Frustration and Dzyaloshinsky-Moriya anisotropy in the kagome francisites ${\mathrm{Cu}}_3\mathrm{Bi}({\mathrm{SeO}}_3{)}_2{\mathrm{O}}_{2}X (X = \mathrm{Br}, \mathrm{Cl})$

We investigate the antiferromagnetic canting instability of the spin-1/2 kagome ferromagnet, as realized in the layered cuprates ${\mathrm{Cu}}_3\mathrm{Bi}({\mathrm{SeO}}_3{)}_2{\mathrm{O}}_{2}X (X = \mathrm{Br}, \mathrm{Cl})$. While the local canting can be explained in terms of competing exchange interactions, the direction of the ferrimagnetic order parameter fluctuates strongly even at short distances on account of frustration which gives rise to an infinite ground state degeneracy at the classical level. In analogy with the kagome antiferromagnet, the accidental degeneracy is fully lifted only by nonlinear $1/S$ corrections, rendering the selected uniform canted phase very fragile even for spins-1/2, as shown explicitly by coupled-cluster calculations. To account for the observed ordering, we show that the minimal description of these systems must include the microscopic Dzyaloshinsky-Moriya interactions, which we obtain from density-functional band-structure calculations. The model explains all qualitative properties of the kagome francisites, including the detailed nature of the ground state and the anisotropic response under a magnetic field. The predicted magnon excitation spectrum and quantitative features of the magnetization process call for further experimental investigations of these compounds.

Figure 1

Left panel: Kagomelike layers in the crystal structure of francisite. Green and brown colors denote the crystallographic positions Cu1 and Cu2, yellow triangles depict trigonal bipyramids ${\mathrm{SeO}}_3$, the Bi and Cl atoms are not shown. Middle panel: Projection of the spin lattice showing only the three leading in-plane couplings ($J_1,J_1^{\prime }$, and $J_2$), with shaded triangles indicating the kagome units. The arrows depict the relative sign of the dominant, $\mathbf{a}$ component of the DM anisotropy on the $J_1^{\prime }$ bonds. Right panel: Stacking of the kagome layers and leading interplane couplings $J_{\perp 1}$ and $J_{\perp 2}$. The crystal structures are visualized using the vesta software [27].

Figure 2

Left: $\mathrm{GGA}+U$ density of states (DOS) for ${\mathrm{Cu}}_3\mathrm{Bi}\left({\mathrm{SeO}}_3{)}_2{\mathrm{O}}_2\mathrm{Bi}\right(U_d=9.5$ eV, $J_d=1$ eV). The Fermi level is at zero energy. Right: GGA bands near the Fermi level (thin light lines) are reproduced with a tight-binding fit (thick dark lines) to obtain hopping integrals $t_i$ that yield the AFM contribution to the isotropic exchange as $J_i^{\text{AFM}}=4t_i^2/U_{\text{eff}}$. The notation of the $k$ points is as follows: $\Gamma (0,0,0),X(\frac{1}{2},0,0),S(\frac{1}{2},\frac{1}{2},0),Y(0,\frac{1}{2},0),Z(0,0,\frac{1}{2}),U(\frac{1}{2},0,\frac{1}{2}),R(\frac{1}{2},\frac{1}{2},\frac{1}{2}),T(0,\frac{1}{2},\frac{1}{2})$, all given in units of the reciprocal lattice parameters. Note that the GGA band structure is metallic because strong correlation effects have not been accounted for in this calculation.

Figure 3

(a) Magnetic 2D planes of the kagome francisites, with two symmetry inequivalent Cu sites, Cu1 (green squares) and Cu2 (orange circles). The shaded cells A and B denote the unit cells of the Hamiltonian with and without anisotropy, with primitive translations $\{\mathbf{a},\mathbf{b}\}$ and $\{{\mathbf{t}}_1=\mathbf{a},{\mathbf{t}}_2=(\mathbf{a}+\mathbf{b})/2\}$, respectively. (b) Classical minimum of an isolated crossed plaquette at the isotropic limit. (c) and (d) Two representative coplanar GSs of the isotropic classical model (there is no modulation along the FM, $J_1$ bonds). (e) The dependence of the in-plane canting angle $\theta$ on $J_2/\left|J_1^{\prime }\right|$ for three different values of $d_a^{\prime }$, and for all other DM components disregarded. The shaded bar indicates the experimentally observed regime [18].

Figure 4

(a) and (b) Linear spin-wave spectra around any coplanar GS of the isotropic model for $J_1=J_1^{\prime }$ and $J_2=0.73\left|J_1^{\prime }\right|$, represented in the BZ (a) of the isotropic model, and (b) of the crystal. [The six spin-wave branches in (b) can be folded back into the three branches of (a).] The spectra in (c) include the effect of a finite DM component $d_a^{\prime }=0.15\left|J_1^{\prime }\right|$, and will be discussed in Sec. 4c. (d) First BZs of ${\mathcal{H}}_{\text{iso}}$ and $\mathcal{H}$.

Figure 5

Key quantities of interest for the isotropic spin-1/2 Heisenberg model at $J_1=J_1^{\prime }$, as a function of the frustration parameter $J_2/\left|J_1^{\prime }\right|$. (a) Difference of the GS energies $\Delta E=E_0^{\mathrm{canted}}-E_0^{\mathrm{spiral}}$ between the canted and the spiral states, as calculated within the CCM $\mathrm{LSUB}m$ approximation with $m=4$ and 6. (b) GS energy $E_0$ per site calculated by linear spin-wave theory, exact diagonalizations on a periodic 36-site cluster, and CCM (extrapolated value, see main text). The classical energy is also shown for comparison. Inset: Classical and quantum canting angle $\theta$. (c) Net GS magnetic moment per site $M$ (in units of $g{\mu }_B=1$), as calculated by exact digonalizations on the 36-site cluster, and CCM ($\mathrm{LSUB}m,m=2,4,6$ and extrapolated value, see main text). For comparison we also show the classical value. Inset: Sublattice magnetizations $M_{\text{Cu1}}$ and $M_{\text{Cu2}}$ from CCM (extrapolated values). In the classical limit $M_{\text{Cu1}}^{\mathrm{cl}}=M_{\text{Cu2}}^{\mathrm{cl}}=\frac{1}{2}$.

Figure 6

Common-origin representation of the ground state configuration of the kagome francisites, showing the six spins per unit cell (see Fig. 3). (a) The dominant DM component $d_a^{\prime }$ forces the spins to lie in the $\mathbf{b}-\mathbf{c}$ plane, but the global direction in this plane (i.e., the angle $\varphi$) is only fixed by the remaining components $d_b^{\prime }$ and $d_c^{\prime }$, as explained in the text. (b) For $|d_b^{\prime }|\gg |d_c^{\prime }|,\varphi \simeq \pi /2$ as observed experimentally. (c) The out-of-plane canting driven by $d_b^{\prime }$ and $d_c^{\prime }$. For $|d_b^{\prime }|\gg |d_c^{\prime }|,{\gamma }_3=-{\gamma }_1$, see text.

Figure 7

Anisotropic response of the kagome francisites under a magnetic field $\mathbf{H}$. Local moments (a)–(c) and total magnetization per site (d) for three different directions of $\mathbf{H}$ (in units of ${\mu }_B$, for $g=2$). The results are obtained by numerical minimizations of the total energy of a finite 2D layer with $8\times 8\times 6$ sites and periodic boundary conditions. The microscopic parameters used are (in K): $J_1=-75,J_1^{\prime }=-67,J_2=49,d_a^{\prime }=12.1,d_b^{\prime }=-4.7,d_c^{\prime }=0$, and we have disregarded the much weaker DM terms from the $J_1$ and $J_2$ bonds. Notation: $(1,4)\mathbf{a}$ stands for the local moments of sites 1 and 4 of Fig. 1 along the $\mathbf{a}$ axis, etc.