Spin-reorientation transitions in the Cairo pentagonal magnet ${\mathbf{Bi}}_4{\mathbf{Fe}}_5{\mathbf{O}}_{13}\mathbf{F}$

We show that interlayer spins play a dual role in the Cairo pentagonal magnet ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$, on one hand mediating the three-dimensional magnetic order, and on the other driving spin-reorientation transitions both within and between the planes. The corresponding sequence of magnetic orders unraveled by neutron diffraction and Mössbauer spectroscopy features two orthogonal magnetic structures described by opposite local vector chiralities, and an intermediate, partly disordered phase with nearly collinear spins. A similar collinear phase has been predicted theoretically to be stabilized by quantum fluctuations, but ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$ is very far from the relevant parameter regime. While the observed in-plane reorientation cannot be explained by any standard frustration mechanism, our ab initio band-structure calculations reveal strong single-ion anisotropy of the interlayer ${\mathrm{Fe}}^{3+}$ spins that turns out to be instrumental in controlling the local vector chirality and the associated interlayer order.

Figure 1

(a) The symmetric Cairo lattice with two exchange couplings, $J_{33}$ and $J_{43}$. (b),(c) Orthogonal and collinear phases. Note the zero moment on half of the threefold sites in (c). The Fe1 sites 1–4 in (b) provide a measure of local vector chirality as $\mathbf{\chi }=\langle \mathbf{\Gamma }\rangle /|\langle \mathbf{\Gamma }\rangle |$, where $\mathbf{\Gamma }={\mathbf{S}}_1\times {\mathbf{S}}_2+{\mathbf{S}}_2\times {\mathbf{S}}_3+{\mathbf{S}}_3\times {\mathbf{S}}_4+{\mathbf{S}}_4\times {\mathbf{S}}_1$. Note that $\mathbf{\chi }$ is defined on the plaquette centered by the $\overline{4}$ rotoinversion axis (and vertical $\mathrm{Fe}{2}_1$ dimers). The adjacent plaquettes have opposite chirality, but they are centered by the $4_2$ screw axis (and horizontal $\mathrm{Fe}{2}_2$ dimers) and thus distinguished by symmetry. The overall magnetic structure is nonchiral. (d)–(f) Magnetic structures of ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$ in phases I, II, and III, respectively. The two types of $J_{43}$ couplings are also indicated. The crystal and magnetic structures are visualized using vesta [43].

Figure 2

Distinct magnetic structures A and B at 1.5 K. The Cairo planes at $z$ = 1/2 are shown.

Figure 3

Left: temperature dependence of the ordered magnetic moments in ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$. Right: temperature dependence of the polar angle $\phi$ showing abrupt rotations of the moments upon the first-order spin-reorientation transitions at $T_1$ and $T_2$. The lines are only guides for the eye.

Figure 4

Mössbauer spectra of ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$ and their fits, as described in the text. For fit parameters, see Table 3.

Figure 5

Fit of the magnetic susceptibility of ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$ [32] using revised exchange parameters reported in this work.

Figure 6

In-plane single-ion anisotropy energies for the Fe1, Fe2, and Fe3 sites. The arrows denote preferred spin directions. Only the Fe3 sites are compatible with the orthogonal structure, because their preferred directions are at ${90}^{\circ }$ to each other. For the Fe1 and Fe2 sites, the angle between preferred directions of the neighboring sites is largely different from ${90}^{\circ }$. Note that we use ${\mathrm{Fe}3}_1$ and ${\mathrm{Fe}3}_2$ for those Fe3 sites that are coupled to ${\mathrm{Fe}1}_1$ and ${\mathrm{Fe}1}_2$, respectively. The angle $\phi$ is measured between the magnetic moment and the ${\mathbf{a}}_m$ axis, and all curves are periodic, $E(\phi +{180}^{\circ })=E\left(\phi \right)$.

Figure 7

Classical phase diagram of the isotropic $J_{43}-J_{43}^{\prime }-J_{33}$ model, with two Fe1 spins on each fourfold site of the lattice; see the text. The two available compounds are shown by the filled blue triangle (based on the parameters of [25]) and the filled red dot (based on the parameters given above).

Figure 8

Left panel: sawtooth-chain model with the couplings $J_{\perp }$ and $J_{\perp 2}$ and two ordered states, incommensurate and commensurate, which are degenerate on the classical level. Right panel: the interlayer order in ${\mathrm{Bi}}_4{\mathrm{Fe}}_5{\mathrm{O}}_{13}\mathrm{F}$.