (triangles), $U=3.0\mathrm{eV}$ (squares), and $U=4.0\mathrm{eV}$ (pentagons). The experimental data (empty circles) is extracted from Fig. 7 of Ref. [1]. For the normalization, all the data are divided by their values at $T=400$ K. The crossing of the line fitted at high temperatures to $1/\chi$ with the horizontal axis gives the Curie-Weiss temperature.

Figure 5

Interlayer spin-spin correlation up to neighbor obtained by (top) MC simulation, (bottom) DMRG (normalized by $S^2$). $J_1,J_2,J_3,J_{1c},J_{3c}$, and $J_{4c}$ are kept fixed at those found by $U=1.5\mathrm{eV}$. The spin-spin correlations at $J_{2c}=1.100\mathrm{meV}$ are obtained by using $U=1.5\mathrm{eV}$ (blue circles) and compared with $J_{2c}=1.045$ (green diamonds) and $J_{2c}=1.155\mathrm{meV}$ (red squares).

Figure 6

Interlayer spin-spin correlation (normalized by $S^2$) up to fourth neighbor obtained by MC simulation. $J_1,J_2,J_3,J_{2c},J_{3c}$, and $J_{4c}$ are kept fixed at the values obtained by $U=1.5\mathrm{eV}$. The spin-spin correlations at $J_{2c}=3.0\mathrm{meV}$ are obtained by using $U=1.5\mathrm{eV}$ (blue circles) and compared with $J_{1c}=3.15$ (red squares) and $J_{1c}=2.85\mathrm{meV}$ (green diamonds).

Figure 7

Interlayer spin-spin correlation (normalized by $S^2$) up to fourth neighbor obtained by MC simulation. $J_1,J_2,J_3$ are kept fixed at the values obtained using $U=1.5\mathrm{eV}$. The spin-spin correlations for the interlayer coupling calculated at $U=1.5$ (blue circles) are compared with the ones increased (red squares) and decreased (green diamonds) by 5 percent.

Figure 8

DMRG results for the variation of spin-spin correlations for first, second, third in-plane neighbors and first interplane neighbors versus $J_{2c}/J_1$. Other couplings are fixed by those obtained by $U=1.5\mathrm{eV}$.

Figure 9

DMRG results for the variation of the second interlayer spin-spin correlations versus $J_{2c}/J_1$, for lattices of size $4\times L^2$ with $L=2,3,4$. Other couplings are fixed by those obtained by $U=1.5\mathrm{eV}$.