${\mathrm{BaFe}}_2\mathrm{S}{\mathrm{e}}_3$: A High $T_C$ Magnetic Multiferroic with Large Ferrielectric Polarization

The iron selenides are important because of their superconducting properties. Here, an unexpected phenomenon is predicted to occur in an iron-selenide compound with a quasi-one-dimensional ladder geometry: ${\mathrm{BaFe}}_2\mathrm{S}{\mathrm{e}}_3$ should be a magnetic ferrielectric system, driven by its magnetic block order via exchange striction. A robust performance (high $T_C$ and large polarization) is expected. Different from most multiferroics, ${\mathrm{BaFe}}_2\mathrm{S}{\mathrm{e}}_3$ is ferrielectric, with a polarization that mostly cancels between ladders. However, its strong magnetostriction still produces a net polarization that is large ($\sim 0.1\mu C/{\mathrm{cm}}^2$) as compared with most magnetic multiferroics. Its fully ferroelectric state, with energy only slightly higher than the ferrielectric, has a giant improper polarization $\sim 2–3\mu \mathrm{C}/{\mathrm{cm}}^2$.

Figure 1

Crystal and magnetic structures of ${\mathrm{BaFe}}_2\mathrm{S}{\mathrm{e}}_3$. (a) Side view along the $b$ axis. Blue: Fe; green: Se; pink: Ba. (b) A Fe-Se ladder along the $b$ axis and its magnetic order. Partial ionic displacements driven by the exchange striction are marked as black arrows. (c) A unit cell considering the AFM order. (d) Spin structures. Left: Block-MF; middle: Block-EX; right: $Cx$. The side arrows denote the local FE $P$’s of each ladder. In (b)–(d), the spins ($\uparrow /\downarrow$) of Fe’s are distinguished by colors. (e) Vector addition of FE $P$’s of ladders $A$ and $B$.

Figure 2

(a)–(c) DFT results varying the effective Hubbard interaction. (a) Energies for various magnetic states, with the Block-EX as reference. (b) Band gaps. NM and FM are metallic (zero gap). (c) The FE $P$’s of Block-MF and Block-EX states. The dashed lines (with solid symbols) are the components along the symmetry expected directions (e.g., $a$ axis for Block-MF, $c$ axis for Block-EX), which are almost identical to the total $P$ and imply a successful prediction by the symmetry analysis. The purple $P_{\mathrm{MF}}^e$’s (solid and open symbols nearly overlapping) are the pure electronic contribution in the Block-MF case. (d) Sketch of switchings between $\pm P_{\mathrm{EX}}$ and $\pm P_{\mathrm{MF}}$ driven by the electric field $E_x$ along the $x$ ($x=a$ or $c$) direction. (e)–(f) DFT demonstration (without $U$) of switching between $\pm P_{\mathrm{EX}}$ via the rotation of Fe-ladder planes. Horizontal axis: the angle between ladder $A$’s and ladder $B$’s planes. The two limits ($\sim \pm 4.6°$) denote the relaxed $\pm P_{\mathrm{EX}}$ states, respectively. The center 0° denotes the relaxed nontilting case. For other angles, the structures are obtained by proportional mixing among these three limits. Vertical axes: (e) energy per Fe; (f) polarization along the $c$ axis.

Figure 3

Two-dimensional profiles of electronic density difference (${\mathrm{BaFe}}_2{\mathrm{S}}_3$ minus ${\mathrm{BaFe}}_2\mathrm{S}{\mathrm{e}}_3$). Left: the Se(5)-Fe(3)-Fe(4)-Se(7) plane. Spheres denote the Se/S sites while multilobe images denote the Fe sites. Right: the Fe-ladder plane.