Ab initio investigation of the exchange interactions in Bi${}_2$Fe${}_4$O${}_9$: The Cairo pentagonal lattice compound

We present the ab initio calculation of the electronic structure and magnetic properties of Bi${}_2$Fe${}_4$O${}_9$. This compound crystallizes in the orthorhombic crystal structure with the Fe${}^{3+}$ ions forming the Cairo pentagonal lattice implying strong geometric frustration. The neutron diffraction measurements reveal nearly orthogonal magnetic configuration, which at first sight is rather unexpected since it does not minimize the total energy of the pair of magnetic ions coupled by the Heisenberg exchange interaction. Here, we calculate the electronic structure and exchange integrals of Bi${}_2$Fe${}_4$O${}_9$ within the $\mathrm{LSDA}+U$ method. We obtain three different in-plane ($J_3=36$ K, $J_4=73$ K, $J_5=23$ K) and two interplane ($J_1=10$ K, $J_2=12$ K) exchange parameters. The derived set of exchange integrals shows that the realistic description of Bi${}_2$Fe${}_4$O${}_9$ needs a more complicated model than the ideal Cairo pentagonal lattice with only two exchange parameters in the plane. However, if one takes into account only the two largest exchange integrals, then according to the ratio $x\equiv J_3/J_4=0.49<\sqrt{2}$ [a critical parameter for the ideal Cairo pentagonal lattice, see Rousochatzakis, Läuchli, and Moessner, Phys. Rev. B 85, 104415 (2012)] the ground state should be the orthogonal magnetic configuration in agreement with experiment. The microscopic origin of different exchange interactions is also discussed.

Figure 1

The crystal structure of Bi${}_2$Fe${}_4$O${}_9$ along the $c$ axis (upper panel) and in the $ab$ plane (lower panel). There are two types of the Fe ions: Fe${}_o$ is placed in the oxygen octahedra (blue), while Fe${}_t$ is in the ligand tetrahedra (green). O and Bi are shown as red and yellow balls, respectively. $J_1$ and $J_2$ are the interplane exchange interactions. We use vesta software[8] for visualization.

Figure 2

The pentagonal magnetic lattice of Bi${}_2$Fe${}_4$O${}_9$ ($ab$ projection) together with exchange interactions notations. The spins on each site are the same ($S=5/2$), but the sites have different surrounding: the octahedral Fe${}_o$ ions are shown in blue, the tetrahedral Fe${}_t$ ions in black. The bonds between tetrahedral Fe${}_t$ are violet ($J_4$), while between octahedral Fe${}_o$ and tetrahedral Fe${}_t$ are green ($J_5$) and blue ($J_3$, dotted line). The spin orientation represents the experimentally detected magnetic structure from Ref. 7. In the case of ideal Cairo pentagonal lattice, Fe${}_t$ ions correspond to the threefold-coordinated sites, while Fe${}_o$ correspond to the fourfold-coordinated sites. Hence, in the notation of Ref. 2, $J_4\to J_{33}$ and $J_3=J_5\to J_{43}$.

Figure 3

The total and partial density of states plot obtained in the $\mathrm{LSDA}+U$ calculation for the magnetic configuration, where all Fe${}_t$ and Fe${}_o$ are antiferromagnetically ordered. The positive (negative) values correspond to the spin up (down). The Fermi energy is in zero.

Figure 4

The sketch illustrating the strongest exchange coupling between $a_{1g}$ orbitals on two tetrahedral Fe${}_t$ ions via the $p_{\sigma }$ (light blue color) orbital, directed to the centers of tetrahedra. Oxygen ions are blue balls.