First-principles study of the magnetic ground state and magnetization process of the kagome francisites ${\mathrm{Cu}}_3\mathrm{Bi}\left({\mathrm{SeO}}_3{)}_2{\mathrm{O}}_{2}X\right(X=\mathrm{Cl},\mathrm{Br})$

We explore the magnetic behavior of the kagome francisites ${\mathrm{Cu}}_3\mathrm{Bi}\left({\mathrm{SeO}}_3{)}_2{\mathrm{O}}_{2}X\right(X=\mathrm{Cl},\mathrm{Br})$ by using first-principles electronic structure calculations. To this end, we propose an approach based on the effective Hubbard model in the Wannier functions basis constructed on the level of local-density approximation. The ground-state spin configuration is determined by a mean-field Hartree-Fock solution of the Hubbard model both in zero magnetic field and in applied magnetic fields. Additionally, parameters of an effective spin Hamiltonian are obtained by taking into account hybridization effects and spin-orbit coupling. We show that only the former approach based on the Hartree-Fock approximation allows for a complete description of the anisotropic magnetization process. While our calculations confirm that the canted zero-field ground state arises from a competition between ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor couplings in the kagome planes, weaker anisotropic terms are crucial for fixing spin directions and for the strong anisotropy of the magnetization. We show that the Hartree-Fock solution of an electronic Hamiltonian is a viable alternative to the analysis of effective spin Hamiltonians when magnetic ground states and their evolution in external field are concerned.

Figure 1

(a) Crystal structure of $\text{CBSOO}X$: $ab$ (left) and $bc$ (right) projections; (b) schematic view of intraplane (left panel) and interplane (right panel) isotropic exchange parameters. Sites 1, 2, 3, 4 and 5, 6 stand for the Cu1 and Cu2 atoms, respectively. Equivalent bonds are shown by the same color. The sign of the $a$ component of the anisotropic exchange coupling ${\mathit{D}}_{15}$ is shown by $+$ and $-$.

Figure 2

(a) Bands located near the Fermi level as obtained from LDA+SO (up) and LDA (down) calculations for CBSOOBr. The red lines show the Wannier parametrization. The notation of $k$ points is $X(\frac{1}{2},0,0), \mathrm{\Gamma }(0,0,0), Y(0,\frac{1}{2},0), S(\frac{1}{2},\frac{1}{2},0)$, and $R(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ in reciprocal lattice units. (b) Wannier functions centered on copper atoms, as obtained from LDA calculations for CBSOOBr.

Figure 3

(a) Magnetic lattice as obtained from the mean-field Hartree-Fock approximation at zero applied magnetic field. Sites 1, 2, 3, 4 and 5, 6 stand for the Cu1 and Cu2 atoms, respectively. (b) Total magnetization per site in CBSOOBr for three different directions of magnetic field $\mathit{B}$ and their comparison with experimental data [7].